Chapter 6 Resource Masters - commackschools.org 6 Worksheets.pdf · Chapter 6 Resource Masters...
Transcript of Chapter 6 Resource Masters - commackschools.org 6 Worksheets.pdf · Chapter 6 Resource Masters...
Chapter 6 Resource Masters
Bothell, WA • Chicago, IL • Columbus, OH • New York, NY
00i_ALG1_A_CRM_C06_TP_660280.indd i00i_ALG1_A_CRM_C06_TP_660280.indd i 12/21/10 8:40 AM12/21/10 8:40 AM
CONSUMABLE WORKBOOKS Many of the worksheets contained in the Chapter Resource Masters booklets are available as consumable workbooks in both English and Spanish.
ISBN10 ISBN13Study Guide and Intervention Workbook 0-07-660292-3 978-0-07-660292-6Homework Practice Workbook 0-07-660291-5 978-0-07-660291-9
Spanish VersionHomework Practice Workbook 0-07-660294-X 978-0-07-660294-0
Answers For Workbooks The answers for Chapter 6 of these workbooks can be found in the back of this Chapter Resource Masters booklet.
ConnectED All of the materials found in this booklet are included for viewing, printing, and editing at connected.mcgraw-hill.com.
Spanish Assessment Masters (MHID: 0-07-660289-3, ISBN: 978-0-07-660289-6) These masters contain a Spanish version of Chapter 6 Test Form 2A and Form 2C.
PDF Pass
connected.mcgraw-hill.com
Copyright © by The McGraw-Hill Companies, Inc.
All rights reserved. The contents, or parts thereof, may be reproduced in print form for non-profit educational use with Glencoe Algebra 1, provided such reproductions bear copyright notice, but may not be reproduced in any form for any other purpose without the prior written consent of The McGraw-Hill Companies, Inc., including, but not limited to, network storage or transmission, or broadcast for distance learning.
Send all inquiries to:McGraw-Hill Education8787 Orion PlaceColumbus, OH 43240
ISBN: 978-0-07-660280-3MHID: 0-07-660280-X
Printed in the United States of America.
1 2 3 4 5 6 7 8 9 DOH 16 15 14 13 12 11
0ii_0vi_ALG1_A_CRM_C06_FM_660280.indd ii0ii_0vi_ALG1_A_CRM_C06_FM_660280.indd ii 12/21/10 8:42 AM12/21/10 8:42 AM
iii
Teacher’s Guide to Using the Chapter 6 Resource Masters ..............................................iv
Chapter Resources Chapter 6 Student-Built Glossary ...................... 1Chapter 6 Anticipation Guide (English) ............. 3Chapter 6 Anticipation Guide (Spanish) ............ 4
Lesson 6-1Graphing Systems of EquationsStudy Guide and Intervention ............................ 5Skills Practice .................................................... 7Practice ............................................................. 8Word Problem Practice ...................................... 9Enrichment ...................................................... 10Graphing Calculator Activity .............................11
Lesson 6-2SubstitutionStudy Guide and Intervention .......................... 12Skills Practice .................................................. 14Practice ........................................................... 15Word Problem Practice .................................... 16Enrichment ...................................................... 17
Lesson 6-3Elimination Using Addition and SubtractionStudy Guide and Intervention .......................... 18Skills Practice .................................................. 20Practice ........................................................... 21Word Problem Practice .................................... 22Enrichment ...................................................... 23
Lesson 6-4Elimination Using MultipcationStudy Guide and Intervention .......................... 24Skills Practice .................................................. 26Practice ........................................................... 27Word Problem Practice .................................... 28Enrichment ...................................................... 29
Lesson 6-5Applying Systems of Linear EquationsStudy Guide and Intervention .......................... 30Skills Practice .................................................. 32Practice ........................................................... 33Word Problem Practice .................................... 34Enrichment ...................................................... 35
Lesson 6-6Systems of InequalitiesStudy Guide and Intervention .......................... 36Skills Practice .................................................. 38Practice ........................................................... 39Word Problem Practice .................................... 40Enrichment ...................................................... 41Graphing Calculator ......................................... 42Spreadsheet .................................................... 43Student Recording Sheet ............................... 45Rubric for Scoring Extended Response .......... 46Chapter 6 Quizzes 1 and 2 ............................. 47Chapter 6 Quizzes 3 and 4 ............................. 48Chapter 6 Mid-Chapter Test ............................ 49Chapter 6 Vocabulary Test ............................... 50Chapter 6 Test, Form 1 .................................... 51Chapter 6 Test, Form 2A ................................. 53Chapter 6 Test, Form 2B ................................. 55Chapter 6 Test, Form 2C ................................. 57Chapter 6 Test, Form 2D ................................. 59Chapter 6 Test, Form 3 .................................... 61Chapter 6 Extended-Response Test ................ 63Standardized Test Practice .............................. 64Unit 2 Test ........................................................ 67
Answers ........................................... A1–A32
ContentsC
op
yrig
ht ©
Gle
nco
e/M
cGra
w-H
ill, a
div
isio
n o
f T
he M
cGra
w-H
ill C
om
pan
ies,
Inc.
PDF Pass0ii_0vi_ALG1_A_CRM_C06_FM_660280.indd iii0ii_0vi_ALG1_A_CRM_C06_FM_660280.indd iii 12/21/10 8:42 AM12/21/10 8:42 AM
iv
Teacher’s Guide to Using the Chapter 6 Resource Masters
Chapter Resources
Student-Built Glossary (pages 1–2) These masters are a student study tool that presents up to twenty of the key vocabulary terms from the chapter. Students are to record definitions and/or examples for each term. You may suggest that students highlight or star the terms with which they are not familiar. Give this to students before beginning Lesson 6-1. Encourage them to add these pages to their mathematics study notebooks. Remind them to complete the appropriate words as they study each lesson.
Anticipation Guide (pages 3–4) This master, presented in both English and Spanish, is a survey used before beginning the chapter to pinpoint what students may or may not know about the concepts in the chapter. Students will revisit this survey after they complete the chapter to see if their perceptions have changed.
Lesson ResourcesStudy Guide and Intervention These masters provide vocabulary, key concepts, additional worked-out examples and Guided Practice exercises to use as a reteaching activity. It can also be used in conjunction with the Student Edition as an instructional tool for students who have been absent.
Skills Practice This master focuses more on the computational nature of the lesson. Use as an additional practice option or as homework for second-day teaching of the lesson.
Practice This master closely follows the types of problems found in the Exercises section of the Student Edition and includes word problems. Use as an additional practice option or as homework for second-day teaching of the lesson.
Word Problem Practice This master includes additional practice in solving word problems that apply the concepts of the lesson. Use as an additional practice or as homework for second-day teaching of the lesson.
Enrichment These activities may extend the concepts of the lesson, offer a historical or multicultural look at the concepts, or widen students’ perspectives on the mathematics they are learning. They are written for use with all levels of students.
Graphing Calculator, TI-Nspire, or Spreadsheet Activities These activities present ways in which technology can be used with the concepts in some lessons of this chapter. Use as an alternative approach to some concepts or as an integral part of your lesson presentation.
The Chapter 6 Resource Masters includes the core materials needed for Chapter 6. These materials include worksheets, extensions, and assessment options. The answers for these pages appear at the back of this booklet.
All of the materials found in this booklet are included for viewing, printing, and editing at connectED.mcgraw-hill.com.
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
PDF Pass0ii_0vi_ALG1_A_CRM_C06_FM_660280.indd iv0ii_0vi_ALG1_A_CRM_C06_FM_660280.indd iv 12/21/10 8:42 AM12/21/10 8:42 AM
v
Assessment OptionsThe assessment masters in the Chapter 6 Resource Masters offer a wide range of assessment tools for formative (monitoring) assessment and summative (final) assessment.
Student Recording Sheet This master cor-responds with the standardized test practice at the end of the chapter.
Extended Response Rubric This master provides information for teachers and students on how to assess performance on open-ended questions.
Quizzes Four free-response quizzes offer assessment at appropriate intervals in the chapter.
Mid-Chapter Test This 1-page test provides an option to assess the first half of the chapter. It parallels the timing of the Mid-Chapter Quiz in the Student Edition and includes both multiple-choice and free-response questions.
Vocabulary Test This test is suitable for all students. It includes a list of vocabulary words and 9 questions to assess students’ knowledge of those words. This can also be used in conjunction with one of the leveled chapter tests.
Leveled Chapter Tests• Form 1 contains multiple-choice
questions and is intended for use with below grade level students.
• Forms 2A and 2B contain multiple-choice questions aimed at on grade level students. These tests are similar in format to offer comparable testing situations.
• Forms 2C and 2D contain free-response questions aimed at on grade level students. These tests are similar in format to offer comparable testing situations.
• Form 3 is a free-response test for use with above grade level students.
All of the above mentioned tests include a free-response Bonus question.
Extended-Response Test Performance assessment tasks are suitable for all students. Sample answers and a scoring rubric are included for evaluation.
Standardized Test Practice These three pages are cumulative in nature. It includes three parts: multiple-choice questions with bubble-in answer format, griddable questions with answer grids, and short-answer free-response questions.
Answers• The answers for the Anticipation Guide
and Lesson Resources are provided as reduced pages.
• Full-size answer keys are provided for the assessment masters.
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
PDF Pass0ii_0vi_ALG1_A_CRM_C06_FM_660280.indd v0ii_0vi_ALG1_A_CRM_C06_FM_660280.indd v 12/21/10 8:42 AM12/21/10 8:42 AM
0ii_0vi_ALG1_A_CRM_C06_FM_660280.indd vi0ii_0vi_ALG1_A_CRM_C06_FM_660280.indd vi 12/21/10 8:42 AM12/21/10 8:42 AM
Ch
apte
r R
eso
urc
es
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
NAME DATE PERIOD
PDF Pass
Chapter 6 1 Glencoe Algebra 1
This is an alphabetical list of the key vocabulary terms you will learn in Chapter 6. As you study the chapter, complete each term’s definition or description. Remember to add the page number where you found the term. Add these pages to your Algebra Study Notebook to review vocabulary at the end of the chapter.
Student-Built Glossary6
Vocabulary TermFound
on PageDefi nition/Description/Example
consistentkuhn·SIHS·tuhnt
dependent
eliminationih·LIH·muh·NAY·shuhn
inconsistent
independent
substitutionSUHB·stuh·TOO·shuhn
system of equations
system of inequalities
001_009_ALG1_A_CRM_C06_CR_660280.indd 1001_009_ALG1_A_CRM_C06_CR_660280.indd 1 12/21/10 8:43 AM12/21/10 8:43 AM
001_009_ALG1_A_CRM_C06_CR_660280.indd 2001_009_ALG1_A_CRM_C06_CR_660280.indd 2 12/21/10 8:43 AM12/21/10 8:43 AM
Ch
apte
r R
eso
urc
es
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
NAME DATE PERIOD
PDF Pass
Chapter 6 3 Glencoe Algebra 1
Before you begin Chapter 6
• Read each statement.
• Decide whether you Agree (A) or Disagree (D) with the statement.
• Write A or D in the first column OR if you are not sure whether you agree or disagree, write NS (Not Sure).
After you complete Chapter 6
• Reread each statement and complete the last column by entering an A or a D.
• Did any of your opinions about the statements change from the first column?
• For those statements that you mark with a D, use a piece of paper to write an example of why you disagree.
6 Anticipation GuideSolving Systems of Linear Equations
STEP 1A, D, or NS
StatementSTEP 2A or D
1. A solution of a system of equations is any ordered pair that satisfies one of the equations
2. A system of equations of parallel lines will have no solutions.
3. A system of equations of two perpendicular lines will have infinitely many solutions.
4. It is not possible to have exactly two solutions to a system of linear equations
5. The most accurate way to solve a system of equations is tograph the equations to see where they intersect.
6. To solve a system of equations, such as 2x - y = 21 and3y = 2x - 6, by substitution, solve one of the equations for one variable and substitute the result into the other equation.
7. When solving a system of equations, a result that is a true statement, such as -5 = -5, means the equations do not share a common solution.
8. Adding the equations 3x - 4y = 8 and 2x + 4y = 7 results in a 0 coefficient for y.
9. The equation 7x - 2y = 12 can be multiplied by 2 so that the coefficient of y is -4.
10. The result of multiplying -7x - 3y = 11 by -3 is -1x + 9y = 11.
Step 1
Step 2
001_009_ALG1_A_CRM_C06_CR_660280.indd 3001_009_ALG1_A_CRM_C06_CR_660280.indd 3 12/21/10 8:43 AM12/21/10 8:43 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
NOMBRE FECHA PERÍODO
PDF Pass
Capítulo 6 4 Álgebra 1 de Glencoe
6
Antes de comenzar el Capítulo 6
• Lee cada enunciado.
• Decide si estás de acuerdo (A) o en desacuerdo (D) con el enunciado.
• Escribe A o D en la primera columna O si no estás seguro(a) de la respuesta, escribe NS (No estoy seguro(a)).
Después de completar el Capítulo 6
• Vuelve a leer cada enunciado y completa la última columna con una A o una D.
• ¿Cambió cualquiera de tus opiniones sobre los enunciados de la primera columna?
• En una hoja de papel aparte, escribe un ejemplo de por qué estás en desacuerdo con los enunciados que marcaste con una D.
Ejercicios preparatoriosResuelve sistemas de ecuaciones lineales
Paso 1
PASO 1A, D o NS
EnunciadoPASO 2A o D
1. Una solución de un sistema de ecuaciones es cualquier par ordenado que satisface una de las ecuaciones
2. Un sistema de ecuaciones de rectas paralelas no tendrá soluciones.
3. Un sistema de ecuaciones de dos rectas perpendiculares tendrá un número infinito de soluciones.
4. No es posible tener exactamente dos soluciones para un sistema de ecuaciones lineales.
5. La forma más precisa de resolver un sistema de ecuaciones es graficar las ecuaciones y ver dónde se intersecan.
6. Para resolver un sistema de ecuaciones, como 2x - y = 21 y 3y = 2x - 6, por sustitución, despeja una variable en una de la ecuaciones y reemplaza el resultado en la otra ecuación.
7. Cuando se resuelve un sistema de ecuaciones, un resultado que es un enunciado verdadero, como -5 = -5, significa que las ecuaciones no comparten una solución.
8. El sumar las ecuaciones 3x - 4y = 8 y 2x + 4y = 7 resulta en un coeficiente de 0 para y.
9. La ecuación 7x - 2y = 12 se puede multiplicar por 2, de modo que el coeficiente de y es -4.
10. El resultado de multiplicar -7x - 3y = 11 por -3 es -1x + 9y = 11.
Paso 2
001_009_ALG1_A_CRM_C06_CR_660280.indd 4001_009_ALG1_A_CRM_C06_CR_660280.indd 4 12/21/10 8:43 AM12/21/10 8:43 AM
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
NAME DATE PERIOD
Less
on
6-1
PDF Pass
Chapter 6 5 Glencoe Algebra 1
Possible Number of Solutions Two or more linear equations involving the same variables form a system of equations. A solution of the system of equations is an ordered pair of numbers that satisfies both equations. The table below summarizes information about systems of linear equations.
Graph of a System intersecting lines same line parallel lines
Number of Solutions exactly one solution infi nitely many solutions no solution
Terminology
consistent and consistent and inconsistent
independent dependent
Use the graph at the right to determine whether each system is consistent or inconsistent and if it is independent or dependent.
a. y = -x + 2 y = x + 1
Since the graphs of y = -x + 2 and y = x + 1 intersect, there is one solution. Therefore, the system is consistent and independent.
b. y = -x + 2 3x + 3y = -3
Since the graphs of y = -x + 2 and 3x + 3y = -3 are parallel, there are no solutions. Therefore, the system is inconsistent.
c. 3x + 3y = -3 y = -x - 1
Since the graphs of 3x + 3y = -3 and y = -x - 1 coincide, there are infinitely many solutions. Therefore, the system is consistent and dependent.
ExercisesUse the graph at the right to determine whether each system is consistent or inconsistent and if it is independent or dependent.
1. y = -x - 3 2. 2x + 2y = -6y = x - 1 y = -x - 3
3. y = -x - 3 4. 2x + 2y = -62x + 2y = 4 3x + y = 3
6-1 Study Guide and InterventionGraphing Systems of Equations
x
y
O
y = -x - 3
3x + y = 3
2x + 2y = -6
2x + 2y = 4
y = x - 1
x
y
O
y = x + 1
y = -x - 1
3x + 3y = -3
y = -x + 2
Example
x
y
Ox
y
O x
y
O
001_009_ALG1_A_CRM_C06_CR_660280.indd 5001_009_ALG1_A_CRM_C06_CR_660280.indd 5 12/21/10 8:43 AM12/21/10 8:43 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
NAME DATE PERIOD
PDF Pass
Chapter 6 6 Glencoe Algebra 1
Solve by Graphing One method of solving a system of equations is to graph the equations on the same coordinate plane.
Graph each system and determine the number of solutions that it has. If it has one solution, name it.
a. x + y = 2 x - y = 4
The graphs intersect. Therefore, there is one solution. The point (3, -1) seems to lie on both lines. Check this estimate by replacing x with 3 and y with -1 in each equation. x + y = 2 3 + (-1) = 2 � x - y = 4 3 - (-1) = 3 + 1 or 4 �The solution is (3, -1).
b. y = 2x + 1 2y = 4x + 2
The graphs coincide. Therefore there are infinitely many solutions.
ExercisesGraph each system and determine the number of solutions it has. If it has one solution, name it.
1. y = -2 2. x = 2 3. y = 1 −
2 x
3x - y = -1 2x + y = 1 x + y = 3
4. 2x + y = 6 5. 3x + 2y = 6 6. 2y = -4x + 4 2x - y = -2 3x + 2y = -4 y = -2x + 2
6-1 Study Guide and Intervention (continued)
Graphing Systems of Equations
x
y
O
y = 2x + 1 2y = 4x + 2
x
y
O(3, –1)
x - y = 4
x + y = 2
Example
x
y
O
x
y
O
x
y
O
x
y
O
x
y
O
x
y
O
001_009_ALG1_A_CRM_C06_CR_660280.indd 6001_009_ALG1_A_CRM_C06_CR_660280.indd 6 12/21/10 8:43 AM12/21/10 8:43 AM
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
NAME DATE PERIOD
Less
on
6-1
PDF Pass
Chapter 6 7 Glencoe Algebra 1
Use the graph at the right to determine whether each system is consistent or inconsistent and if it is independent or dependent.
1. y = x - 1 2. x - y = -4
y = -x + 1 y = x + 4
3. y = x + 4 4. y = 2x - 32x - 2y = 2 2x - 2y = 2
Graph each system and determine the number of solutions that it has. If it has one solution, name it.
5. 2x - y = 1 6. x = 1 7. 3x + y = -3 y = -3 2x + y = 4 3x + y = 3
x
y
O
8. y = x + 2 9. x + 3y = -3 10. y - x = -1x - y = -2 x - 3y = -3 x + y = 3
x
y
11. x - y = 3 12. x + 2y = 4 13. y = 2x + 3
x - 2y = 3 y = -
1 −
2 x + 2 3y = 6x - 6
x
y
O
6-1 Skills PracticeGraphing Systems of Equations
x
y
y = -x + 1
x - y = -4 2x - 2y = 2
y = 2x - 3
y = x - 1
y = x + 4
x
y
O x
y
O
x
y
O x
y
x
y
x
y
O
001_009_ALG1_A_CRM_C06_CR_660280.indd 7001_009_ALG1_A_CRM_C06_CR_660280.indd 7 12/21/10 8:43 AM12/21/10 8:43 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
NAME DATE PERIOD
PDF Pass
Chapter 6 8 Glencoe Algebra 1
Use the graph at the right to determine whether each system is consistent or inconsistent and if it is independent or dependent.
1. x + y = 3 2. 2x - y = -3x + y = -3 4x - 2y = -6
3. x + 3y = 3 4. x + 3y = 3x + y = -3 2x - y = -3
Graph each system and determine the number of solutions that it has. If it has one solution, name it.
5. 3x - y = -2 6. y = 2x - 3 7. x + 2y = 3 3x - y = 0 4x = 2y + 6 3x - y = -5
x
y
O
8. BUSINESS Nick plans to start a home-based business producing and selling gourmet dog treats. He figures it will cost $20 in operating costs per week plus $0.50 to produce each treat. He plans to sell each treat for $1.50.
a. Graph the system of equations y = 0.5x + 20 and y = 1.5x to represent the situation.
b. How many treats does Nick need to sell per week to break even?
9. SALES A used book store also started selling used CDs and videos. In the first week, the store sold 40 used CDs and videos, at $4.00 per CD and $6.00 per video. The sales for both CDs and videos totaled $180.00
a. Write a system of equations to represent the situation.
b. Graph the system of equations.
c. How many CDs and videos did the store sell in the first week?
6-1 PracticeGraphing Systems of Equations
x
y
x + y = -3
x + 3y = 3 2x - y = -3
4x - 2y = -6
x + y = 3
x
y
x
y
O
Sales ($)
Dog Treats
Co
st (
$)
5 1510 20 25 30 35 40 450
40
35
30
25
20
15
10
5
CD Sales ($)
CD and Video Sales
Vid
eo S
ales
($)
5 1510 20 25 30 35 40 450
40
35
30
25
20
15
10
5
001_009_ALG1_A_CRM_C06_CR_660280.indd 8001_009_ALG1_A_CRM_C06_CR_660280.indd 8 12/21/10 8:43 AM12/21/10 8:43 AM
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
NAME DATE PERIOD
Less
on
6-1
PDF 2nd
Chapter 6 9 Glencoe Algebra 1
1. BUSINESS The widget factory will sell a total of y widgets after x days according to the equation y = 200x + 300. The gadget factory will sell y gadgets after x days according to the equation y = 200x + 100. Look at the graph of the system of equations and determine whether it has no solution, one solution, or infinitely many solutions.
2. ARCHITECTURE An office building has two elevators. One elevator starts out on the 4th floor, 35 feet above the ground, and is descending at a rate of 2.2 feet per second. The other elevator starts out at ground level and is rising at a rate of 1.7 feet per second. Write a system of equations to represent the situation.
3. FITNESS Olivia and her brother William had a bicycle race. Olivia rode at a speed of 20 feet per second while William rode at a speed of 15 feet per second. To be fair, Olivia decided to give William a 150-foot head start. The race ended in a tie. How far away was the finish line from where Olivia started?
4. AVIATION Two planes are in flight near a local airport. One plane is at an altitude of 1000 meters and is ascending at a rate of 400 meters per minute. The second plane is at an altitude of 5900 meters and is descending at a rate of 300 meters per minute.
a. Write a system of equations that represents the progress of each plane
b. Make a graph that represents the progress of each plane.
6-1 Word Problem PracticeGraphing Systems of Equations
Days210 43
y
x5 6
Item
s so
ld
300
400
200
100
500
800
900
700
600
Gadgets
Widgetsy = 200x + 300
y = 200x + 100
Time (min.)0
y
x
Alt
itu
de
(m)
001_009_ALG1_A_CRM_C06_CR_660280.indd 9001_009_ALG1_A_CRM_C06_CR_660280.indd 9 12/24/10 11:41 AM12/24/10 11:41 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
NAME DATE PERIOD
PDF Pass
Chapter 6 10 Glencoe Algebra 1
Enrichment
Graphing a TripThe distance formula, d = rt, is used to solve many types of
t
d
O
slope is 50
slope is –25
2
50
problems. If you graph an equation such as d = 50t, the graph is a model for a car going at 50 mph. The time the car travels is t; the distance in miles the car covers is d. The slope of the line is the speed.Suppose you drive to a nearby town and return. You average 50 mph on the trip out but only 25 mph on the trip home. The round trip takes 5 hours. How far away is the town?The graph at the right represents your trip. Notice that the return trip is shown with a negative slope because you are driving in the opposite direction.
Solve each problem.
1. Estimate the answer to the problem in the above example. About how far away is the town?
Graph each trip and solve the problem.
2. An airplane has enough fuel for 3 hours of safe flying. On the trip out the pilot averages 200 mph flying against a headwind. On the trip back, the pilot averages 250 mph. How long a trip out can the pilot make?
3. You drive to a town 100 miles away. 4. You drive at an average speed of 50 mph On the trip out you average 25 mph. to a discount shopping plaza, spend On the trip back you average 50 mph. 2 hours shopping, and then return at an How many hours do you spend driving? average speed of 25 mph. The entire trip takes 8 hours. How far away is the shopping plaza?
t
d
O 2
50
t
d
O
50
2
6-1
t
d
O 1
100
010_018_ALG1_A_CRM_C06_CR_660280.indd 10010_018_ALG1_A_CRM_C06_CR_660280.indd 10 12/21/10 8:43 AM12/21/10 8:43 AM
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
NAME DATE PERIOD
Less
on
6-1
PDF 2nd
Chapter 6 11 Glencoe Algebra 1
Graphing Calculator ActivitySolution to a System of Linear Equations
A graphing calculator can be used to solve a system of linear equations graphically. The solution of a system of linear equations can be found by using the TRACE feature or by using the intersect command under the CALC menu.
ExercisesSolve each system of linear equations.
1. y = 2 2. y = -x + 3 3. x + y = -1 5x + 4y = 18 y = x + 1 2x - y = -8
4. -3x + y = 10 5. -4x + 3y = 10 6. 5x + 3y = 11 -x + 2y = 0 7x + y = 20 x - 5y = 5
7. 3x - 2y = -4 8. 3x + 2y = 4 9. 4x - 5y = 0 -4x + 3y = 5 -6x - 4y = -8 6x - 5y = 10
Solve each system of linear equations.
a. x + y = 0 x - y = -4
Using TRACE: Solve each equation for y and enter each equation into Y=. Then graph using Zoom 8: ZInteger. Use TRACE to find the solution.Keystrokes: Y= (–) ENTER + 4 ZOOM 6
ZOOM 8 ENTER TRACE .
The solution is (-2, 2).
b. 2x + y = 4 4x + 3y = 3
Using CALC: Solve each equation for y, enter each into the calculator, and graph. Use CALC to determine the solution.
Keystrokes: Y= (–) 2 + 4 ENTER ( (–) 4 ÷ 3 )
+ 1 ZOOM 6 2nd [CALC] 5 ENTER ENTER ENTER .
To change the x-value to a fraction, press 2nd [QUIT]
MATH ENTER ENTER .
The solution is (4.5, -5) or (
9 −
2 , -5
)
.
[-47, 47] scl:10 by [-31, 31]
scl:10
[-10, 10] scl:1 by [-10, 10]
scl:1
6-1
Example
010_018_ALG1_A_CRM_C06_CR_660280.indd 11010_018_ALG1_A_CRM_C06_CR_660280.indd 11 12/24/10 7:07 AM12/24/10 7:07 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
NAME DATE PERIOD
PDF Pass
Chapter 6 12 Glencoe Algebra 1
Study Guide and InterventionSubstitution
Solve by Substitution One method of solving systems of equations is substitution.
Use substitution to solve the system of equations.y = 2x4x - y = -4
Substitute 2x for y in the second equation. 4x - y = -4 Second equation
4x - 2x = -4 y = 2x
2x = -4 Combine like terms.
x = -2 Divide each side by 2
and simplify.
Use y = 2x to find the value of y.y = 2x First equation
y = 2(-2) x = -2
y = -4 Simplify.
The solution is (-2, -4).
Solve for one variable, then substitute.x + 3y = 72x - 4y = -6
Solve the first equation for x since the coefficient of x is 1. x + 3y = 7 First equation
x + 3y - 3y = 7 - 3y Subtract 3y from each side.
x = 7 - 3y Simplify.
Find the value of y by substituting 7 - 3y for x in the second equation. 2x - 4y = -6 Second equation
2(7 - 3y) - 4y = -6 x = 7 - 3y
14 - 6y - 4y = -6 Distributive Property
14 - 10y = -6 Combine like terms.
14 - 10y - 14 = -6 - 14 Subtract 14 from each side.
-10y = -20 Simplify.
y = 2 Divide each side by -10
and simplify.
Use y = 2 to find the value of x.x = 7 - 3yx = 7 - 3(2)x = 1The solution is (1, 2).
ExercisesUse substitution to solve each system of equations.
1. y = 4x 2. x = 2y 3. x = 2y - 33x - y = 1 y = x - 2 x = 2y + 4
4. x - 2y = -1 5. x - 4y = 1 6. x + 2y = 03y = x + 4 2x - 8y = 2 3x + 4y = 4
7. 2b = 6a - 14 8. x + y = 16 9. y = -x + 3 3a - b = 7 2y = -2x + 2 2y + 2x = 4
10. x = 2y 11. x - 2y = -5 12. -0.2x + y = 0.50.25x + 0.5y = 10 x + 2y = -1 0.4x + y = 1.1
6-2
Example 1 Example 2
010_018_ALG1_A_CRM_C06_CR_660280.indd 12010_018_ALG1_A_CRM_C06_CR_660280.indd 12 12/21/10 8:43 AM12/21/10 8:43 AM
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
NAME DATE PERIOD
Less
on
6-2
PDF Pass
Chapter 6 13 Glencoe Algebra 1
Study Guide and Intervention (continued)
Substitution
Solve Real-World Problems Substitution can also be used to solve real-world problems involving systems of equations. It may be helpful to use tables, charts, diagrams, or graphs to help you organize data.
CHEMISTRY How much of a 10% saline solution should be mixed with a 20% saline solution to obtain 1000 milliliters of a 12% saline solution?Let s = the number of milliliters of 10% saline solution.Let t = the number of milliliters of 20% saline solution.Use a table to organize the information.
10% saline 20% saline 12% saline
Total milliliters s t 1000
Milliliters of saline 0.10 s 0.20 t 0.12(1000)
Write a system of equations.s + t = 10000.10s + 0.20t = 0.12(1000)Use substitution to solve this system. s + t = 1000 First equation
s = 1000 - t Solve for s.
0.10s + 0.20t = 0.12(1000) Second equation
0.10(1000 - t) + 0.20t = 0.12(1000) s = 1000 - t
100 - 0.10t + 0.20t = 0.12(1000) Distributive Property
100 + 0.10t = 0.12(1000) Combine like terms.
0.10t = 20 Simplify.
0.10t −
0.10 = 20 −
0.10 Divide each side by 0.10.
t = 200 Simplify.
s + t = 1000 First equation
s + 200 = 1000 t = 200
s = 800 Solve for s.
800 milliliters of 10% solution and 200 milliliters of 20% solution should be used.
Exercises 1. SPORTS At the end of the 2007–2008 football season, 38 Super Bowl games had been
played with the current two football leagues, the American Football Conference (AFC) and the National Football Conference (NFC). The NFC won two more games than the AFC. How many games did each conference win?
2. CHEMISTRY A lab needs to make 100 gallons of an 18% acid solution by mixing a 12% acid solution with a 20% solution. How many gallons of each solution are needed?
3. GEOMETRY The perimeter of a triangle is 24 inches. The longest side is 4 inches longer than the shortest side, and the shortest side is three-fourths the length of the middle side. Find the length of each side of the triangle.
6-2
Example
010_018_ALG1_A_CRM_C06_CR_660280.indd 13010_018_ALG1_A_CRM_C06_CR_660280.indd 13 12/21/10 8:43 AM12/21/10 8:43 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
NAME DATE PERIOD
PDF Pass
Chapter 6 14 Glencoe Algebra 1
Use substitution to solve each system of equations.
1. y = 4x 2. y = 2xx + y = 5 x + 3y = -14
3. y = 3x 4. x = -4y2x + y = 15 3x + 2y = 20
5. y = x - 1 6. x = y - 7x + y = 3 x + 8y = 2
7. y = 4x - 1 8. y = 3x + 8y = 2x - 5 5x + 2y = 5
9. 2x - 3y = 21 10. y = 5x - 8y = 3 - x 4x + 3y = 33
11. x + 2y = 13 12. x + 5y = 43x - 5y = 6 3x + 15y = -1
13. 3x - y = 4 14. x + 4y = 82x - 3y = -9 2x - 5y = 29
15. x - 5y = 10 16. 5x - 2y = 142x - 10y = 20 2x - y = 5
17. 2x + 5y = 38 18. x - 4y = 27x - 3y = -3 3x + y = -23
19. 2x + 2y = 7 20. 2.5x + y = -2x - 2y = -1 3x + 2y = 0
6-2 Skills PracticeSubstitution
010_018_ALG1_A_CRM_C06_CR_660280.indd 14010_018_ALG1_A_CRM_C06_CR_660280.indd 14 12/21/10 8:43 AM12/21/10 8:43 AM
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
NAME DATE PERIOD
Less
on
6-2
PDF Pass
Chapter 6 15 Glencoe Algebra 1
Use substitution to solve each system of equations.
1. y = 6x 2. x = 3y 3. x = 2y + 72x + 3y = -20 3x - 5y = 12 x = y + 4
4. y = 2x - 2 5. y = 2x + 6 6. 3x + y = 12y = x + 2 2x - y = 2 y = -x - 2
7. x + 2y = 13 8. x - 2y = 3 9. x - 5y = 36 - 2x - 3y = -18 4x - 8y = 12 2x + y = -16
10. 2x - 3y = -24 11. x + 14y = 84 12. 0.3x - 0.2y = 0.5x + 6y = 18 2x - 7y = -7 x - 2y = -5
13. 0.5x + 4y = -1 14. 3x - 2y = 11 15. 1 −
2 x + 2y = 12
x + 2.5y = 3.5 x - 1 −
2 y = 4 x - 2y = 6
16. 1 −
3 x - y = 3 17. 4x - 5y = -7 18. x + 3y = -4
2x + y = 25 y = 5x 2x + 6y = 5
19. EMPLOYMENT Kenisha sells athletic shoes part-time at a department store. She can earn either $500 per month plus a 4% commission on her total sales, or $400 per month plus a 5% commission on total sales.
a. Write a system of equations to represent the situation.
b. What is the total price of the athletic shoes Kenisha needs to sell to earn the same income from each pay scale?
c. Which is the better offer?
20. MOVIE TICKETS Tickets to a movie cost $7.25 for adults and $5.50 for students. A group of friends purchased 8 tickets for $52.75.
a. Write a system of equations to represent the situation.
b. How many adult tickets and student tickets were purchased?
6-2 PracticeSubstitution
010_018_ALG1_A_CRM_C06_CR_660280.indd 15010_018_ALG1_A_CRM_C06_CR_660280.indd 15 12/21/10 8:43 AM12/21/10 8:43 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
NAME DATE PERIOD
PDF Pass
Chapter 6 16 Glencoe Algebra 1
1. BUSINESS Mr. Randolph finds that the supply and demand for gasoline at his station are generally given by the following equations.
x - y = -2x + y = 10
Use substitution to find the equilibrium point where the supply and demand lines intersect.
2. GEOMETRY The measures of complementary angles have a sum of 90 degrees. Angle A and angle B are complementary, and their measures have a difference of 20°. What are the measures of the angles?
AB
3. MONEY Harvey has some $1 bills and some $5 bills. In all, he has 6 bills worth $22. Let x be the number of $1 bills and let y be the number of $5 bills. Write a system of equations to represent the information and use substitution to determine how many bills of each denomination Harvey has.
4. POPULATION Sanjay is researching population trends in South America. He found that the population of Ecuador to increased by 1,000,000 and the population of Chile to increased by 600,000 from 2004 to 2009. The table displays the information he found.
Source: World Almanac
If the population growth for each country continues at the same rate, in what year are the populations of Ecuador and Chile predicted to be equal?
5. CHEMISTRY Shelby and Calvin are doing a chemistry experiment. They need 5 ounces of a solution that is 65% acid and 35% distilled water. There is no undiluted acid in the chemistry lab, but they do have two flasks of diluted acid: Flask A contains 70% acid and 30% distilled water. Flask B contains 20% acid and 80% distilled water.
a. Write a system of equations that Shelby and Calvin could use to determine how many ounces they need to pour from each flask to make their solution.
b. Solve your system of equations. How many ounces from each f lask do Shelby and Calvin need?
Word Problem PracticeSubstitution
6-2
Country2004
Population
5-Year
Population
Change
Ecuador 13,000,000 +1,000,000
Chile 16,000,000 +600,000
010_018_ALG1_A_CRM_C06_CR_660280.indd 16010_018_ALG1_A_CRM_C06_CR_660280.indd 16 12/21/10 11:48 PM12/21/10 11:48 PM
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
NAME DATE PERIOD
Less
on
6-2
PDF Pass
Chapter 6 17 Glencoe Algebra 1
Enrichment
Intersection of Two Parabolas
Substitution can be used to find the intersection of two parabolas. Replace the y-value in one of the equations with the y-value in terms of x from the other equation.
Find the intersection of the two parabolas. y = x2 + 5x + 6 y = x2 + 4x + 3
Graph the equations.
37.5
y
x-10 10-5 5
12.5
25.0
50.0
From the graph, notice that the two graphs intersect in one point.
Use substitution to solve for the point of intersection. x2 + 5x + 6 = x2 + 4x + 3 5x + 6 = 4x + 3 Subtract x2 from each side.
x + 6 = 3 Subtract 4x from each side.
x = -3 Subtract 6 from each side.
The graphs intersect at x = -3.
Replace x with -3 in either equation to find the y-value. y = x2 + 5x + 6 Original equation
y = (-3)2 + 5(-3) + 6 Replace x with -3.
y = 9 – 15 + 6 or 0 Simplify.
The point of intersection is (-3, 0).
ExercisesUse substitution to find the point of intersection of the graphs of each pair of equations.
1. y = x2 + 8x + 7 2. y = x2 + 6x + 8 3. y = x2 + 5x + 6y = x2 + 2x + 1 y = x2 + 4x + 4 y = x2 + 7x + 6
6-2
Example
010_018_ALG1_A_CRM_C06_CR_660280.indd 17010_018_ALG1_A_CRM_C06_CR_660280.indd 17 12/21/10 8:43 AM12/21/10 8:43 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
NAME DATE PERIOD
PDF Pass
Chapter 6 18 Glencoe Algebra 1
Study Guide and InterventionElimination Using Addition and Subtraction
Elimination Using Addition In systems of equations in which the coefficients of the x or y terms are additive inverses, solve the system by adding the equations. Because one of the variables is eliminated, this method is called elimination.
Use elimination to solve the system of equations.x - 3y = 73x + 3y = 9
Write the equations in column form and add to eliminate y. x - 3y = 7 (+) 3x + 3y = 9 4x = 16Solve for x. 4x −
4 = 16 −
4
x = 4Substitute 4 for x in either equation and solve for y. 4 - 3y = 7 4 - 3y - 4 = 7 - 4 -3y = 3
-3y
−
-3 = 3 −
-3
y = -1The solution is (4, -1).
The sum of two numbers is 70 and their difference is 24. Find the numbers.
Let x represent one number and y represent the other number. x + y = 70 (+) x - y = 24 2x = 94 2x −
2 = 94 −
2
x = 47Substitute 47 for x in either equation. 47 + y = 70 47 + y - 47 = 70 - 47 y = 23The numbers are 47 and 23.
ExercisesUse elimination to solve each system of equations.
1. x + y = -4 2. 2x - 3y = 14 3. 3x - y = -9x - y = 2 x + 3y = -11 -3x - 2y = 0
4. -3x - 4y = -1 5. 3x + y = 4 6. -2x + 2y = 93x - y = -4 2x - y = 6 2x - y = -6
7. 2x + 2y = -2 8. 4x - 2y = -1 9. x - y = 23x - 2y = 12 -4x + 4y = -2 x + y = -3
10. 2x - 3y = 12 11. -0.2x + y = 0.5 12. 0.1x + 0.3y = 0.9 4x + 3y = 24 0.2x + 2y = 1.6 0.1x - 0.3y = 0.2
13. Rema is older than Ken. The difference of their ages is 12 and the sum of their ages is 50. Find the age of each.
14. The sum of the digits of a two-digit number is 12. The difference of the digits is 2. Find the number if the units digit is larger than the tens digit.
6-3
Example 1 Example 2
010_018_ALG1_A_CRM_C06_CR_660280.indd 18010_018_ALG1_A_CRM_C06_CR_660280.indd 18 12/21/10 8:43 AM12/21/10 8:43 AM
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
NAME DATE PERIOD
Less
on
6-3
PDF Pass
Chapter 6 19 Glencoe Algebra 1
Study Guide and Intervention (continued)
Elimination Using Addition and Subtraction
Elimination Using Subtraction In systems of equations where the coefficients of the x or y terms are the same, solve the system by subtracting the equations.
Use elimination to solve the system of equations.2x - 3y = 115x - 3y = 14
2x - 3y = 11 Write the equations in column form and subtract.
(-) 5x - 3y = 14 -3x = -3 Subtract the two equations. y is eliminated.
-3x −
-3 = -3 −
-3 Divide each side by -3.
x = 1 Simplify.
2(1) - 3y = 11 Substitute 1 for x in either equation.
2 - 3y = 11 Simplify.
2 - 3y - 2 = 11 - 2 Subtract 2 from each side.
-3y = 9 Simplify.
-3y
−
-3 = 9 −
-3 Divide each side by -3.
y = -3 Simplify.
The solution is (1, -3).
ExercisesUse elimination to solve each system of equations.
1. 6x + 5y = 4 2. 3m - 4n = -14 3. 3a + b = 16x - 7y = -20 3m + 2n = -2 a + b = 3
4. -3x - 4y = -23 5. x - 3y = 11 6. x - 2y = 6-3x + y = 2 2x - 3y = 16 x + y = 3
7. 2a - 3b = -13 8. 4x + 2y = 6 9. 5x - y = 62a + 2b = 7 4x + 4y = 10 5x + 2y = 3
10. 6x - 3y = 12 11. x + 2y = 3.5 12. 0.2x + y = 0.74x - 3y = 24 x - 3y = -9 0.2x + 2y = 1.2
13. The sum of two numbers is 70. One number is ten more than twice the other number. Find the numbers.
14. GEOMETRY Two angles are supplementary. The measure of one angle is 10° more than three times the other. Find the measure of each angle.
6-3
Example
019_027_ALG1_A_CRM_C06_CR_660280.indd 19019_027_ALG1_A_CRM_C06_CR_660280.indd 19 12/21/10 8:43 AM12/21/10 8:43 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
NAME DATE PERIOD
PDF Pass
Chapter 6 20 Glencoe Algebra 1
Skills PracticeElimination Using Addition and Subtraction
Use elimination to solve each system of equations.
1. x - y = 1 2. -x + y = 1x + y = 3 x + y = 11
3. x + 4y = 11 4. -x + 3y = 6x - 6y = 11 x + 3y = 18
5. 3x + 4y = 19 6. x + 4y = -83x + 6y = 33 x - 4y = -8
7. 3x + 4y = 2 8. 3x - y = -14x - 4y = 12 -3x - y = 5
9. 2x - 3y = 9 10. x - y = 4-5x - 3y = 30 2x + y = -4
11. 3x - y = 26 12. 5x - y = -6-2x - y = -24 -x + y = 2
13. 6x - 2y = 32 14. 3x + 2y = -194x - 2y = 18 -3x - 5y = 25
15. 7x + 4y = 2 16. 2x - 5y = -287x + 2y = 8 4x + 5y = 4
17. The sum of two numbers is 28 and their difference is 4. What are the numbers?
18. Find the two numbers whose sum is 29 and whose difference is 15.
19. The sum of two numbers is 24 and their difference is 2. What are the numbers?
20. Find the two numbers whose sum is 54 and whose difference is 4.
21. Two times a number added to another number is 25. Three times the first number minus the other number is 20. Find the numbers.
6-3
019_027_ALG1_A_CRM_C06_CR_660280.indd 20019_027_ALG1_A_CRM_C06_CR_660280.indd 20 12/21/10 8:43 AM12/21/10 8:43 AM
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
NAME DATE PERIOD
Less
on
6-3
PDF Pass
Chapter 6 21 Glencoe Algebra 1
Practice Elimination Using Addition and Subtraction
Use elimination to solve each system of equations.
1. x - y = 1 2. p + q = -2 3. 4x + y = 23x + y = -9 p - q = 8 3x - y = 12
4. 2x + 5y = -3 5. 3x + 2y = -1 6. 5x + 3y = 222x + 2y = 6 4x + 2y = -6 5x - 2y = 2
7. 5x + 2y = 7 8. 3x - 9y = -12 9. -4c - 2d = -2-2x + 2y = -14 3x - 15y = -6 2c - 2d = -14
10. 2x - 6y = 6 11. 7x + 2y = 2 12. 4.25x - 1.28y = -9.22x + 3y = 24 7x - 2y = -30 x + 1.28y = 17.6
13. 2x + 4y = 10 14. 2.5x + y = 10.7 15. 6m - 8n = 3x - 4y = -2.5 2.5x + 2y = 12.9 2m - 8n = -3
16. 4a + b = 2 17. -
1 −
3 x - 4 −
3 y = -2 18. 3 −
4 x - 1 −
2 y = 8
4a + 3b = 10 1 −
3 x - 2 −
3 y = 4 3 −
2 x + 1 −
2 y = 19
19. The sum of two numbers is 41 and their difference is 5. What are the numbers?
20. Four times one number added to another number is 36. Three times the first number minus the other number is 20. Find the numbers.
21. One number added to three times another number is 24. Five times the first number added to three times the other number is 36. Find the numbers.
22. LANGUAGES English is spoken as the first or primary language in 78 more countries than Farsi is spoken as the first language. Together, English and Farsi are spoken as a first language in 130 countries. In how many countries is English spoken as the first language? In how many countries is Farsi spoken as the first language?
23. DISCOUNTS At a sale on winter clothing, Cody bought two pairs of gloves and four hats for $43.00. Tori bought two pairs of gloves and two hats for $30.00. What were the prices for the gloves and hats?
6-3
019_027_ALG1_A_CRM_C06_CR_660280.indd 21019_027_ALG1_A_CRM_C06_CR_660280.indd 21 12/21/10 8:43 AM12/21/10 8:43 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
NAME DATE PERIOD
PDF Pass
Chapter 6 22 Glencoe Algebra 1
Word Problem PracticeElimination Using Addition and Subtraction
1. NUMBER FUN Ms. Simms, the sixth grade math teacher, gave her students this challenge problem.
Twice a number added to another number is 15. The sum of the two numbers is 11.
Lorenzo, an algebra student who was Ms. Simms aide, realized he could solve the problem by writing the following system of equations.
2x + y = 15x + y = 11
Use the elimination method to solve the system and find the two numbers.
2. GOVERNMENT The Texas State Legislature is comprised of state senators and state representatives. The sum of the number of senators and representatives is 181. There are 119 more representatives than senators. How many senators and how many representatives make up the Texas State Legislature?
3. RESEARCH Melissa wondered how much it would cost to send a letter by mail in 1990, so she asked her father. Rather than answer directly, Melissa’s father gave her the following information. It would have cost $3.70 to send 13 postcards and 7 letters, and it would have cost $2.65 to send 6 postcards and 7 letters. Use a system of equations and elimination to find how much it cost to send a letter in 1990.
4. SPORTS As of 2010, the New York Yankees had won more World Series Championships than any other team. In fact, the Yankees had won 3 fewer than 3 times the number of World Series championships won by the second most-winning team, the St. Louis Cardinals. The sum of the two teams’ World Series championships is 37. How many times has each team won the World Series?
5. BASKETBALL In 2005, the average ticket prices for Dallas Mavericks games and Boston Celtics games are shown in the table below. The change in price is from the 2004 season to the 2005 season.
Source: Team Marketing Report
a. Assume that tickets continue to change at the same rate each year after 2005. Let x be the number of years after 2005, and y be the price of an average ticket. Write a system of equations to represent the information in the table.
b. In how many years will the average ticket price for Dallas approximately equal that of Boston?
6-3
Team Average
Ticket Price
Change in
Price
Dallas $53.60 $0.53
Boston $55.93 -$1.08
019_027_ALG1_A_CRM_C06_CR_660280.indd 22019_027_ALG1_A_CRM_C06_CR_660280.indd 22 12/21/10 8:43 AM12/21/10 8:43 AM
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
NAME DATE PERIOD
Less
on
6-3
PDF Pass
Chapter 6 23 Glencoe Algebra 1
Systems of equations can involve more than 2 equations and 2 variables. It is possible to solve a system of 3 equations and 3 variables using elimination.
Solve the following system. x + y + z = 6 3x - y + z = 8 x - z = 2
Step 1: Use elimination to get rid of the y in the first two equations. x + y + z = 6 3x – y + z = 8
4x + 2z = 14
Step 2: Use the equation you found in step 1 and the third equation to eliminate the z. 4x + 2z = 14
x – z = 2
Multiply the second equation by 2 so that the z’s will eliminate. 4x + 2z = 14 2x – 2z = 4 6x = 18 So, x = 3.
Step 3: Replace x with 3 in the third original equation to determine z. 3 – z = 2, so z = 1.
Step 4: Replace x with 3 and z with 1 in either of the first two original equation to determine the value of y. 3 + y + 1 = 6 or 4 + y = 6. So, y = 2.So, the solution to the system of equations is (3, 2, 1).
ExercisesSolve each system of equations.
1. 3x + 2y + z = 42 2. x + y + z = -3 3. x + y + z = 7 2y + z + 12 = 3x 2x + 3y + 5z = -4 x + 2y + z = 10
x – 3y = 0 2y – z = 4 2y + z = 5
EnrichmentSolving Systems of Equations in Three Variables
6-3
Example
019_027_ALG1_A_CRM_C06_CR_660280.indd 23019_027_ALG1_A_CRM_C06_CR_660280.indd 23 12/21/10 8:43 AM12/21/10 8:43 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
NAME DATE PERIOD
PDF Pass
Chapter 6 24 Glencoe Algebra 1
Study Guide and InterventionElimination Using Multiplication
Elimination Using Multiplication Some systems of equations cannot be solved simply by adding or subtracting the equations. In such cases, one or both equations must first be multiplied by a number before the system can be solved by elimination.
Use elimination to solve the system of equations.x + 10y = 34x + 5y = 5
If you multiply the second equation by -2, you can eliminate the y terms. x + 10y = 3 (+) -8x - 10y = -10 -7x = -7 -7x −
-7 = -7 −
-7
x = 1Substitute 1 for x in either equation. 1 + 10y = 3 1 + 10y - 1 = 3 - 1 10y = 2
10y
−
10 = 2 −
10
y =
1 −
5
The solution is (
1, 1 −
5 )
.
Use elimination to solve the system of equations.3x - 2y = -72x - 5y = 10
If you multiply the first equation by 2 and the second equation by -3, you can eliminate the x terms. 6x - 4y = -14(+) -6x + 15y = -30 11y = -44
11y
−
11 = -
44 −
11
y = -4Substitute -4 for y in either equation. 3x - 2(-4) = -7 3x + 8 = -7 3x + 8 -8 = -7 -8 3x = -15
3x −
3 = -
15 −
3
x = -5The solution is (-5, -4).
ExercisesUse elimination to solve each system of equations.
1. 2x + 3y = 6 2. 2m + 3n = 4 3. 3a - b = 2x + 2y = 5 -m + 2n = 5 a + 2b = 3
4. 4x + 5y = 6 5. 4x - 3y = 22 6. 3x - 4y = -46x - 7y = -20 2x - y = 10 x + 3y = -10
7. 4x - y = 9 8. 4a - 3b = -8 9. 2x + 2y = 55x + 2y = 8 2a + 2b = 3 4x - 4y = 10
10. 6x - 4y = -8 11. 4x + 2y = -5 12. 2x + y = 3.5 4x + 2y = -3 -2x - 4y = 1 -x + 2y = 2.5
13. GARDENING The length of Sally’s garden is 4 meters greater than 3 times the width. The perimeter of her garden is 72 meters. What are the dimensions of Sally’s garden?
6-4
Example 1 Example 2
019_027_ALG1_A_CRM_C06_CR_660280.indd 24019_027_ALG1_A_CRM_C06_CR_660280.indd 24 12/21/10 8:43 AM12/21/10 8:43 AM
Less
on
6-4
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
NAME DATE PERIOD
Less
on
6-4
PDF Pass
Chapter 6 25 Glencoe Algebra 1
6-4 Study Guide and Intervention (continued)
Elimination Using Multiplication
Solve Real-World Problems Sometimes it is necessary to use multiplication before elimination in real-world problems.
CANOEING During a canoeing trip, it takes Raymond 4 hours to paddle 12 miles upstream. It takes him 3 hours to make the return trip paddling downstream. Find the speed of the canoe in still water.
Read You are asked to find the speed of the canoe in still water.
Solve Let c = the rate of the canoe in still water.Let w = the rate of the water current.
r t d r · t = d
Against the Current c – w 4 12 (c – w)4 = 12
With the Current c + w 3 12 (c + w)3 = 12
So, our two equations are 4c - 4w = 12 and 3c + 3w = 12.Use elimination with multiplication to solve the system. Since the problem asks for c, eliminate w.4c - 4w = 12 ⇒ Multiply by 3 ⇒ 12c - 12w = 363c + 3w = 12 ⇒ Multiply by 4 ⇒ (+) 12c + 12w = 48 24c = 84 w is eliminated.
24c−
24=
84−
24Divide each side by 24.
c = 3.5 Simplify.The rate of the canoe in still water is 3.5 miles per hour.
Exercises 1. FLIGHT An airplane traveling with the wind flies 450 miles in 2 hours. On the return
trip, the plane takes 3 hours to travel the same distance. Find the speed of the airplane if the wind is still.
2. FUNDRAISING Benji and Joel are raising money for their class trip by selling gift wrapping paper. Benji raises $39 by selling 5 rolls of red wrapping paper and 2 rolls of foil wrapping paper. Joel raises $57 by selling 3 rolls of red wrapping paper and 6 rolls of foil wrapping paper. For how much are Benji and Joel selling each roll of red and foil wrapping paper?
Example
019_027_ALG1_A_CRM_C06_CR_660280.indd 25019_027_ALG1_A_CRM_C06_CR_660280.indd 25 12/21/10 8:43 AM12/21/10 8:43 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
NAME DATE PERIOD
PDF Pass
Chapter 6 26 Glencoe Algebra 1
Skills PracticeElimination Using Multiplication
Use elimination to solve each system of equations.
1. x + y = -9 2. 3x + 2y = -95x - 2y = 32 x - y = -13
3. 2x + 5y = 3 4. 2x + y = 3-x + 3y = -7 -4x - 4y = -8
5. 4x - 2y = -14 6. 2x + y = 03x - y = -8 5x + 3y = 2
7. 5x + 3y = -10 8. 2x + 3y = 143x + 5y = -6 3x - 4y = 4
9. 2x - 3y = 21 10. 3x + 2y = -265x - 2y = 25 4x - 5y = -4
11. 3x - 6y = -3 12. 5x + 2y = -32x + 4y = 30 3x + 3y = 9
13. Two times a number plus three times another number equals 13. The sum of the two numbers is 7. What are the numbers?
14. Four times a number minus twice another number is -16. The sum of the two numbers is -1. Find the numbers.
15. FUNDRAISING Trisha and Byron are washing and vacuuming cars to raise money for a class trip. Trisha raised $38 washing 5 cars and vacuuming 4 cars. Byron raised $28 by washing 4 cars and vacuuming 2 cars. Find the amount they charged to wash a car and vacuum a car.
6-4
019_027_ALG1_A_CRM_C06_CR_660280.indd 26019_027_ALG1_A_CRM_C06_CR_660280.indd 26 12/21/10 8:43 AM12/21/10 8:43 AM
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
NAME DATE PERIOD
Less
on
6-4
PDF Pass
Chapter 6 27 Glencoe Algebra 1
PracticeElimination Using Multiplication
Use elimination to solve each system of equations.
1. 2x - y = -1 2. 5x - 2y = -10 3. 7x + 4y = -43x - 2y = 1 3x + 6y = 66 5x + 8y = 28
4. 2x - 4y = -22 5. 3x + 2y = -9 6. 4x - 2y = 323x + 3y = 30 5x - 3y = 4 -3x - 5y = -11
7. 3x + 4y = 27 8. 0.5x + 0.5y = -2 9. 2x - 3 −
4 y = -7
5x - 3y = 16 x - 0.25y = 6 x + 1 −
2 y = 0
10. 6x - 3y = 21 11. 3x + 2y = 11 12. -3x + 2y = -152x + 2y = 22 2x + 6y = -2 2x - 4y = 26
13. Eight times a number plus five times another number is -13. The sum of the two numbers is 1. What are the numbers?
14. Two times a number plus three times another number equals 4. Three times the first number plus four times the other number is 7. Find the numbers.
15. FINANCE Gunther invested $10,000 in two mutual funds. One of the funds rose 6% in one year, and the other rose 9% in one year. If Gunther’s investment rose a total of $684 in one year, how much did he invest in each mutual fund?
16. CANOEING Laura and Brent paddled a canoe 6 miles upstream in four hours. The return trip took three hours. Find the rate at which Laura and Brent paddled the canoe in still water.
17. NUMBER THEORY The sum of the digits of a two-digit number is 11. If the digits are reversed, the new number is 45 more than the original number. Find the number.
6-4
019_027_ALG1_A_CRM_C06_CR_660280.indd 27019_027_ALG1_A_CRM_C06_CR_660280.indd 27 12/21/10 8:43 AM12/21/10 8:43 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
NAME DATE PERIOD
PDF Pass
Chapter 6 28 Glencoe Algebra 1
1. SOCCER Suppose a youth soccer field has a perimeter of 320 yards and its length measures 40 yards more than its width. Ms. Hughey asks her players to determine the length and width of their field. She gives them the following system of equations to represent the situation. Use elimination to solve the system to find the length and width of the field.
2L + 2W = 320 L – W = 40
2. SPORTS The Fan Cost Index (FCI) tracks the average costs for attending sporting events, including tickets, drinks, food, parking, programs, and souvenirs. According to the FCI, a family of four would spend a total of $592.30 to attend two Major League Baseball (MLB) games and one National Basketball Association (NBA) game. The family would spend $691.31 to attend one MLB and two NBA games. Write and solve a system of equations to find the family’s costs for each kind of game according to the FCI.
3. ART Mr. Santos, the curator of the children’s museum, recently made two purchases of clay and wood for a visiting artist to sculpt. Use the table to find the cost of each product per kilogram.
Clay (kg) Wood (kg) Total Cost
5 4 $35.50
3.5 6 $50.45
4. TRAVEL Antonio flies from Houston to Philadelphia, a distance of about 1340 miles. His plane travels with the wind and takes 2 hours and 20 minutes. At the same time, Paul is on a plane from Philadelphia to Houston. Since his plane is heading against the wind, Paul’s flight takes 2 hours and 50 minutes. What was the speed of the wind in miles per hour?
5. BUSINESS Suppose you start a business assembling and selling motorized scooters. It costs you $1500 for tools and equipment to get started, and the materials cost $200 for each scooter. Your scooters sell for $300 each.
a. Write and solve a system of equations representing the total costs and revenue of your business.
b. Describe what the solution means in terms of the situation.
c. Give an example of a reasonable number of scooters you could assemble and sell in order to make a profit, and find the profit you would make for that number of scooters.
Word Problem PracticeElimination Using Multiplication
6-4
028_036_ALG1_A_CRM_C06_CR_660280.indd 28028_036_ALG1_A_CRM_C06_CR_660280.indd 28 12/21/10 8:43 AM12/21/10 8:43 AM
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
NAME DATE PERIOD
Less
on
6-4
PDF Pass
Chapter 6 29 Glencoe Algebra 1
Enrichment
George Washington Carver and Percy Julian In 1990, George Washington Carver and Percy Julian became the first African Americans elected to the National Inventors Hall of Fame. Carver (1864–1943) was an agricultural scientist known worldwide for developing hundreds of uses for the peanut and the sweet potato. His work revitalized the economy of the southern United States because it was no longer dependent solely upon cotton. Julian (1898–1975) was a research chemist who became famous for inventing a method of making a synthetic cortisone from soybeans. His discovery has had many medical applications, particularly in the treatment of arthritis.There are dozens of other African American inventors whose accomplishments are not as well known. Their inventions range from common household items like the ironing board to complex devices that have revolutionized manufacturing. The exercises that follow will help you identify just a few of these inventors and their inventions.
Match the inventors with their inventions by matching each system with its solution. (Not all the solutions will be used.)
1. Sara Boone x + y = 2 A. (1, 4) automatic traffic signal x - y = 10
2. Sarah Goode x = 2 - y B. (4, -2) eggbeater 2y + x = 9
3. Frederick M. y = 2x + 6 C. (-2, 3) fire extinguisher Jones y = -x - 3
4. J. L. Love 2x + 3y = 8 D. (-5, 7) folding cabinet bed 2x - y = -8
5. T. J. Marshall y - 3x = 9 E. (6, -4) ironing board 2y + x = 4
6. Jan Matzeliger y + 4 = 2x F. (-2, 4) pencil sharpener 6x - 3y = 12
7. Garrett A. 3x - 2y = -5 G. (-3, 0) portable x-ray machine Morgan 3y - 4x = 8
8. Norbert Rillieux 3x - y = 12 H. (2, -3) player piano y - 3x = 15
I. no solution evaporating pan for refining sugar
J. infinitely lasting (shaping) many machine for solutions manufacturing shoes
6-4
028_036_ALG1_A_CRM_C06_CR_660280.indd 29028_036_ALG1_A_CRM_C06_CR_660280.indd 29 12/21/10 8:43 AM12/21/10 8:43 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
NAME DATE PERIOD
PDF Pass
Chapter 6 30 Glencoe Algebra 1
Study Guide and Intervention Applying Systems of Linear Equations
Determine The Best Method You have learned five methods for solving systems of linear equations: graphing, substitution, elimination using addition, elimination using subtraction, and elimination using multiplication. For an exact solution, an algebraic method is best.
At a baseball game, Henry bought 3 hotdogs and a bag of chips for $14. Scott bought 2 hotdogs and a bag of chips for $10. Each of the boys paid the same price for their hotdogs, and the same price for their chips. The following system of equations can be used to represent the situation. Determine the best method to solve the system of equations. Then solve the system.
3x + y = 142x + y = 10
• Since neither the coefficients of x nor the coefficients of y are additive inverses, you cannot use elimination using addition.
• Since the coefficient of y in both equations is 1, you can use elimination using subtraction. You could also use the substitution method or elimination using multiplication
The following solution uses elimination by subtraction to solve this system. 3x + y = 14 Write the equations in column form and subtract. (-) 2x + (-) y = (-)10
x = 4 The variable y is eliminated. 3(4) + y = 14 Substitute the value for x back into the first equation.
y = 2 Solve for y.
This means that hot dogs cost $4 each and a bag of chips costs $2.
ExercisesDetermine the best method to solve each system of equations. Then solve the system.
1. 5x + 3y = 16 2. 3x - 5y = 73x - 5y = -4 2x + 5y = 13
3. y + 3x = 24 4. -11x - 10y = 175x - y = 8 5x – 7y = 50
6-5
Example
028_036_ALG1_A_CRM_C06_CR_660280.indd 30028_036_ALG1_A_CRM_C06_CR_660280.indd 30 12/21/10 8:43 AM12/21/10 8:43 AM
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
NAME DATE PERIOD
Less
on
6-5
PDF Pass
Chapter 6 31 Glencoe Algebra 1
Apply Systems Of Linear Equations When applying systems of linear equations to problem situations, it is important to analyze each solution in the context of the situation.
BUSINESS A T-shirt printing company sells T-shirts for $15 each. The company has a fixed cost for the machine used to print the T-shirts and an additional cost per T-shirt. Use the table to estimate the number of T-shirts the company must sell in order for the income to equal expenses.
Understand You know the initial income and the initial expense and the rates of change of each quantity with each T-shirt sold.
Plan Write an equation to represent the income and the expenses. Then solve to find how many T-shirts need to be sold for both values to be equal.
Solve Let x = the number of T-shirts sold and let y = the total amount.
income y = 0 + 15x
expenses y = 3000 + 5x
You can use substitution to solve this system.
y = 15x The fi rst equation.
15x = 3000 + 5x Substitute the value for y into the second equation.
10x = 3000 Subtract 10x from each side and simplify.
x = 300 Divide each side by 10 and simplify.
This means that if 300 T-shirts are sold, the income and expenses of the T-shirt company are equal.
Check Does this solution make sense in the context of the problem? After selling 300 T-shirts, the income would be about 300 × $15 or $4500. The costs would be about $3000 + 300 × $5 or $4500.
ExercisesRefer to the example above. If the costs of the T-shirt company change to the given values and the selling price remains the same, determine the number of T-shirts the company must sell in order for income to equal expenses.
1. printing machine: $5000.00; 2. printing machine: $2100.00; T-shirt: $10.00 each T-shirt: $8.00 each
3. printing machine: $8800.00; 4. printing machine: $1200.00;T-shirt: $4.00 each T-shirt: $12.00 each
Study Guide and Intervention (continued)
Applying Systems of Linear Equations
6-5
T-shirt Printing Cost
printing
machine $3000.00
blank
T-shirt $5.00
Example
total
amount
initial
amount
rate of change times
number of T-shirts sold
028_036_ALG1_A_CRM_C06_CR_660280.indd 31028_036_ALG1_A_CRM_C06_CR_660280.indd 31 12/21/10 8:43 AM12/21/10 8:43 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
NAME DATE PERIOD
PDF Pass
Chapter 6 32 Glencoe Algebra 1
Determine the best method to solve each system of equations. Then solve the system.
1. 5x + 3y = 16 2. 3x – 5y = 73x – 5y = -4 2x + 5y = 13
3. y = 3x - 24 4. -11x – 10y = 175x - y = 8 5x – 7y = 50
5. 4x + y = 24 6. 6x – y = -1455x - y = 12 x = 4 – 2y
7. VEGETABLE STAND A roadside vegetable stand sells pumpkins for $5 each and squashes for $3 each. One day they sold 6 more squash than pumpkins, and their sales totaled $98. Write and solve a system of equations to find how many pumpkins and squash they sold?
8. INCOME Ramiro earns $20 per hour during the week and $30 per hour for overtime on the weekends. One week Ramiro earned a total of $650. He worked 5 times as many hours during the week as he did on the weekend. Write and solve a system of equations to determine how many hours of overtime Ramiro worked on the weekend.
9. BASKETBALL Anya makes 14 baskets during her game. Some of these baskets were worth 2-points and others were worth 3-points. In total, she scored 30 points. Write and solve a system of equations to find how 2-points baskets she made.
6-5 Skills PracticeApplying Systems of Linear Equations
028_036_ALG1_A_CRM_C06_CR_660280.indd 32028_036_ALG1_A_CRM_C06_CR_660280.indd 32 12/21/10 8:43 AM12/21/10 8:43 AM
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
NAME DATE PERIOD
Less
on
6-5
PDF Pass
Chapter 6 33 Glencoe Algebra 1
Determine the best method to solve each system of equations. Then solve the system.
1. 1.5x – 1.9y = -29 2. 1.2x – 0.8y = -6 3. 18x –16y = -312x – 0.9y = 4.5 4.8x + 2.4y = 60 78x –16y = 408
4. 14x + 7y = 217 5. x = 3.6y + 0.7 6. 5.3x – 4y = 43.514x + 3y = 189 2x + 0.2y = 38.4 x + 7y = 78
7. BOOKS A library contains 2000 books. There are 3 times as many non-fiction books as fiction books. Write and solve a system of equations to determine the number of non-fiction and fiction books.
8. SCHOOL CLUBS The chess club has 16 members and gains a new member every month. The film club has 4 members and gains 4 new members every month. Write and solve a system of equations to find when the number of members in both clubs will be equal.
9. Tia and Ken each sold snack bars and magazine subscriptions for a school fund-raiser, as shown in the table. Tia earned $132 and Ken earned $190.
a. Define variable and formulate a system of linear equation from this situation.
b. What was the price per snack bar? Determine the reasonableness of your solution.
PracticeApplying Systems of Linear Equations
6-5
ItemNumber Sold
Tia Ken
snack bars 16 20
magazine
subscriptions4 6
028_036_ALG1_A_CRM_C06_CR_660280.indd 33028_036_ALG1_A_CRM_C06_CR_660280.indd 33 12/21/10 8:43 AM12/21/10 8:43 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
NAME DATE PERIOD
PDF Pass
Chapter 6 34 Glencoe Algebra 1
Word Problem PracticeApplying Systems of Linear Equations
1. MONEY Veronica has been saving dimes and quarters. She has 94 coins in all, and the total value is $19.30. How many dimes and how many quarters does she have?
2. CHEMISTRY How many liters of 15% acid and 33% acid should be mixed to make 40 liters of 21% acid solution?
Concentration
of Solution
Amount of
Solution (L)
Amount
of Acid
15% x
33% y
21% 40
3. BUILDINGS The Sears Tower in Chicago is the tallest building in North America. The total height of the tower t and the antenna that stands on top of it a is 1729 feet. The difference in heights between the building and the antenna is 279 feet. How tall is the Sears Tower?
4. PRODUCE Roger and Trevor went shopping for produce on the same day. They each bought some apples and some potatoes. The amount they bought and the total price they paid are listed in the table below.
What was the price of apples and potatoes per pound?
5. SHOPPING Two stores are having a sale on T-shirts that normally sell for $20. Store S is advertising an s percent discount, and Store T is advertising at dollar discount. Rose spends $63 for three T-shirts from Store S and one from Store T. Manny spends $140 on five T-shirts from Store S and four from Store T. Find the discount at each store.
6. TRANSPORTATION A Speedy River barge bound for New Orleans leaves Baton Rouge, Louisiana, at 9:00 A.M. and travels at a speed of 10 miles per hour. A Rail Transport freight train also bound for New Orleans leaves Baton Rouge at 1:30 P.M. the same day. The train travels at 25 miles per hour, and the river barge travels at 10 miles per hour. Both the barge and the train will travel 100 miles to reach New Orleans.
a. How far will the train travel before catching up to the barge?
b. Which shipment will reach New Orleans first? At what time?
c. If both shipments take an hour to unload before heading back to Baton Rouge, what is the earliest time that either one of the companies can begin to load grain to ship in Baton Rouge?
6-5
Apples
(Ib)
Potatoes
(Ib)
Total Cost
($)
Roger 8 7 18.85
Trevor 2 10 12.88
028_036_ALG1_A_CRM_C06_CR_660280.indd 34028_036_ALG1_A_CRM_C06_CR_660280.indd 34 12/21/10 8:43 AM12/21/10 8:43 AM
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
NAME DATE PERIOD
Less
on
6-5
PDF Pass
Chapter 6 35 Glencoe Algebra 1
Cramer’s RuleCramer’s Rule is a method for solving a system of equations. To use Cramer’s Rule, set up a matrix to represent the equations. A matrix is a way of organizing data.
Solve the following system of equations using Cramer’s Rule. 2x + 3y = 13 x + y = 5
Step 1: Set up a matrix representing the coefficients of x and y.
A = x y
Step 2: Find the determinant of matrix A.
If a matrix A = ⎪
a c b d ⎥
, then the determinant, det(A) = ad – bc.
det(A) = 2(1) – 1(3) = -1
Step 3: Replace the first column in A with 13 and 5 and find the determinant of the new matrix.
A1 = ⎪
13 5 3 1 ⎥
; det (A1) = 13(1) – 5(3) = -2
Step 4: To find the value of x in the solution to the system of equations,
determine the value of det(A1) −
det(A) .
det(A1) −
det(A) = -2 −
-1 or 2
Step 5: Repeat the process to find the value of y. This time, replace the second column with 13 and 5 and find the determinant.
A2 = ⎪
2 1 13
5 ⎥
; det (A2) = 2(5) – 1(13) = -3 and det(A2) −
det(A) = -3 −
-1 or 3.
So, the solution to the system of equations is (2, 3).
Exercises Use Cramer’s Rule to solve each system of equations.
1. 2x + y = 1 2. x + y = 4 3. x – y = 43x + 5y = 5 2x – 3y = -2 3x – 5y = 8
4. 4x – y = 3 5. 3x – 2y = 7 6. 6x – 5y = 1x + y = 7 2x + y = 14 3x + 2y = 5
Enrichment6-5
Example
⎪
2 1 3 1
⎥
028_036_ALG1_A_CRM_C06_CR_660280.indd 35028_036_ALG1_A_CRM_C06_CR_660280.indd 35 12/21/10 8:43 AM12/21/10 8:43 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
NAME DATE PERIOD
PDF Pass
Chapter 6 36 Glencoe Algebra 1
Study Guide and InterventionSystems of Inequalities
Systems of Inequalities The solution of a system of inequalities is the set of all ordered pairs that satisfy both inequalities. If you graph the inequalities in the same coordinate plane, the solution is the region where the graphs overlap.
Solve the system of inequalities by graphing.y > x + 2y ≤ -2x - 1
The solution includes the ordered pairs in the intersection of the graphs. This region is shaded at the right. The graphs of y = x + 2 and y = -2x - 1 are boundaries of this region. The graph of y = x + 2 is dashed and is not included in the graph of y > x + 2.
Solve the system of inequalities by graphing.x + y > 4x + y < -1
The graphs of x + y = 4 and x + y = -1 are parallel. Because the two regions have no points in common, the system of inequalities has no solution.
ExercisesSolve each system of inequalities by graphing.
1. y > -1 2. y > -2x + 2 3. y < x + 1x < 0 y ≤ x + 1 3x + 4y ≥ 12
x
y
O
x
y
O
x
y
O
4. 2x + y ≥ 1 5. y ≤ 2x + 3 6. 5x - 2y < 6x - y ≥ -2 y ≥ -1 + 2x y > -x + 1
x
y
O
x
y
O
x
y
O
x
y
O
x + y = 1
x + y = 4
x
y
O
y = x + 2
y = -2x - 1
6-6
Example 1
Example 2
028_036_ALG1_A_CRM_C06_CR_660280.indd 36028_036_ALG1_A_CRM_C06_CR_660280.indd 36 12/21/10 8:43 AM12/21/10 8:43 AM
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
NAME DATE PERIOD
Less
on
6-6
PDF Pass
Chapter 6 37 Glencoe Algebra 1
Study Guide and Intervention (continued)
Systems of Inequalities
Apply Systems of Inequalities In real-world problems, sometimes only whole numbers make sense for the solution, and often only positive values of x and y make sense.
BUSINESS AAA Gem Company produces necklaces and bracelets. In a 40-hour week, the company has 400 gems to use. A necklace requires 40 gems and a bracelet requires 10 gems. It takes 2 hours to produce a necklace and a bracelet requires one hour. How many of each type can be produced in a week?
Let n = the number of necklaces that will be produced and b = the number of bracelets that will be produced. Neither n or b can be a negative number, so the following system of inequalities represents the conditions of the problems.
n ≥ 0b ≥ 0b + 2n ≤ 4010b + 40n ≤ 400The solution is the set ordered pairs in the intersection of the graphs. This region is shaded at the right. Only whole-number solutions, such as (5, 20) make sense in this problem.
ExercisesFor each exercise, graph the solution set. List three possible solutions to the problem.
1. HEALTH Mr. Flowers is on a restricted 2. RECREATION Maria had $150 in gift diet that allows him to have between certificates to use at a record store. She 1600 and 2000 Calories per day. His bought fewer than 20 recordings. Each daily fat intake is restricted to between tape cost $5.95 and each CD cost $8.95. 45 and 55 grams. What daily Calorie How many of each type of recording might and fat intakes are acceptable? she have bought?
60
50
40
30
20
10
Calories
Fat
Gra
ms
1000 2000 30000
30
25
20
15
10
5
Tapes
Co
mp
act
Dis
cs
5 10 15 20 25 300
Necklaces
Bra
cele
ts
10 20 30 40 500
50
40
30
20
10
10b + 40n = 400
b + 2n = 40
6-6
Example
037_044_ALG1_A_CRM_C06_CR_660280.indd 37037_044_ALG1_A_CRM_C06_CR_660280.indd 37 12/21/10 8:43 AM12/21/10 8:43 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
NAME DATE PERIOD
PDF 2nd
Chapter 6 38 Glencoe Algebra 1
Skills PracticeSystems of Inequalities
Solve each system of inequalities by graphing.
1. x > -1 2. y > 2 3. y > x + 3y ≤ -3 x < -2 y ≤ -1
x
y
O
x
y
O
x
y
O
4. x < 2 5. x + y ≤ -1 6. y - x > 4y - x ≤ 2 x + y ≥ 3 x + y > 2
x
y
O
y
xO
x
y
O
7. y > x + 1 8. y ≥ -x + 2 9. y < 2x + 4y ≥ -x + 1 y < 2x - 2 y ≥ x + 1
x
y
O
x
y
O
x
y
O
Write a system of inequalities for each graph.
10.
x
y
O
11.
x
y
O
12.
x
y
O
6-6
037_044_ALG1_A_CRM_C06_CR_660280.indd 38037_044_ALG1_A_CRM_C06_CR_660280.indd 38 12/24/10 12:38 PM12/24/10 12:38 PM
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
NAME DATE PERIOD
Less
on
6-6
PDF Pass
Chapter 6 39 Glencoe Algebra 1
PracticeSystems of Inequalities
$4 Stones
Turquoise Stones
$6 S
ton
es
1 32 4 5 6 70
6
5
4
3
2
1
Gym (hours)
Diego’s RoutineW
alki
ng
(m
iles)
1 32 4 5 6 7 80
16
14
12
10
8
6
4
2
6-6
Solve each system of inequalities by graphing.
1. y > x - 2 2. y ≥ x + 2 3. x + y ≥ 1y ≤ x y > 2x + 3 x + 2y > 1
x
y
O
x
y
O
x
y
O
4. y < 2x - 1 5. y > x - 4 6. 2x - y ≥ 2y > 2 - x 2x + y ≤ 2 x - 2y ≥ 2
x
y
O
x
y
O
x
y
O
7. FITNESS Diego started an exercise program in which each week he works out at the gym between 4.5 and 6 hours and walks between 9 and 12 miles.
a. Make a graph to show the number of hours Diego works out at the gym and the number of miles he walks per week.
b. List three possible combinations of working out and walking that meet Diego’s goals.
8. SOUVENIRS Emily wants to buy turquoise stones on her trip to New Mexico to give to at least 4 of her friends. The gift shop sells stones for either $4 or $6 per stone. Emily has no more than $30 to spend.
a. Make a graph showing the numbers of each price of stone Emily can purchase.
b. List three possible solutions.
037_044_ALG1_A_CRM_C06_CR_660280.indd 39037_044_ALG1_A_CRM_C06_CR_660280.indd 39 12/21/10 8:43 AM12/21/10 8:43 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
NAME DATE PERIOD
PDF Pass
Chapter 6 40 Glencoe Algebra 1
Word Problem PracticeSystems of Inequalities
1. PETS Renée’s Pet Store never has more than a combined total of 20 cats and dogs and never more than 8 cats. This is represented by the inequalities x ≤ 8 and x + y ≤ 20. Solve the system of inequalities by graphing.
2. WAGES The minimum wage for one group of workers in Texas is $7.25 per hour effective Sept. 1, 2008. The graph below shows the possible weekly wages for a person who makes at least minimum wages and works at most 40 hours. Write the system of inequalities for the graph.
3. FUND RAISING The Camp Courage Club plans to sell tins of popcorn and peanuts as a fundraiser. The Club members have $900 to spend on products to sell and want to order up to 200 tins in all. They also want to order at least as many tins of popcorn as tins of peanuts. Each tin of popcorn costs $3 and each tin of peanuts costs $4. Write a system of equations to represent the conditions of this problem.
4. BUSINESS For maximum efficiency, a factory must have at least 100 workers, but no more than 200 workers on a shift. The factory also must manufacture at least 30 units per worker.
a. Let x be the number of workers and let y be the number of units. Write four inequalities expressing the conditions in the problem given above.
b. Graph the systems of inequalities.
c. List at least three possible solutions.
O
Do
gs
Cats
4
6
2
8
10
12
14
16
18
20y
642 10 14 16 18 208 12 x
x
y
25
50
75
100
Wag
es (
$)
125
150
175
200
225
250
275
300
Hours5 10 15 20 25 30 35 40 45 50
O
Un
its
(100
0)
2
3
1
4
5
6
7
Workers
y
150100
50 250 350200 300 400
x
6-6
037_044_ALG1_A_CRM_C06_CR_660280.indd 40037_044_ALG1_A_CRM_C06_CR_660280.indd 40 12/21/10 8:43 AM12/21/10 8:43 AM
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
NAME DATE PERIOD
Less
on
6-6
PDF Pass
Chapter 6 41 Glencoe Algebra 1
Enrichment
Describing Regions
The shaded region inside the triangle can be described with a system of three inequalities. y < -x + 1 y > 1 −
3 x - 3
y > -9x - 31
Write systems of inequalities to describe each region. You may first need to divide a region into triangles or quadrilaterals.
1.
x
y
O
2.
x
y
O
3.
x
y
O
6-6
x
y
O
037_044_ALG1_A_CRM_C06_CR_660280.indd 41037_044_ALG1_A_CRM_C06_CR_660280.indd 41 12/21/10 8:43 AM12/21/10 8:43 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
NAME DATE PERIOD
PDF Pass
Chapter 6 42 Glencoe Algebra 1
Graphing Calculator ActivityShading Absolute Value Inequalities
Absolute value inequalities of the form y ≤ a|x - h| + k or y ≥ a|x - h| + k can be graphed using the SHADE command or by using the shading feature in the Y= screen.
Graph y ≤ 2|x - 3| + 2.
Method 1 SHADE command
Enter the boundary equation y = 2|x - 3| + 2 into Y1. From the home screen, use SHADE to shade the region under the boundary equation because the inequality uses <. Use Ymin as the lower boundary and the boundary equation, Y1, as the upper boundary.Keystrokes: 2nd [DRAW] 7 VARS ENTER 4 , VARS ENTER ENTER ) ENTER .
Method 2 Y=
On the Y= screen, move the cursor to the far left and repeatedly press ENTER to toggle through the graph choices. Be sure to select the option to shade below . Then enter the boundary equation and graph.
The same techniques can be used to solve a system of absolute value inequalities.
ExercisesSolve each system of inequalities by graphing.
1. y ≤ 2|x - 4| + 1 2. y < 1 −
2 |x - 3| + 2 3. y ≤ 4|x - 4| + 3 4. y ≤
1 −
4 |x + 1| + 1
y > |x + 3| + 6 y ≤ |x + 1| + 4 y ≥ 3|x - 4| - 2 y ≥ 2|x + 3| + 5
Solve the system y ≥ |x + 5| - 4 and y ≤ 3|x - 6| + 1.
Enter y = |x + 5| - 4 in Y1 and set the graph to shade above. Then enter y = 3|x - 6| + 1 in Y2 and set the graph to shade below. Graph the inequalities. Notice that a different pattern is used for the second inequality. The region where the two patterns overlap is the solution to the system.
[-10, 10] scl:1 by [-10, 10] scl:1
[-10, 10] scl:1 by [-10, 10] scl:1
[-20, 20] scl:2 by [-20, 20] scl:2
6-6
Example 1
Example 2
037_044_ALG1_A_CRM_C06_CR_660280.indd 42037_044_ALG1_A_CRM_C06_CR_660280.indd 42 12/21/10 8:43 AM12/21/10 8:43 AM
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
NAME DATE PERIOD
Less
on
6-6
PDF Pass
Chapter 6 43 Glencoe Algebra 1
6-6 Spreadsheet ActivitySystems of Inequalities
ExercisesUse the spreadsheet to determine whether TopSport can make the following combinations of shoes.
1. 1000 Runners, 1100 Flyers 2. 1200 Runners, 1000 Flyers
3. 2000 Runners, 500 Flyers 4. 300 Runners, 1500 Flyers
4. 950 Runners, 1400 Flyers 5. 1800 Runners, 1300 Flyers
6. If TopSport can either buy another cutting machine or hire more assemblers, which would make more combinations of shoe production possible? Explain.
TopSport Shoe Company has a total of 9600 minutes of machine time each week to cut the materials for the two types of athletic shoes they make. There are a total of 28,000 minutes of worker time per week for assembly. It takes 3 minutes to cut and 12 minutes to assemble a pair of Runners and 2 minutes to cut and 10 minutes to assemble a pair of Flyers. Is it possible for the company to make 1200 pairs of Runners and 1400 pairs of Flyers in a week?
Step 1 Represent the situation using a system of inequalities. Let r represent the number of Runners and f represent the number of Flyers.
3r + 2f ≤ 9600 12r + 10f ≤ 28,000
Step 2 Columns A and B contain the values of r and f. Columns C and D contain the formulas for the inequalities. The formulas will return TRUE or FALSE. If both inequalities are true for an ordered pair, then the ordered pair is a solution to the system of inequalities.
Since one of the inequalities is false for (1200, 1400), the ordered pair is not part of the solution set. The company cannot make 1200 pairs of Runners and 1400 pairs of Flyers in a week.The formula in cell D2 is
12*A2 + 10*B2 <= 28,000.
+ ≤ + ≤
Example
037_044_ALG1_A_CRM_C06_CR_660280.indd 43037_044_ALG1_A_CRM_C06_CR_660280.indd 43 12/21/10 8:43 AM12/21/10 8:43 AM
037_044_ALG1_A_CRM_C06_CR_660280.indd 44037_044_ALG1_A_CRM_C06_CR_660280.indd 44 12/21/10 11:49 PM12/21/10 11:49 PM
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
NAME DATE PERIOD
Ass
essm
ent
PDF Pass
Chapter 6 45 Glencoe Algebra 1
Student Recording SheetUse this recording sheet with pages 386–387 of the Student Edition.
6
Read each question. Then fill in the correct answer.
1. A B C D
2. F G H J
3. A B C D
4. F G H J
5. A B C D
6. F G H J
7. A B C D
8. F G H J
Multiple Choice
Extended Response
Record your answers for Question 18 on the back of this paper.
9.
Record your answer in the blank.
For gridded response questions, also enter your answer in the grid by writing each number or symbol in a box. Then fill in the corresponding circle for that number or symbol.
9. (grid in)
10.
11. (grid in)
12.
13.
14. (grid in)
15.
16.
17.
Short Response/Gridded Response
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
. . . . .
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
11.
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
. . . . .
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
14.
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
. . . . .
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
045_052_ALG1_A_CRM_C06_CR_660280.indd 45045_052_ALG1_A_CRM_C06_CR_660280.indd 45 12/22/10 12:03 AM12/22/10 12:03 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
NAME DATE PERIOD
PDF Pass
Chapter 6 46 Glencoe Algebra 1
General Scoring Guidelines• If a student gives only a correct numerical answer to a problem but does not show how he
or she arrived at the answer, the student will be awarded only 1 credit. All extended-response questions require the student to show work.
• A fully correct answer for a multiple-part question requires correct responses for all parts of the question. For example, if a question has three parts, the correct response to one or two parts of the question that required work to be shown is not considered a fully correct response.
• Students who use trial and error to solve a problem must show their method. Merely showing that the answer checks or is correct is not considered a complete response for full credit.
Exercise 18 Rubric
6 Rubric for Scoring Extended Response
Score Specifi c Criteria
4 The equation and calculations written in part a correctly represent the information provided. They show that 250 canned goods were collected on Day 1 and that after 5 days, 5(250) or 1250 cans will be collected. In part b, students should note that the total number of cans collected each day will vary and that the total after Day 5 is highly unlikely to be exactly 1250.
3 A generally correct solution, but may contain minor flaws in reasoning or computation.
2 A partially correct interpretation and/or solution to the problem.
1 A correct solution with no evidence or explanation.
0 An incorrect solution indicating no mathematical understanding of the concept or task, or no solution is given.
045_052_ALG1_A_CRM_C06_CR_660280.indd 46045_052_ALG1_A_CRM_C06_CR_660280.indd 46 12/22/10 12:03 AM12/22/10 12:03 AM
Ass
essm
ent
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
NAME DATE PERIOD
Chapter 6 Quiz 2(Lessons 6-3 and 6-4)
CCCh tt QQ iiNAME DATE PERIOD
6
PDF Pass
Chapter 6 47 Glencoe Algebra 1
SCORE
Graph each system of equations. Then determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it.
1. y =
3 −
2 x 2. x - 2y =
-2 y = -x + 5 x - 2y = 3
For Questions 3 and 4, use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solutions or infinitely many solutions.
3. 3x - 2y = -7 4. -6x - 2y = -20y = x + 4 y = -3x + 10
5. MULTIPLE CHOICE In order for José and Marty to compete against each other during the wrestling season next year they need to be in the same weight category. José weighs 180 pounds and plans to gain 2 pounds per week. Marty weighs 249 pounds and plans to lose 1 pound per week. In how manyweeks will they weigh the same?
A 23 B 34.5 C 69 D 226
For Questions 1–4, use elimination to solve each system of equations.
1. x + y = 4 2. -2x + y = 5x - y = 7 2x + 3y = 3
3. 4x + 6y = -10 4. 2x + 3y = 18x - 3y = 25 5x - 4y = 14
5. MULTIPLE CHOICE If 5x - 3y = 7 and -3x - 5y = 23, what is the value of x?
A (-1, -4) B (-4, -1) C -4 D -1
1.
2.
3.
4.
5.
y
x
y
x
1.
2.
3.
4.
5.
6 Chapter 6 Quiz 1(Lessons 6-1 and 6-2)
SCORE
045_052_ALG1_A_CRM_C06_CR_660280.indd 47045_052_ALG1_A_CRM_C06_CR_660280.indd 47 12/22/10 12:03 AM12/22/10 12:03 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
NAME DATE PERIOD
PDF Pass
Chapter 6 48 Glencoe Algebra 1
SCORE
NAME DATE PERIOD
Chapter 6 Quiz 4(Lessons 6-6)
6
Determine the best method to solve each system of equations. Then solve the system.
1. 2y + 1 = x 2. 7x + 5y = 293x + y = 17 21x - 25y = -33
3. -2x + 5y = 14 4. 4x + 2y = 1 -2x - 3y = -2 -4x + 4y = 5
5. MULTIPLE CHOICE Timothy and his brothers are selling lemonade and cookies to raise funds for a camping trip. Paul bought two cups of lemonade and two cookies for $3 and Nancy bought three cookies and two cups of lemonade for $4.00. What is the cost of a cup of lemonade?
A $2 B $1 C $0.75 D $0.50
6 Chapter 6 Quiz 3(Lessons 6-5)
1.
2.
3.
4.
5.
1. Solve the system of inequalities by graphing. y < -3x + 2
y - 5x ≤ 3
2. Solve the system of inequalities by graphing. y < 1 −
2 x + 2
y ≥ -3x -1
3. MULTIPLE CHOICE LaShawn designs websites for local businesses. He charges $25 an hour to build a website, and charges $15 an hour to update websites once he builds them. He wants to earn at least $100 every week, but he does not want to work more than 6 hours each week. What is a possible solution to describe how many hours LaShawn can spend building a website (x) and updating a website (y) in a week?
A (1, 4) B (1, 6) C (2, 3) D (3, 3)
1.
2.
3.
x
y
y
x
SCORE
045_052_ALG1_A_CRM_C06_CR_660280.indd 48045_052_ALG1_A_CRM_C06_CR_660280.indd 48 12/22/10 12:03 AM12/22/10 12:03 AM
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
NAME DATE PERIOD
Ass
essm
ent
PDF Pass
Chapter 6 49 Glencoe Algebra 1
SCORE
Part I Write the letter for the correct answer in the blank at the right of each question.
Use the graph for Questions 1 and 2.For Questions 1 and 2, determine how many solutions exist for each system of equations.
A no solutionB one solution
C infinitely many solutionsD cannot be determined
1. x + y = 3 2. x + y = 3 x - y = 3 x + y = -2
3. The solution to which system of equations has a positive y value? F x + y = 3 G x + y = 3 H x - 3y = -2 J x + y = 3
x - 3y = -2 x + y = -2 x + y = -2 x - y = 3
4. If y = 5x - 3 and 3x - y = -1, what is the value of y? A 2 B -1 C 7 D -8
5. Use elimination to solve the system x - 5y = -6of equations for y. x + 2y = 8
F 2−
3G 2 H 4 J 4 2−
3
6. How many solutions does the system 2y = 10x - 14 and 5x - y = 7 have? A one B two C none D infinitely many
Part II
7. Graph the system of equations. Then determine whether it has no solution, one solution, or infinitely many solutions. If the system has one solution name it.
x - 3y = -3 x + 3y = 9
8. Use substitution to solve the 4x + y = 0system of equations. 2y + x = -7
For Questions 9–11, use elimination to solve each system of equations.
9. x - 4y = 11 10. 2r + 3t = 9 11. 9x + 2y = -17 5x - 7y = -10 3r + 2t = 12 -11x + 2y = 3
12. At the end of a recent WNBA regular season, the Phoenix Mercury had 12 more victories than losses. The number of victories they had was one more than two times the number of losses. How many regular season games did the Phoenix Mercury play during that season?
y
x
x + y = 3
x - 3y = -2
x - y = 3x + y = -2
y = x + 13
23
7.
8.
9.
10.
11.
12.
y
x
Chapter 6 Mid-Chapter Test(Lessons 6-1 through 6-4)
6
y
x
1.
2.
3.
4.
5.
6.
045_052_ALG1_A_CRM_C06_CR_660280.indd 49045_052_ALG1_A_CRM_C06_CR_660280.indd 49 12/22/10 12:04 AM12/22/10 12:04 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
NAME DATE PERIOD
PDF Pass
Chapter 6 50 Glencoe Algebra 1
SCORE 6 Chapter 6 Vocabulary Test
Choose from the terms above to complete each sentence.
1. Two equations, such as y = 3x - 6 and y = 12 + 4x, together are
called a(n) .
2. If the graphs of the equations of a system intersect or coincide,
the system of equations is .
3. If a system of equations has an infinite number of solutions,
the system is .
4. Sometimes adding two equations of a system will give an equation with only one variable. This is helpful when you are
solving the system by .
Define the terms in your own words.
5. substitution
6. independent
consistentdependent
eliminationinconsistent
independentsubstitution
system of equationssystem of inequalities
?
?
?
?
1.
2.
3.
4.
045_052_ALG1_A_CRM_C06_CR_660280.indd 50045_052_ALG1_A_CRM_C06_CR_660280.indd 50 12/22/10 12:04 AM12/22/10 12:04 AM
Ass
essm
ent
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
NAME DATE PERIOD
PDF Pass
Chapter 6 51 Glencoe Algebra 1
SCORE
Write the letter for the correct answer in the blank at the right of each question.
Use the graph for Questions 1–4.
1. The ordered pair (3, 2) is the solution of which system of equations?
A y = -
1 −
3 x + 3 C y = -
1 −
3 x + 3
y = -3x + 2 y = 2x -4
B y = -
1 −
3 x + 3 D y = -3x + 2
y = 2x y = 2x - 4
For Questions 2–4, find how many solutions exist for each system of equations. F no solution H one solution G infinitely many solutions J cannot be determined 2. y = 2x 3. y = 2x 4. y = -3x + 2
y = 2x - 4 y - 2x = 0 y = 2x
5. When solving the system of equations, r = 4 - twhich expression could be substituted 3r + 2t = 15for r in the second equation?
A 4 - t B 4 - r C t - 4 D 4 − t
6. If x = 2 and 3x + y = 5, what is the value of y? F 0 G -1 H 11 J 10
7. Use substitution to solve the system n = 3m -11of equations. 2m + 3n = 0
A (-2, 3) B (-3, 2) C (3, -2) D (2, -3)
8. Use elimination to solve the system x - y = 5of equations. x + y = 3
F (4, 1) G (4, -1) H (-4, 1) J (-4, -1)
9. Use elimination to solve the system x + 6y = 10of equations. x + 5y = 9
A (1, 4) B (4, 1) C (-1, -4) D (-4, -1)
10. Use elimination to find the value of x in 2x + 2y = 10the solution for the system of equations. 2x - 3y = 5
F 1 G 10 H 4 J -2
11. To eliminate the variable y in the system of equations, 6x + 4y = 22multiply the second equation by which number? 2x - y = 1
A 3 B 9 C 22 D 4
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
Chapter 6 Test, Form 1
y
xO
y - 2x = 0
y = 2x - 4
y =-3x + 2
y = 2x
y = - x + 313
6
045_052_ALG1_A_CRM_C06_CR_660280.indd 51045_052_ALG1_A_CRM_C06_CR_660280.indd 51 12/22/10 12:04 AM12/22/10 12:04 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
NAME DATE PERIOD
PDF Pass
Chapter 6 52 Glencoe Algebra 1
6 Chapter 6 Test, Form 1 (continued)
For Questions 12 and 13, determine the best method to solve the system of equations. A substitution C elimination using subtraction B elimination using addition D elimination using multiplication
12. 5x - 2y = 4 13. y = 3x + 122x + 2y = 8 2x + y = 16
14. The length of a rectangle is three times the width. The sum of the length and the width is 24 inches. What is the length of the rectangle?F 3 inches G 6 inches H 9 inches J 18 inches
15. An airport shuttle company owns sedans that have a maximum capacity of 3 passengers and vans that have a maximum capacity of 8 passengers. Their 12 vehicles have a combined maximum capacity of 61 passengers. How many vans does the company own?
A 5 B 8 C 12 D 7
For Questions 16 and 17, use the graph. 16. What is a solution to the system of inequalities?
A (0, 2) C (1, 1)
B (0, 0) D (2, 4)
17. What system of inequalities is represented? F y < x + 2 G y > x + 2 H y > x + 2 J y < x + 2 y ≤ -x + 1 y ≤ -x + 1 y ≥ -x + 1 y ≥ -x + 1
Bonus Determine the best method to solve x - 3y = 0 the system of equations. Then solve 2x - 7y = 0the system.
12. 13.
14.
15.
16.
17.
B:
x
y
045_052_ALG1_A_CRM_C06_CR_660280.indd 52045_052_ALG1_A_CRM_C06_CR_660280.indd 52 12/22/10 12:04 AM12/22/10 12:04 AM
Ass
essm
ent
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
NAME DATE PERIOD
PDF Pass
Chapter 6 53 Glencoe Algebra 1
SCORE
Write the letter for the correct answer in the blank at the right of each question.
Use the graph for Questions 1–4.For Questions 1 and 2, determine how many solutions exist for each system of equations.
A no solutionB one solutionC infinitely many solutionsD cannot be determined
1. y = 3x + 3 2. x + 2y = -13x - y = 2 2x + 3y = 0
3. The solution to which system of equations has an x value of 3? F x + 2y = -1 G 3x - y = 2 H y = 3x + 3 J 2x + 3y = 0
y = 3x + 3 x + 2y = -1 2x + 3y = 0 x + 2y = -1
4. The solution to which system of equations has a y value of 0? A x + 2y = -1 B 3x - y = 2 C y = 3x + 3 D 2x + 3y = 0
y = 3x + 3 x + 2y = -1 2x + 3y = 0 x + 2y = -1
5. When solving the system of equations, x + 2y = 15which expression could be substituted 5x + y = 21for x in the second equation?
F 15 - 2y G 21 - 5x H 15 - x −
2 J
21- 2y −
5
6. If x = 2y + 3 and 4x - 5y = 9, what is the value of y? A 2 B 1 C -1 D -2
7. Use elimination to solve the system x + 7y = 16 and 3x - 7y = 4 for x. F 3 G 4 H 5 J -6
8. Use elimination to solve the system x - 5y = 20 and x + 3y = -4 for x. A 5 B -3 C 10 D -40
9. Use elimination to solve the system 8x - 7y = 5 and 3x - 5y = 9 for y. F -2 G 8 H -3 J -1
10. Use elimination to solve the system 4x + 6y = 10 and 2x + 5y = 1 for x.
A 11 B 5 1 −
2 C -2 D -
1 −
2
11. The substitution method should be used to solve which system of equations?
F 4x + 3y = 6 G 2x + 5y = 1 H 6x + 2y = 1 J y = 3x + 15x - 3y = 2 2x - 3y = 4 3x + 4y = 7 2x + 4y = 5
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
y
xO
2x + 3y = 0
x + 2y = -1
y = 3x + 3
y = 3x - 2
3x - y = 2
6 Chapter 6 Test, Form 2A
053_060_ALG1_A_CRM_C06_AS_660280.indd 53053_060_ALG1_A_CRM_C06_AS_660280.indd 53 12/22/10 12:04 AM12/22/10 12:04 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
NAME DATE PERIOD
PDF Pass
Chapter 6 54 Glencoe Algebra 1
6 Chapter 6 Test, Form 2A (continued)
12.
13.
14.
15.
16.
17.
12. Use substitution to solve the system x + 2y = 1 and 2x + 5y = 3.A (-1, 1) B (1, -1) C (-5, 3) D (-1, -1)
For Questions 13 and 14, solve the system and find values of y. 13. 3x - 5y = -35 F 4 G 4 −
5 H -4 J -
4
−
5
2x - 5y = -30
14. 3x + 4y = -30 A -6 B 6 C -12 D 12 2x - 5y = 72
15. Coffee Cafe makes 90 pounds of coffee that costs $6 per pound. The types of coffee used to make this mixture cost $7 per pound and $4 per pound. How many pounds of the $7-per-pound coffee should be used in this mixture?
F 30 lb G 40 lb H 50 lb J 60 lb
16. Your teacher is giving a test that has 5 more four-point questions than six-point questions. The test is worth 120 points. Which system represents this information? A x + 5 = y B x + y = 5 C x - y = 5 D x - y = 5
4x + 6y = 120 6x + 4y =120 6x + 4y = 120 4x + 6y = 120
17. What system of inequalities is represented in the graph?
F y < -2x + 1 H y < -2x + 1 y ≤ 1 −
5 x - 1 y ≥ 1 −
5 x - 1
G y > -2x + 1 J y > -2x + 1 y ≤ 1 −
5 x - 1 y ≥ 1 −
5 x - 1
Bonus Where on the graph of 2x - 6y = 7 is the x-coordinate twice the y-coordinate? B:
x
y
053_060_ALG1_A_CRM_C06_AS_660280.indd 54053_060_ALG1_A_CRM_C06_AS_660280.indd 54 12/22/10 12:10 AM12/22/10 12:10 AM
Ass
essm
ent
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
NAME DATE PERIOD
PDF Pass
Chapter 6 55 Glencoe Algebra 1
SCORE
Write the letter for the correct answer in the blank at the right of each question.
Use the graph for Questions 1–4. For Questions 1 and 2, determine how many solutions exist for each system of equations.
A no solutionB one solutionC infinitely many solutionsD cannot be determined
1. 3x - 3y = -6 2. x + y = 0y = x + 2 3x - y = 2
3. The solution to which system of equations has an x value of -1?F x + y = 0 G 3x - y = 2 H x + y = 0 J y = -x - 2
3x - y = 2 y = -x - 2 3x - 3y = -6 x + y = 0
4. The solution to which system of equations has a y value of -2? A x + y = 0 B 3x - y = 2 C x + y = 0 D y = -x - 2 3x - y = 2 y = -x - 2 3x - 3y = -6 x + y = 0
5. When solving the system of equations, 3x + y = 14which expression could be substituted x + 4y = 3for y in the second equation?
F 3 - 4y G 3 - x −
4 H
14 - y −
3 J 14 - 3x
6. If x = 5y - 1 and 2x + 5y = -32, what is the value of y? A -2 B 2 C 1 D -1
7. Use elimination to solve the system 3x + 5y = 16 and 8x - 5y = 28 for x. F -6 G 5 H 4 J 4 −
5
8. Use elimination to solve the system x - 4y = 1 and x + 2y = 19 for x. A -11 B 3 C 25 D 13
9. Use elimination to solve the system 4x + 7y = -14 and 8x + 5y = 8 for x. F 3
1 −
2 G -1
1 −
2 H 8 J -4
10. Use elimination to solve the system 5x + 4y = -10 and 3x + 6y = -6 for y. A -2 B -5 C 0 D 2
11. The substitution method should be used to solve which system of equations? F 5x - 7y = 16 G 4x + 3y = -5 H x = 3y + 1 J 2x + 6y = 3 2x - 7y = 12 6x - 3y = 2 2x + y = 7 3x + 2y = -1
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
y
xO
y = x + 2
y = -x - 2
x + y = 0
3x - y = 2
3x - 3y = - 6
6 Chapter 6 Test, Form 2B
053_060_ALG1_A_CRM_C06_AS_660280.indd 55053_060_ALG1_A_CRM_C06_AS_660280.indd 55 12/22/10 12:10 AM12/22/10 12:10 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
NAME DATE PERIOD
PDF Pass
Chapter 6 56 Glencoe Algebra 1
6 Chapter 6 Test, Form 2B (continued)
12. Use substitution to solve the system x - 2y = 1 and 6x - 5y = 20.A (2, 5) B (-5, -2) C (5, 2) D (-2, -5)
For Questions 13 and 14, solve the system and find the value of y. 13. 2x + 3y = 1 5x - 4y = -32 F -8 3 −
7 G 8 3 −
7 H -3 J 3
14. 6x + 3y = 12 5x + 3y = 0
A -20 B -1 9 −
11 C 20 D 1 9 −
11
15. Colortime Bakers wants to make 30 pounds of a berry mix that costs $3 per pound to use in their pancake mix. They are using blueberries that cost $2 per pound and blackberries that cost $3.50 per pound. How many pounds of blackberries should be used in this mixture?
F 15 lb G 20 lb H 10 lb J 30 lb
16. Your teacher is giving a test that has 12 more three-point questions than five-point questions. The test is worth 100 points. Which system represents this information? A x + y = 12 B x + y = 12 C x - y = 12 D x - y = 12
3x + 5y = 100 5x + 3y = 100 3x + 5y = 100 5x + 3y = 100
17. What system of inequalities is represented in the graph?
F y ≥ - 3x G y ≥ - 3x H y ≤ - 3x J y ≤ - 3x
y < - 1 −
2 x - 2 y > - 1 −
2 x - 2 y < - 1 −
2 x - 2 y > - 1 −
2 x - 2
x
y
Bonus Manuel is 8 years older than his sister. Three years ago he was 3 times older than his sister. How old is each now?
12.
13.
14.
15.
16.
17.
B:
053_060_ALG1_A_CRM_C06_AS_660280.indd 56053_060_ALG1_A_CRM_C06_AS_660280.indd 56 12/22/10 12:10 AM12/22/10 12:10 AM
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
NAME DATE PERIOD
Ass
essm
ent
PDF Pass
Chapter 6 57 Glencoe Algebra 1
SCORE
y
xO
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
Use the graph at the right to determine whether each system has no solution, one solution, or infinitely many solutions.
1. y = -x x + y = 3
2. x - 2y = -34x + y = 6
Graph each system of equations. Then determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it.
3. y = -x + 4 4. 2x - y = -3y = x - 4 6x - 3y = -9
Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.
5. y = 3x 6. 5x - y = 10x + y = 4 7x - 2y = 11
Use elimination to solve each system of equations. 7. x + 4y = -8 8. 2x + 5y = 3
x - 4y = -8 -x + 3y = -7
9. 2x - 5y = -16 10. 2x + 5y = 9-2x + 3y = 12 2x + y = 13
Determine the best method to solve each system of equations. Then solve the system.
11. x = 2y - 1 12. 5x - y = 173x + y = 11 3x - y = 13
13. The sum of two numbers is 17 and their difference is 29. What are the two numbers?
14. Adult tickets for the school musical sold for $3.50 and student tickets sold for $2.50. Three hundred twenty-one tickets were sold altogether for $937.50. How many of each kind of ticket were sold?
15. Ayana has $2.35 in nickels and dimes. If she has 33 coins in all, find the number of nickels and dimes.
y
xO
y = -x
x + y = 0
x + y = 3
x - 2y = -3
4x + y = 6
y
xO
6 Chapter 6 Test, Form 2C
053_060_ALG1_A_CRM_C06_AS_660280.indd 57053_060_ALG1_A_CRM_C06_AS_660280.indd 57 12/22/10 12:10 AM12/22/10 12:10 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
NAME DATE PERIOD
PDF Pass
Chapter 6 58 Glencoe Algebra 1
For Questions 16 and 17, solve the system of inequalities by graphing.
16. y > x + 2
y ≤ -2x - 1
17. y ≤ 1 −
4 x + 2
y ≥ 2x
18. To qualify for a certain need-based scholarship, a student must get a score of at least 75 on a qualification test. In addition, the student’s family must make less than $40,000 a year. Define the variables, write a system of inequalities to represent this situation, and name one possible solution.
Bonus Mavis is 5 years older than her brother. Five years ago she was 2 times older than her brother. How old is each now?
16.
x
y
17.
x
y
18.
6 Chapter 6 Test, Form 2C (continued)
B:
053_060_ALG1_A_CRM_C06_AS_660280.indd 58053_060_ALG1_A_CRM_C06_AS_660280.indd 58 12/22/10 12:11 AM12/22/10 12:11 AM
Ass
essm
ent
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
NAME DATE PERIOD
PDF Pass
Chapter 6 59 Glencoe Algebra 1
SCORE
Use the graph at the right to determine whether each system has no solution, one solution, or infinitely many solutions.
1. x + y = 3y = -x + 3
2. 3x - y = -32x - 3y = -2
Graph each system of equations. Then determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it.
3. y = -x + 3 4. 2x - y = 5y = x - 3 4x - 2y = 10
Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.
5. y = 2x 6. 2x - y = 32x + y = 8 5x + 7y = 17
Use elimination to solve each system of equations.
7. 2x + 3y = 19 8. 6x + 4y = 202x - 3y = 1 4x - 2y = 4
9. 2x + 2y = 6 10. 7x + 3y = 13x - 2y = -11 9x + 3y = -3
Determine the best method to solve each system of equations. Then solve the system.
11. y = 3x + 1 12. 5x - 15y = -20x - 2y = 8 5x - 4y = -9
13. The sum of two numbers is 16 and their difference is 20. What are the two numbers?
14. Kyle just started a new job that pays $7 per hour. He had been making $5 per hour at his old job. Kyle worked a total of 54 hours last month and made $338 before deductions. How many hours did he work at his new job?
15. Brent has $3.35 in quarters and dimes. If he has 23 coins in all, find the number of quarters and dimes.
1.
2. 3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
y
x
y
x
y
xO
2x - 3y = -2x + y = 3
y = -x + 3
x + y = -1
3x - y = -3
6 Chapter 6 Test, Form 2D
053_060_ALG1_A_CRM_C06_AS_660280.indd 59053_060_ALG1_A_CRM_C06_AS_660280.indd 59 12/22/10 12:04 AM12/22/10 12:04 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
NAME DATE PERIOD
PDF Pass
Chapter 6 60 Glencoe Algebra 1
For Questions 16 and 17, solve the system of inequalities by graphing.
16. y < 1 −
3 x + 1
y ≤ 2x - 3
17. y ≤ x + 3
y > - 1 −
2 x - 2
18. To qualify for a certain car loan, a customer must have a credit score of at least 600. In addition, the cost of the car must be at least $5000. Define the variables, write a system of inequalities to represent this situation, and name one possible solution.
Bonus Find the point on the graph of 3x - 4y = 9 where the y-coordinate is 3 times the x-coordinate.
16.
x
y
17.
x
y
18.
6 Chapter 6 Test, Form 2D (continued)
B:
053_060_ALG1_A_CRM_C06_AS_660280.indd 60053_060_ALG1_A_CRM_C06_AS_660280.indd 60 12/22/10 12:11 AM12/22/10 12:11 AM
Ass
essm
ent
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
NAME DATE PERIOD
PDF 2nd
Chapter 6 61 Glencoe Algebra 1
SCORE
Graph each system of equations. Determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it.
1. 1 −
3 y = x 2. x + 3y = 3
y + x + 4 = 0 3y = -x + 9
Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.
3. y = 2x - 7 4. 4y - 3x = 5
3x - 4y = 8 3 −
4 x = y - 4
5. x - 2y = -3 6. y = -x + 3y = 3x - 1 x + y = -1
Use elimination to solve each system of equations.
7. 6x - 7y = 21 8. 0.2x + 0.5y = 0.73x + 7y = 6 -0.2x - 0.6y = -1.4
9. 2x + 2 −
3 y = -8 10. 1 −
2 x + 2 −
5 y = -10
1 −
2 x - 1 −
3 y = 1 3x + 6y = -6
Determine the best method to solve each system of equations. Then solve the system.
11. x + y = 147 12. 7y = 2 1 −
2 - 2x
25x + 10y = 2415 5x = 3y - 4
13. Three times one number added to five times a second number is 68. Three times the second number minus four times the first number is 6. What are the two numbers?
14. The difference of two numbers is 5. Five times the lesser number minus the greater number is 9. What are the two numbers?
15. A trail mix that costs $2.45 per pound is mixed with a trail mix that costs $2.30 per pound. How much of each type of trail mix must be used to have 30 pounds of a trail mix that costs $2.35 per pound?
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
y
x
6 Chapter 6 Test, Form 3
y
x
061_068_ALG1_A_CRM_C06_AS_660280.indd 61061_068_ALG1_A_CRM_C06_AS_660280.indd 61 12/24/10 7:26 AM12/24/10 7:26 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
NAME DATE PERIOD
PDF Pass
Chapter 6 62 Glencoe Algebra 1
16. A boat travels 60 miles downstream in the same time it takes to go 36 miles upstream. The speed of the boat in still water is 15 mph greater than the speed of the current. Find the speed of the current.
For Questions 17 and 18, solve the system of inequalities by graphing.
17. y < - 1 −
4 x + 2
y ≤ x - 1
18. -2x - y > 1
1 −
2 x - y ≤ 1
19. Mr. Nordmann gets a commission of $2.50 on each pair of women’s shoes he sells, and a commission of $3 on each pair of men’s shoes he sells. To meet his sales targets, he must sell at least 10 pairs of women’s shoes and at least 5 pairs of men’s shoes. He also wants to make at least $60 a week in commissions. Define the variables, write a system of inequalities to represent this situation, and name one possible solution.
Bonus Find the area of the polygon formed by the system of equations y = 0, y = 2x + 4 and -3y - 4x = -12.
16.
17.
x
y
18.
x
y
19.
B:
6 Chapter 6 Test, Form 3 (continued)
061_068_ALG1_A_CRM_C06_AS_660280.indd 62061_068_ALG1_A_CRM_C06_AS_660280.indd 62 12/22/10 12:04 AM12/22/10 12:04 AM
Ass
essm
ent
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
NAME DATE PERIOD
PDF Pass
Chapter 6 63 Glencoe Algebra 1
SCORE
Demonstrate your knowledge by giving a clear, concise solution to each problem. Be sure to include all relevant drawings and justify your answers. You may show your solution in more than one way or investigate beyond the requirements of the problem.
1. Ruth invests $10,000 in two accounts. One account has an annual interest rate of 7%, and the other account has an annual interest rate of 5%. Let I represent the total interest earned in one year. Then Ruth’s investments can be modeled by the system of equations x + y = 10,000 and 0.07x + 0.05y = I.
a. Determine a solution for this system of equations that represents a possible investment, and find the value of I that corresponds to your solution.
b. Find the solution for the system if I = 800. Explain why this solution does not represent a possible investment.
2. A car rental company wants to charge a rate of $A per day plus $B per mile to rent a compact car. Their leading competitor charges $15 per day plus $0.25 per mile to rent a compact car.
a. Explain how the system of equations y = A + Bx and y = 15 + 0.25x compares the total cost of renting a compact car for one day from this company and from their leading competitor.
b. Describe how the value of A and the value of B affects the comparison of the total cost of any one-day rental from these two companies.
3. A bookstore makes a profit of $2.50 on each book they sell, and $0.75 on each magazine they sell. Each week the store sells x books and y magazines. Let $P be the weekly profit, and $S be the weekly sales for the bookstore.
a. Write a system of equations that models the possible weekly sales and weekly profit for the book store. Then describe possible values for P and S.
b. Choose a value for P and a value for S, and substitute the values into the system of equations from part a. Make a graph of this system of equations, and describe what the graph represents.
4. A bird observatory is keeping a tally on the number of adult and baby condors nesting on their grounds. There are 21 birds in total. The number of adult condors is 3 fewer than 5 times the number of babies.
a. Write a system of linear equations to model the situation. b. How many adult and baby condors are there?
6 Chapter 6 Extended-Response Test
061_068_ALG1_A_CRM_C06_AS_660280.indd 63061_068_ALG1_A_CRM_C06_AS_660280.indd 63 12/22/10 12:04 AM12/22/10 12:04 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
NAME DATE PERIOD
PDF Pass
Chapter 6 64 Glencoe Algebra 1
SCORE 6 Standardized Test Practice(Chapters 1–6)
1. Evaluate [1 + 4(5)] + [3(9) - 7]. (Lesson 1-2)
A 45 B 27 C 41 D 31
2. Rewrite 3(4a - 2b + c) using the Distributive Property. (Lesson 1-4)
F 12a - 2b + c G 12a -2b + 3c H 12a - 6b + c J 12a - 6b + 3c
3. How many liters of a 10% saline solution must be added to 4 liters of a 40% saline solution to obtain a 15% saline solution? (Lesson 2-9)
A 20 L B 4 L C 2 L D 48 L
4. When Nick was traveling in Montreal, the currency exchange rate between the U.S. and Canada could be modeled by d = 1.02c whered represents the number of U.S. dollars and c represents the numberof Canadian dollars. Solve the equation for Canadian dollar amountsof $1, $2, $5, and $20. (Lesson 3-1)
F {(1, 1.02), (2, 2.04), (5, 5.10), (20, 20.40)} G {(1, 1), (2, 2), (5, 5), (20, 20)} H {(1, 0.98), (2, 1.96), (5, 4.9), (20, 19.61)} J {(1, 2.02), (2, 3.02), (5, 6.02), (20, 21.02)}
5. If a line passes through (0, -6) and has a slope of -3, what is the equation of the line? (Lesson 4-2)
A y = -6x - 3 B x = -6y - 3 C y = -3x - 6 D x = -3y - 6
6. If r is the slope of a line, and m is the slope of a line perpendicular to that line, what is the relationship between r and m? (Lesson 4-4)
F There is no relationship. H r = m G r = -m J r = - 1
−
m
7. Which inequality does not have the solution {t | t > 4}? (Lesson 5-2)
A -t < -4 B 3t > 12 C t −
2 > 2 D - t −
8 > -
1 −
2
8. Solve 4 - 2r ≥ 3(5 - r) + 7(r + 1). (Lesson 5-3)
F {
r | r ≤ - 3 −
2 }
G {r | r ≤ -3} H {r | r ≤ -2} J {
r | r ≤ -
9 −
4 }
9. Write a compound inequality for the graph. (Lesson 5-4)
A x < -1 and x ≥ 2 C x ≤ -1 or x > 2 B x < -1 or x ≥ 2 D x ≤ -1 and x > 2
10. Evaluate x2 + 5(y - 3) when x = -3 and y = 14. (Lesson 1-2)
F 64 G 58 H 61 J 29
Part 1: Multiple Choice
Instructions: Fill in the appropriate circle for the best answer.
-1-2-3-4 0 1 2 43
1. A B C D
2. F G H J
3. A B C D
4. F G H J
5. A B C D
6. F G H J
7. A B C D
8.
9. A B C D
10. F G H J
F G H J
061_068_ALG1_A_CRM_C06_AS_660280.indd 64061_068_ALG1_A_CRM_C06_AS_660280.indd 64 12/22/10 12:11 AM12/22/10 12:11 AM
Ass
essm
ent
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
NAME DATE PERIOD
PDF Pass
Chapter 6 65 Glencoe Algebra 1
6 Standardized Test Practice (continued)
11. Solve 8x - 5 = 19. (Lesson 2-3)
A 7 −
2 B -3 C 3 D 6
12. Solve -
4 − x = 6 −
9 . (Lesson 2-3)
F -6 G 6 H 12 J -12
13. Use the graph to determine how many solutions exist for the system -4x + 3y = 12 and x + y = 2. (Lesson 6-1)
A 0 C 2 B 1 D infinitely many
14. Use elimination to solve the system 2x + y = -8 and -2x + 3y = -8 for x. (Lesson 6-3)
F -2 G 2 H -4 J 4
15. The substitution method could be used to solve which system of equations? (Lesson 6-2)
A 5x - 7y = 8 C 3x - 3y = 5 2x + 6y = 6 -2x + 2y = -6
B 3x + 2y = 8 D x = 4y + 6 4x + 3y = 5 3x - 2y = 3
16. Which graph represents the solution of � n + 5 � > 1? (Lesson 5-5)
F H
G J
For Questions 17 and 18, determine the value that is missing. 17. The solution set 17. 18.
is {n � n ≥ 15} for the inequality n - 7 ≥ ___. (Lesson 5-1)
18. If � a - 8 � = 17, then a = ___ or a = -9. (Lesson 2-5)
-4-3-2-1 0 2 3 4 5 6 71-1 0 1 2 3 4 5 6 7 8 9 10
-9-8-7-6-5-4-3-2-1 0 21-1-2-3-5-6 -4 0 1 2 3 4 5
y
xO
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
11. A B C D
12. F G H J
13. A B C D
14. F G H J
15. A B C D
16. F G H J
Part 2: Gridded ResponsePart 2: Gridded Response
Instructions: Enter your answer by writing each digit of the answer in a column
box and then shading in the appropriate circle that corresponds to that entry.
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
061_068_ALG1_A_CRM_C06_AS_660280.indd 65061_068_ALG1_A_CRM_C06_AS_660280.indd 65 12/22/10 12:32 AM12/22/10 12:32 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
NAME DATE PERIOD
PDF Pass
Chapter 6 66 Glencoe Algebra 1
19.
20.
21.
22.
23.
24.
25.
26.
27.
28a.
28b.
Part 3: Short Response
Instructions: Write your answers in the space provided.
19. State whether the percent of change is a percent of increase or a percent of decrease. Then find the percent of change. original: 76; new: 57 (Lesson 2-7)
20. Express the relation {(-2, 1), (3, -1), (2, -2), (-2, 0)} as a mapping. (Lesson 1-6)
21. Determine whether -6, -3, 0, 3 … is an arithmetic sequence. If it is, state the common difference. (Lesson 3-4)
22. Find the slope of the line that passes through (-2, 0) and (5, -8). (Lesson 3-3)
23. The Lopez family drove 165 miles in 3 hours. Write a direct variation equation for the distance driven in any time. How far can the Lopez family drive in 5 hours? (Lesson 3-4)
24. Write an equation of a line that passes through (-2, -1) with slope 3. (Lesson 4-3)
25. Solve the system of inequalities by graphing. (Lesson 6-6) 2x - y ≥ 4 x - 2y < 4
26. Solve 12 + r < 15. Then graph the solution. (Lesson 5-1)
27. Define a variable, write a compound inequality, and solve the problem. (Lesson 5-4)
Seven less than twice a number is greater than 13 or less than or equal to -5.
28. Three times a first number minus a second number equals negative forty. The first number plus twice the second number equals negative four. (Lesson 6-5)
a. Define variables and formulate a system of linear equations from this situation.
b. What are the numbers?
y
xO
-1-2-3-4 0 1 2 43
6 Standardized Test Practice (continued)
061_068_ALG1_A_CRM_C06_AS_660280.indd 66061_068_ALG1_A_CRM_C06_AS_660280.indd 66 12/22/10 12:04 AM12/22/10 12:04 AM
Ass
essm
ent
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
NAME DATE PERIOD
PDF Pass
Chapter 6 67 Glencoe Algebra 1
SCORE
1. Graph 3x - y = 1.
2. Solve 4x + 9 = 4x + 13.
3. Find the value of r so that the line through (2, -3) and (-4, r) has a slope of - 1 −
2 .
4. A giraffe can travel 800 feet in 20 seconds. Write a direct variation equation for the distance traveled in any time.
5. Find the 25th term of the arithmetic sequence with first term 7 and common difference -2.
6. Write an equation of the line whose slope is 2 and whose y-intercept is 9.
7. Write an equation of the line that passes through (-1, -7) and (1, 3).
8. Write y - 4 = - 3 −
2 (x + 6) in standard form.
9. Write the slope-intercept form of an equation of the linethat passes through (-2, 0) and is parallel to the graph of y = -3x - 2.
10. The table below shows the distance driven during four different trips and the duration of each trip. Draw a scatter plot and determine what relationship exists, if any, in the data. Write an equation for a line of fit for the data.
Time (hours) 1 2 2.5 4
Distance (miles) 50 85 120 180
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
y
xO
6 Unit 2 Test (Chapters 3–6)
40
0
80
120
160
1 2 3 4
Dis
tan
ce (
mile
s)
Time (hours)
061_068_ALG1_A_CRM_C06_AS_660280.indd 67061_068_ALG1_A_CRM_C06_AS_660280.indd 67 12/22/10 12:12 AM12/22/10 12:12 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
NAME DATE PERIOD
PDF Pass
Chapter 6 68 Glencoe Algebra 1
11. The table below shows the cost to ride the New York City subway in various years.
Year 1985 1987 1990 1994 2000 2007
Subway Fare $0.90 $1.00 $1.15 $1.25 $1.50 $2.00
Source: Metropolitan Transportation Authority (MTA)
Use a regression line to estimate the cost of a subway ride in 2014.
Solve each inequality. 12. 4x - 5 < 7x + 10 13. 2(5a - 4) - 3(6 + 2a) ≤ 6
Solve each compound inequality. 14. 5 < 2t + 7 < 11 15. 13 < 4 - 3v or 2v - 14 > 8
For Questions 16 and 17, solve each open sentence. Then graph the solution set.
16. |3b - 5| ≤ 7 17. |w + 5| > 1
18. Use a graph to determine whether the system x - y = 4 and y = x has no solution, one solution, or infinitely many solutions.
For Questions 19–22, determine the best method to solve each system of equations. Then solve the system.
19. x + y = 2 20. -x - 5y = 7y = 2x -1 x + y = 1
21. 3x + y = 26 22. 4x - 8y = 523x + 3y = 18 7x + 4y = 1
23. Solve the system of inequalities by graphing. 2x - y > 3x + 2y ≤ 4
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23. y
xO
-3-2-1 0 1 2 4 53
y
xO
6 Unit 2 Test (continued)
-7-6-5-4-8 -3-2-1 0
061_068_ALG1_A_CRM_C06_AS_660280.indd 68061_068_ALG1_A_CRM_C06_AS_660280.indd 68 12/22/10 12:04 AM12/22/10 12:04 AM
An
swer
s
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
PDF Pass
Chapter 6 A1 Glencoe Algebra 1
Chapter Resources
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DAT
E
P
ER
IOD
Cha
pte
r 6
3 G
lenc
oe A
lgeb
ra 1
B
efor
e yo
u b
egin
Ch
ap
ter
6
•
Rea
d ea
ch s
tate
men
t.
•
Dec
ide
wh
eth
er y
ou A
gree
(A
) or
Dis
agre
e (D
) w
ith
th
e st
atem
ent.
•
Wri
te A
or
D i
n t
he
firs
t co
lum
n O
R i
f yo
u a
re n
ot s
ure
wh
eth
er y
ou a
gree
or
disa
gree
, wri
te N
S (
Not
Su
re).
Aft
er y
ou c
omp
lete
Ch
ap
ter
6
•
Rer
ead
each
sta
tem
ent
and
com
plet
e th
e la
st c
olu
mn
by
ente
rin
g an
A o
r a
D.
•
Did
an
y of
you
r op
inio
ns
abou
t th
e st
atem
ents
ch
ange
fro
m t
he
firs
t co
lum
n?
•
For
th
ose
stat
emen
ts t
hat
you
mar
k w
ith
a D
, use
a p
iece
of
pape
r to
wri
te a
n
exam
ple
of w
hy
you
dis
agre
e.
6A
ntic
ipat
ion
Gui
deS
olv
ing
Syste
ms o
f Lin
ear
Eq
uati
on
s
ST
EP
1A
, D, o
r N
SS
tate
men
tS
TE
P 2
A o
r D
1.
A s
olu
tion
of
a sy
stem
of
equ
atio
ns
is a
ny
orde
red
pair
th
at
sati
sfie
s on
e of
th
e eq
uat
ion
s
2.
A s
yste
m o
f eq
uat
ion
s of
par
alle
l li
nes
wil
l h
ave
no
solu
tion
s.
3.
A s
yste
m o
f eq
uat
ion
s of
tw
o pe
rpen
dicu
lar
lin
es w
ill
hav
e in
fin
itel
y m
any
solu
tion
s.
4. I
t is
not
pos
sibl
e to
hav
e ex
actl
y tw
o so
luti
ons
to a
sys
tem
of
lin
ear
equ
atio
ns
5.
Th
e m
ost
accu
rate
way
to
solv
e a
syst
em o
f eq
uat
ion
s is
to
grap
h t
he
equ
atio
ns
to s
ee w
her
e th
ey i
nte
rsec
t.
6.
To
solv
e a
syst
em o
f eq
uat
ion
s, s
uch
as
2x -
y =
21
and
3y =
2x
- 6
, by
subs
titu
tion
, sol
ve o
ne
of t
he
equ
atio
ns
for
one
vari
able
an
d su
bsti
tute
th
e re
sult
in
to t
he
oth
er e
quat
ion
.
7.
Wh
en s
olvi
ng
a sy
stem
of
equ
atio
ns,
a r
esu
lt t
hat
is
a tr
ue
stat
emen
t, s
uch
as
-5
= -
5, m
ean
s th
e eq
uat
ion
s do
not
sh
are
a co
mm
on s
olu
tion
.
8.
Add
ing
the
equ
atio
ns
3x -
4y
= 8
an
d 2x
+ 4
y =
7 r
esu
lts
in a
0 c
oeff
icie
nt
for
y.
9.
Th
e eq
uat
ion
7x
- 2
y =
12
can
be
mu
ltip
lied
by
2 so
th
at t
he
coef
fici
ent
of y
is
-4.
10.
Th
e re
sult
of
mu
ltip
lyin
g -
7x -
3y
= 1
1 by
-3
is
-1x
+ 9
y =
11.
Step
1
Step
2
D A D A D A D A A D
001_
009_
ALG
1_A
_CR
M_C
06_C
R_6
6028
0.in
dd
312
/21/
10
8:43
AM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DAT
E
P
ER
IOD
Lesson 6-1
Cha
pte
r 6
5 G
lenc
oe A
lgeb
ra 1
Poss
ible
Nu
mb
er o
f So
luti
on
s T
wo
or m
ore
lin
ear
equ
atio
ns
invo
lvin
g th
e sa
me
vari
able
s fo
rm a
sys
tem
of
equ
atio
ns.
A s
olu
tion
of
the
syst
em o
f eq
uat
ion
s is
an
ord
ered
pa
ir o
f n
um
bers
th
at s
atis
fies
bot
h e
quat
ion
s. T
he
tabl
e be
low
su
mm
ariz
es i
nfo
rmat
ion
ab
out
syst
ems
of l
inea
r eq
uat
ion
s.
Gra
ph
of
a S
yste
m
inte
rsec
ting
lines
sa
me
line
para
llel l
ines
N
um
ber
of
So
luti
on
s ex
actly
one
sol
utio
n in
fi nite
ly m
any
solu
tions
no
sol
utio
n
Te
rmin
olo
gy
cons
iste
nt a
nd
cons
iste
nt a
nd
inco
nsis
tent
inde
pend
ent
depe
nden
t
U
se t
he
grap
h a
t th
e ri
ght
to d
eter
min
e
wh
eth
er e
ach
sys
tem
is
con
sist
ent
or i
nco
nsi
sten
t an
d
if i
t is
in
dep
end
ent
or d
epen
den
t.
a. y
= -
x +
2
y =
x +
1S
ince
th
e gr
aph
s of
y =
-x
+ 2
an
d y
= x
+ 1
in
ters
ect,
th
ere
is o
ne
solu
tion
. Th
eref
ore,
th
e sy
stem
is
con
sist
ent
and
inde
pen
den
t.b
. y
= -
x +
2
3x +
3y
= -
3S
ince
th
e gr
aph
s of
y =
-x
+ 2
an
d 3x
+ 3
y =
-3
are
para
llel
, th
ere
are
no
solu
tion
s. T
her
efor
e, t
he
syst
em i
s in
con
sist
ent.
c. 3
x +
3y
= -
3
y =
-x
- 1
Sin
ce t
he
grap
hs
of 3
x +
3y
= -
3 an
d y
= -
x -
1 c
oin
cide
, th
ere
are
infi
nit
ely
man
y so
luti
ons.
Th
eref
ore,
th
e sy
stem
is
con
sist
ent
and
depe
nde
nt.
Exer
cise
sU
se t
he
grap
h a
t th
e ri
ght
to d
eter
min
e w
het
her
eac
h
syst
em i
s co
nsi
sten
t or
in
con
sist
ent
and
if
it i
s in
dep
end
ent
or d
epen
den
t.
1. y
= -
x -
3
2. 2
x +
2y
= -
6y
= x
- 1
y
= -
x -
3
3. y
= -
x -
3
4. 2
x +
2y
= -
62x
+ 2
y =
4
3x
+ y
= 3
6-1
Stud
y G
uide
and
Inte
rven
tion
Gra
ph
ing
Syste
ms o
f E
qu
ati
on
s
x
y
O
y =
-x
- 33x
+ y
= 3
2x +
2y
= -
6
2x +
2y
= 4
y =
x -
1
x
y O
y =
x +
1
y =
-x
- 1
3x +
3y
= -
3
y =
-x
+ 2
Exam
ple
x
y
Ox
y
Ox
y
O
con
sist
ent;
ind
epen
den
tco
nsi
sten
t; d
epen
den
t
inco
nsi
sten
tco
nsi
sten
t; in
dep
end
ent
001_
009_
ALG
1_A
_CR
M_C
06_C
R_6
6028
0.in
dd
512
/21/
10
8:43
AM
Answers (Anticipation Guide and Lesson 6-1)
A01_A21_ALG1_A_CRM_C06_AN_660280.indd A1A01_A21_ALG1_A_CRM_C06_AN_660280.indd A1 12/21/10 8:49 AM12/21/10 8:49 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
PDF Pass
Chapter 6 A2 Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DAT
E
P
ER
IOD
Cha
pte
r 6
6 G
lenc
oe A
lgeb
ra 1
Solv
e b
y G
rap
hin
g O
ne
met
hod
of
solv
ing
a sy
stem
of
equ
atio
ns
is t
o gr
aph
th
e eq
uat
ion
s on
th
e sa
me
coor
din
ate
plan
e.
G
rap
h e
ach
sys
tem
an
d d
eter
min
e th
e n
um
ber
of
solu
tion
s th
at i
t h
as. I
f it
has
on
e so
luti
on, n
ame
it.
a. x
+ y
= 2
x -
y =
4T
he
grap
hs
inte
rsec
t. T
her
efor
e, t
her
e is
on
e so
luti
on. T
he
po
int
(3, -
1) s
eem
s to
lie
on
bot
h l
ines
. Ch
eck
this
est
imat
e by
rep
laci
ng
x w
ith
3 a
nd
y w
ith
-1
in e
ach
equ
atio
n.
x
+ y
= 2
3 +
(-
1) =
2 �
x
- y
= 4
3 -
(-
1) =
3 +
1 o
r 4
�
Th
e so
luti
on i
s (3
, -1)
.b
. y
= 2
x +
1
2y =
4x
+ 2
Th
e gr
aph
s co
inci
de. T
her
efor
e th
ere
are
infi
nit
ely
man
y so
luti
ons.
Exer
cise
sG
rap
h e
ach
sys
tem
an
d d
eter
min
e th
e n
um
ber
of
solu
tion
s it
has
. If
it h
as o
ne
solu
tion
, nam
e it
.
1. y
= -
2 o
ne;
(-
1, -
2)
2. x
= 2
on
e; (
2, -
3)
3. y
= 1 −
2 x
on
e; (
2, 1
)
3
x -
y =
-1
2
x +
y =
1
x +
y =
3
4. 2
x +
y =
6 o
ne;
(1,
4)
5. 3
x +
2y
= 6
no
so
luti
on
6.
2y
= -
4x +
4 i
nfi
nit
ely
2x -
y =
-2
3x
+ 2
y =
-4
y =
-2x
+ 2
m
any
6-1
Stud
y G
uide
and
Inte
rven
tion
(co
nti
nu
ed)
Gra
ph
ing
Syste
ms o
f E
qu
ati
on
s
x
y O
y =
2x
+ 1
2y =
4x
+ 2x
y
O( 3
, –1) x -
y =
4
x +
y =
2
Exam
ple
x
y
O
3x -
y =
-1
( –1,
–2)
y =
-2
x
y
O
2x +
y =
6
2x -
y =
-2
( 1, 4
)
x
y
O
x =
2
2x +
y =
1( 2
, –3)
x
y
O
3x +
2y
= 6
3x +
2y
= -
4
x
y
O
x +
y =
3
y =
1 2x( 2
, 1)
x
y
O
2y =
-4x
+ 4
y =
-2x
+ 2
001_
009_
ALG
1_A
_CR
M_C
06_C
R_6
6028
0.in
dd
612
/21/
10
8:43
AM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DAT
E
P
ER
IOD
Lesson 6-1
Cha
pte
r 6
7 G
lenc
oe A
lgeb
ra 1
Use
th
e gr
aph
at
the
righ
t to
det
erm
ine
wh
eth
er
ea
ch s
yste
m i
s co
nsi
sten
t or
in
con
sist
ent
and
if
it i
s
ind
epen
den
t or
dep
end
ent.
1. y
= x
- 1
co
nsi
sten
t;
2. x
- y
= -
4 co
nsi
sten
t;
y =
-x
+ 1
in
dep
end
ent
y =
x +
4 d
epen
den
t
3. y
= x
+ 4
4.
y =
2x
- 3
2x -
2y
= 2
2
x -
2y
= 2
i
nco
nsi
sten
t c
on
sist
ent;
in
dep
end
ent
Gra
ph
eac
h s
yste
m a
nd
det
erm
ine
the
nu
mb
er o
f so
luti
ons
that
it
has
. If
it
has
on
e so
luti
on, n
ame
it.
5. 2
x -
y =
1
6. x
= 1
7.
3x
+ y
= -
3 n
o
y =
-3
on
e; (
-1,
-3)
2
x +
y =
4 o
ne;
(1,
2)
3x
+ y
= 3
so
luti
on
x
y
O
x =
1 2x +
y =
4
( 1, 2
)
8. y
= x
+ 2
in
fi n
itel
y
9. x
+ 3
y =
-3
on
e;
10. y
- x
= -
1x
- y
= -
2 m
any
x -
3y
= -
3 (-
3, 0
) x
+ y
= 3
on
e; (
2, 1
)
x
y
y =
x +
2
x -
y =
-2
11
. x -
y =
3
12. x
+ 2
y =
4 i
nfi
nit
ely
13
. y =
2x
+ 3
no
so
luti
on
x
- 2
y =
3 o
ne;
(3,
0)
y =
- 1 −
2 x
+ 2
man
y 3
y =
6x
- 6
x
y
O
y =
2x
+ 3
3y =
6x
- 6
6-1
Skill
s Pr
acti
ceG
rap
hin
g S
yste
ms o
f E
qu
ati
on
s
x
y
y =
-x
+ 1
x -
y =
-4
2x -
2y
= 2
y =
2x
- 3
y =
x -
1
y =
x +
4
x
y
O
2x -
y =
1
y =
-3
( -1,
-3)
x
y
O
3x +
y =
3
3x +
y =
-3
x
y
O
x -
3y
= -
3
x +
3y
= -
3
( -3,
0)
x
y
x +
y =
3
( 2, 1
)
y -
x =
-1
x
y
x -
2y
= 3
x -
y =
3
( 3, 0
)
x
y
O
y =
- 1 – 2
x +
2
x +
2y
= 4
001_
009_
ALG
1_A
_CR
M_C
06_C
R_6
6028
0.in
dd
712
/21/
10
8:43
AM
Answers (Lesson 6-1)
A01_A21_ALG1_A_CRM_C06_AN_660280.indd A2A01_A21_ALG1_A_CRM_C06_AN_660280.indd A2 12/21/10 8:49 AM12/21/10 8:49 AM
An
swer
s
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
PDF 2nd
Chapter 6 A3 Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DAT
E
P
ER
IOD
Cha
pte
r 6
8 G
lenc
oe A
lgeb
ra 1
Use
th
e gr
aph
at
the
righ
t to
det
erm
ine
wh
eth
er
ea
ch s
yste
m i
s co
nsi
sten
t or
in
con
sist
ent
and
if
it
is i
nd
epen
den
t or
dep
end
ent.
1. x
+ y
= 3
2.
2x
- y
= -
3x
+ y
= -
3 4
x -
2y
= -
6in
con
sist
ent
co
nsi
sten
t an
d
3. x
+ 3
y =
3
4. x
+ 3
y =
3x
+ y
= -
3 2
x -
y =
-3
con
sist
ent
and
c
on
sist
ent
and
ind
epen
den
t
Gra
ph
eac
h s
yste
m a
nd
det
erm
ine
the
nu
mb
er o
f so
luti
ons
that
it
has
. If
it h
as
one
solu
tion
, nam
e it
.
5. 3
x -
y =
-2
no
6.
y =
2x
- 3
in
fi n
itel
y
7. x
+ 2
y =
3
on
e;
3x -
y =
0
solu
tio
n
4x
= 2
y +
6
man
y 3
x -
y =
-5
(-1,
2)
x
y
O
3x -
y =
-2
3x -
y =
0
8. B
USI
NES
S N
ick
plan
s to
sta
rt a
hom
e-ba
sed
busi
nes
s pr
odu
cin
g an
d se
llin
g go
urm
et d
og t
reat
s. H
e fi
gure
s it
w
ill
cost
$20
in
ope
rati
ng
cost
s pe
r w
eek
plu
s $0
.50
to
prod
uce
eac
h t
reat
. He
plan
s to
sel
l ea
ch t
reat
for
$1.
50.
a.
Gra
ph t
he
syst
em o
f eq
uat
ion
s y
= 0
.5x
+ 2
0 an
d y
= 1
.5x
to r
epre
sen
t th
e si
tuat
ion
.
b.
How
man
y tr
eats
doe
s N
ick
nee
d to
sel
l pe
r w
eek
to
brea
k ev
en?
20
9. S
ALE
S A
use
d bo
ok s
tore
als
o st
arte
d se
llin
g u
sed
CD
s an
d vi
deos
. In
th
e fi
rst
wee
k, t
he
stor
e so
ld 4
0 u
sed
CD
s an
d vi
deos
, at
$4.0
0 pe
r C
D a
nd
$6.0
0 pe
r vi
deo.
Th
e sa
les
for
both
CD
s an
d vi
deos
tot
aled
$18
0.00
a.
Wri
te a
sys
tem
of
equ
atio
ns
to
re
pres
ent
the
situ
atio
n.
b.
Gra
ph t
he
syst
em o
f eq
uat
ion
s.
c.
How
man
y C
Ds
and
vide
os d
id t
he
stor
e se
ll i
n t
he
firs
t w
eek?
6-1
Prac
tice
Gra
ph
ing
Syste
ms o
f E
qu
ati
on
s
x
y
x +
y =
-3
x +
3y
= 3
2x -
y =
-3
4x -
2y
= -
6
x +
y =
3
x
y
3x -
y =
-5
x +
2y
= 3
( –1,
2)
x
y
O
y =
2x
- 3
4x =
2y
+ 6
Sale
s ($
)
Do
g T
reats
Cost ($)
515
1020
2530
3540
450
40 35 30 25 20 15 10 5
y =
0.5
x +
20
y =
1.5
x
( 20,
30)
CD
Sal
es (
$)
CD
an
d V
ideo
Sale
s
Video Sales ($)
515
1020
2530
3540
450
40 35 30 25 20 15 10 5
4c +
6v
= 1
80( 3
0, 1
0)
c +
v =
40
c +
v =
40,
4c +
6v =
180
30 C
Ds
and
10
vid
eos
ind
epen
den
t
dep
end
ent
001_
009_
ALG
1_A
_CR
M_C
06_C
R_6
6028
0.in
dd
812
/21/
10
8:43
AM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DAT
E
P
ER
IOD
Lesson 6-1
Cha
pte
r 6
9 G
lenc
oe A
lgeb
ra 1
1. B
USI
NES
S T
he
wid
get
fact
ory
wil
l se
ll a
tot
al o
f y
wid
gets
aft
er x
day
s ac
cord
ing
to t
he
equ
atio
n y
= 2
00x
+ 3
00.
Th
e ga
dget
fac
tory
wil
l se
ll y
gad
gets
af
ter
x da
ys a
ccor
din
g to
th
e eq
uat
ion
y
= 2
00x
+ 1
00. L
ook
at t
he
grap
h o
f th
e sy
stem
of
equ
atio
ns
and
dete
rmin
e w
het
her
it
has
no
solu
tion
, on
e so
luti
on,
or i
nfi
nit
ely
man
y so
luti
ons.
n
o s
olu
tio
n
2. A
RC
HIT
ECTU
RE
An
off
ice
buil
din
g h
as
two
elev
ator
s. O
ne
elev
ator
sta
rts
out
on
the
4th
flo
or, 3
5 fe
et a
bove
th
e gr
oun
d,
and
is d
esce
ndi
ng
at a
rat
e of
2.2
fee
t pe
r se
con
d. T
he
oth
er e
leva
tor
star
ts o
ut
at
grou
nd
leve
l an
d is
ris
ing
at a
rat
e of
1.
7 fe
et p
er s
econ
d. W
rite
a s
yste
m o
f eq
uat
ion
s to
rep
rese
nt
the
situ
atio
n.
Sam
ple
an
swer
: y =
35
- 2
.2x;
y =
1.7
x
3. F
ITN
ESS
Oli
via
and
her
bro
ther
Wil
liam
h
ad a
bic
ycle
rac
e. O
livi
a ro
de a
t a
spee
d of
20
feet
per
sec
ond
wh
ile
Wil
liam
rod
e at
a s
peed
of
15 f
eet
per
seco
nd.
To
be
fair
, Oli
via
deci
ded
to g
ive
Wil
liam
a
150-
foot
hea
d st
art.
Th
e ra
ce e
nde
d in
a
tie.
How
far
aw
ay w
as t
he
fin
ish
lin
e fr
om w
her
e O
livi
a st
arte
d?
600
ft
4. A
VIA
TIO
N T
wo
plan
es a
re i
n f
ligh
t n
ear
a lo
cal
airp
ort.
On
e pl
ane
is a
t an
al
titu
de o
f 10
00 m
eter
s an
d is
asc
endi
ng
at a
rat
e of
400
met
ers
per
min
ute
. Th
e se
con
d pl
ane
is a
t an
alt
itu
de o
f 59
00 m
eter
s an
d is
des
cen
din
g at
a r
ate
of 3
00 m
eter
s pe
r m
inu
te.
a. W
rite
a s
yste
m o
f eq
uat
ion
s th
at
repr
esen
ts t
he
prog
ress
of
each
pla
ne
y =
40
0x +
10
00;
y =
590
0 -
30
0x
b.
Mak
e a
grap
h t
hat
rep
rese
nts
th
e pr
ogre
ss o
f ea
ch p
lan
e.
6-1
Wor
d Pr
oble
m P
ract
ice
Gra
ph
ing
Syste
ms o
f E
qu
ati
on
s
Day
s2
10
43
y
x5
6
Items sold
300
400
200
100
500
800
900
700
600
Gad
get
s
Wid
get
sy
= 2
00x
+ 3
00
y =
200
x +
100
Tim
e (m
in.)
0
y
x
Altitude (m)
Firs
t Pla
ne
Seco
nd P
lane
21
43
56
78
910
2000
1000
4000
5000
6000
3000
001_
009_
ALG
1_A
_CR
M_C
06_C
R_6
6028
0.in
dd
912
/24/
10
11:4
1 A
M
Answers (Lesson 6-1)
A01_A21_ALG1_A_CRM_C06_AN_660280.indd A3A01_A21_ALG1_A_CRM_C06_AN_660280.indd A3 12/24/10 11:45 AM12/24/10 11:45 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
PDF 2nd
Chapter 6 A4 Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DAT
E
P
ER
IOD
Cha
pte
r 6
10
G
lenc
oe A
lgeb
ra 1
Enri
chm
ent
Gra
ph
ing
a T
rip
Th
e di
stan
ce f
orm
ula
, d =
rt,
is u
sed
to s
olve
man
y ty
pes
of
t
d O
slop
e is
50
slop
e is
–25
2
50
prob
lem
s. I
f yo
u g
raph
an
equ
atio
n s
uch
as
d =
50t
, th
e gr
aph
is
a m
odel
for
a c
ar g
oin
g at
50
mph
. Th
e ti
me
the
car
trav
els
is t
; th
e di
stan
ce i
n m
iles
th
e ca
r co
vers
is
d. T
he
slop
e of
th
e li
ne
is
the
spee
d.S
upp
ose
you
dri
ve t
o a
nea
rby
tow
n a
nd
retu
rn. Y
ou a
vera
ge
50 m
ph o
n t
he
trip
ou
t bu
t on
ly 2
5 m
ph o
n t
he
trip
hom
e. T
he
rou
nd
trip
tak
es 5
hou
rs. H
ow f
ar a
way
is
the
tow
n?
Th
e gr
aph
at
the
righ
t re
pres
ents
you
r tr
ip. N
otic
e th
at t
he
retu
rn
trip
is
show
n w
ith
a n
egat
ive
slop
e be
cau
se y
ou a
re d
rivi
ng
in t
he
oppo
site
dir
ecti
on.
Sol
ve e
ach
pro
ble
m.
1. E
stim
ate
the
answ
er t
o th
e pr
oble
m i
n t
he
abov
e ex
ampl
e. A
bou
t h
ow f
ar a
way
is
the
tow
n?
a
bo
ut
80 m
iles
Gra
ph
eac
h t
rip
an
d s
olve
th
e p
rob
lem
.
2. A
n a
irpl
ane
has
en
ough
fu
el f
or 3
hou
rs o
f sa
fe f
lyin
g. O
n t
he
trip
ou
t th
e pi
lot
aver
ages
20
0 m
ph f
lyin
g ag
ain
st a
hea
dwin
d. O
n t
he
trip
bac
k, t
he
pilo
t av
erag
es 2
50 m
ph. H
ow
lon
g a
trip
ou
t ca
n t
he
pilo
t m
ake?
a
bo
ut
1 2
−
3 h
ou
rs a
nd
330 m
iles
3. Y
ou d
rive
to
a to
wn
100
mil
es a
way
. 4.
You
dri
ve a
t an
ave
rage
spe
ed o
f 50
mph
O
n t
he
trip
ou
t yo
u a
vera
ge 2
5 m
ph.
to
a di
scou
nt
shop
pin
g pl
aza,
spe
nd
On
th
e tr
ip b
ack
you
ave
rage
50
mph
. 2
hou
rs s
hop
pin
g, a
nd
then
ret
urn
at
an
How
man
y h
ours
do
you
spe
nd
driv
ing?
a
vera
ge s
peed
of
25 m
ph. T
he
enti
re t
rip
t
akes
8 h
ours
. How
far
aw
ay i
s th
e
sh
oppi
ng
plaz
a?
6
ho
urs
1
00 m
iles
t
d
O2
50
t
d
O50
2
6-1
t
d
O1
100
010_
018_
ALG
1_A
_CR
M_C
06_C
R_6
6028
0.in
dd
1012
/21/
10
8:43
AM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DAT
E
P
ER
IOD
Lesson 6-1
Cha
pte
r 6
11
Gle
ncoe
Alg
ebra
1
Gra
phin
g Ca
lcul
ator
Act
ivit
yS
olu
tio
n t
o a
Syste
m o
f Lin
ear
Eq
uati
on
s
A g
raph
ing
calc
ula
tor
can
be
use
d to
sol
ve a
sys
tem
of
lin
ear
equ
atio
ns
grap
hic
ally
. Th
e so
luti
on o
f a
syst
em o
f li
nea
r eq
uat
ion
s ca
n b
e fo
un
d by
usi
ng
the
TR
AC
E f
eatu
re o
r by
u
sin
g th
e in
ters
ect
com
man
d u
nde
r th
e C
AL
C m
enu
.
Exer
cise
sS
olve
eac
h s
yste
m o
f li
nea
r eq
uat
ion
s.
1.
y =
2
2. y
= -
x +
3
3. x
+ y
= -
1
5x
+ 4
y =
18
y =
x +
1
2x
- y
= -
8
(2
, 2)
(1, 2)
(-
3, 2)
4. -
3x +
y =
10
5. -
4x +
3y
= 1
0 6.
5x
+ 3
y =
11
-
x +
2y
= 0
7
x +
y =
20
x -
5y
= 5
(-
4,
-2)
(2, 6)
(2.5
, -
0.5
)
7. 3
x -
2y
= -
4 8.
3x
+ 2
y =
4
9. 4
x -
5y
= 0
-
4x +
3y
= 5
-
6x -
4y
= -
8 6
x -
5y
= 1
0
(-
2,
-1)
in
fi n
ite s
olu
tio
ns
(5, 4)
S
olve
eac
h s
yste
m o
f li
nea
r eq
uat
ion
s.
a. x
+ y
= 0
x
- y
= -
4
Usi
ng T
RA
CE:
Sol
ve e
ach
equ
atio
n f
or y
an
d en
ter
each
equ
atio
n
into
Y=.
Th
en g
raph
usi
ng
Zoo
m 8
: ZIn
tege
r. U
se T
RA
CE
to
fin
d th
e so
luti
on.
Key
stro
kes:
Y=
(–)
EN
TE
R
+
4
ZO
OM
6
ZO
OM
8 E
NT
ER
T
RA
CE
.
Th
e so
luti
on i
s (-
2, 2
).
b.
2x +
y =
4
4x +
3y
= 3
Usi
ng
CA
LC:
Sol
ve e
ach
equ
atio
n f
or y
, en
ter
each
in
to t
he
calc
ula
tor,
and
grap
h. U
se C
AL
C t
o de
term
ine
the
solu
tion
.
Key
stro
kes:
Y=
(–) 2
+ 4
EN
TE
R
( (
–)
4 ÷
3
)
+
1
ZO
OM
6
2nd
[C
AL
C]
5 E
NT
ER
E
NT
ER
E
NT
ER
.
To
chan
ge t
he
x-va
lue
to a
fra
ctio
n, p
ress
2nd
[Q
UIT
]
MA
TH
EN
TE
R E
NT
ER
.
Th
e so
luti
on i
s (4
.5, -
5) o
r ( 9 −
2 , -
5 ) .
[-47, 47]
scl:
10 b
y [
-31, 31]
scl:
10
[-10, 10]
scl:
1 b
y [
-10, 10]
scl:
1
6-1
Exam
ple
010_
018_
ALG
1_A
_CR
M_C
06_C
R_6
6028
0.in
dd
1112
/24/
10
7:07
AM
Answers (Lesson 6-1)
A01_A21_ALG1_A_CRM_C06_AN_660280.indd A4A01_A21_ALG1_A_CRM_C06_AN_660280.indd A4 12/24/10 7:13 AM12/24/10 7:13 AM
An
swer
s
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
PDF Pass
Chapter 6 A5 Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DAT
E
P
ER
IOD
Cha
pte
r 6
12
Gle
ncoe
Alg
ebra
1
Stud
y G
uide
and
Inte
rven
tion
Su
bsti
tuti
on
Solv
e b
y Su
bst
itu
tio
n O
ne
met
hod
of
solv
ing
syst
ems
of e
quat
ion
s is
su
bst
itu
tion
.
U
se s
ub
stit
uti
on t
o so
lve
the
syst
em o
f eq
uat
ion
s.y
= 2
x4x
- y
= -
4
Su
bsti
tute
2x
for
y in
th
e se
con
d eq
uat
ion
.
4x -
y =
-4
Sec
ond
equa
tion
4x
- 2
x =
-4
y =
2x
2x
= -
4 C
ombi
ne li
ke te
rms.
x
= -
2 Di
vide
eac
h si
de b
y 2
and
sim
plify
.
Use
y =
2x
to f
ind
the
valu
e of
y.
y =
2x
Firs
t eq
uatio
n
y =
2(-
2)
x =
-2
y =
-4
Sim
plify
.
Th
e so
luti
on i
s (-
2, -
4).
S
olve
for
on
e va
riab
le,
then
su
bst
itu
te.
x +
3y
= 7
2x -
4y
= -
6
Sol
ve t
he
firs
t eq
uat
ion
for
x s
ince
th
e co
effi
cien
t of
x i
s 1.
x
+ 3
y =
7
F
irst
equa
tion
x +
3y
- 3
y =
7 -
3y
Sub
trac
t 3y
fro
m e
ach
side
.
x
= 7
- 3
y S
impl
ify.
Fin
d th
e va
lue
of y
by
subs
titu
tin
g 7
- 3
y fo
r x
in t
he
seco
nd
equ
atio
n.
2x
- 4
y =
-6
Sec
ond
equa
tion
2(7
- 3
y) -
4y
= -
6 x
= 7
- 3
y
14
- 6
y -
4y
= -
6 D
istr
ibut
ive
Pro
pert
y
14
- 1
0y =
-6
Com
bine
like
term
s.
14 -
10y
- 1
4 =
-6
- 1
4 S
ubtr
act
14 f
rom
eac
h si
de.
-10
y =
-20
S
impl
ify.
y =
2
Divi
de e
ach
side
by
-10
and
sim
plify
.
Use
y =
2 t
o fi
nd
the
valu
e of
x.
x =
7 -
3y
x =
7 -
3(2
)x
= 1
Th
e so
luti
on i
s (1
, 2).
Exer
cise
sU
se s
ub
stit
uti
on t
o so
lve
each
sys
tem
of
equ
atio
ns.
1. y
= 4
x 2.
x =
2y
3. x
= 2
y -
33x
- y
= 1
(-
1, -
4)
y =
x -
2 (
4, 2
) x
= 2
y +
4 n
o s
olu
tio
n
4. x
- 2
y =
-1
5. x
- 4
y =
1
infi
nit
ely
6.
x +
2y
= 0
3y =
x +
4 (
5, 3
) 2
x -
8y
= 2
man
y 3
x +
4y
= 4
(4,
-2)
7. 2
b =
6a
- 1
4 in
fi n
itel
y
8. x
+ y
= 1
6 9.
y =
-x
+ 3
n
o3a
- b
= 7
m
any
2y
= -
2x +
2 n
o s
olu
tio
n
2y
+ 2
x =
4 s
olu
tio
n
10. x
= 2
y (2
0, 1
0)
11. x
- 2
y =
-5
12. -
0.2x
+ y
= 0
.50.
25x
+ 0
.5y
= 1
0
x +
2y
= -
1 (-
3, 1
) 0
.4x
+ y
= 1
.1 (
1, 0
.7)
6-2
Exam
ple
1Ex
amp
le 2
010_
018_
ALG
1_A
_CR
M_C
06_C
R_6
6028
0.in
dd
1212
/21/
10
8:43
AM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DAT
E
P
ER
IOD
Lesson 6-2
Cha
pte
r 6
13
Gle
ncoe
Alg
ebra
1
Stud
y G
uide
and
Inte
rven
tion
(co
nti
nu
ed)
Su
bsti
tuti
on
Solv
e R
eal-
Wo
rld
Pro
ble
ms
Su
bsti
tuti
on c
an a
lso
be u
sed
to s
olve
rea
l-w
orld
pr
oble
ms
invo
lvin
g sy
stem
s of
equ
atio
ns.
It
may
be
hel
pfu
l to
use
tab
les,
ch
arts
, dia
gram
s,
or g
raph
s to
hel
p yo
u o
rgan
ize
data
.
C
HEM
ISTR
Y H
ow m
uch
of
a 10
% s
alin
e so
luti
on s
hou
ld b
e m
ixed
w
ith
a 2
0% s
alin
e so
luti
on t
o ob
tain
100
0 m
illi
lite
rs o
f a
12%
sal
ine
solu
tion
?L
et s
= t
he
nu
mbe
r of
mil
lili
ters
of
10%
sal
ine
solu
tion
.L
et t
= t
he
nu
mbe
r of
mil
lili
ters
of
20%
sal
ine
solu
tion
.U
se a
tab
le t
o or
gan
ize
the
info
rmat
ion
.
10%
sal
ine
20%
sal
ine
12%
sal
ine
Tota
l mill
ilite
rss
t10
00
Mill
ilite
rs o
f sa
line
0.10
s0.
20 t
0.12
(100
0)
Wri
te a
sys
tem
of
equ
atio
ns.
s +
t =
100
00.
10s
+ 0
.20t
= 0
.12(
1000
)U
se s
ubs
titu
tion
to
solv
e th
is s
yste
m.
s
+ t
= 1
000
Firs
t eq
uatio
n
s
= 1
000
- t
S
olve
for
s.
0.
10s
+ 0
.20t
= 0
.12(
1000
) S
econ
d eq
uatio
n
0.10
(100
0 -
t)
+ 0
.20t
= 0
.12(
1000
) s
= 1
000
- t
10
0 -
0.1
0t +
0.2
0t =
0.1
2(10
00)
Dis
trib
utiv
e P
rope
rty
10
0 +
0.1
0t =
0.1
2(10
00)
Com
bine
like
term
s.
0.
10t
= 2
0 S
impl
ify.
0.
10t
−
0.10
=
20
−
0.10
D
ivid
e ea
ch s
ide
by 0
.10.
t
= 2
00
Sim
plify
.
s
+ t
= 1
000
Firs
t eq
uatio
n
s +
200
= 1
000
t =
200
s
= 8
00
Sol
ve fo
r s.
800
mil
lili
ters
of
10%
sol
uti
on a
nd
200
mil
lili
ters
of
20%
sol
uti
on s
hou
ld b
e u
sed.
Exer
cise
s 1
. SPO
RTS
At
the
end
of t
he
2007
–200
8 fo
otba
ll s
easo
n, 3
8 S
upe
r B
owl
gam
es h
ad b
een
pl
ayed
wit
h t
he
curr
ent
two
foot
ball
lea
gues
, th
e A
mer
ican
Foo
tbal
l C
onfe
ren
ce (
AF
C)
and
the
Nat
ion
al F
ootb
all
Con
fere
nce
(N
FC
). T
he
NF
C w
on t
wo
mor
e ga
mes
th
an t
he
AF
C. H
ow m
any
gam
es d
id e
ach
con
fere
nce
win
? A
FC
18;
NF
C 2
0
2. C
HEM
ISTR
Y A
lab
nee
ds t
o m
ake
100
gall
ons
of a
n 1
8% a
cid
solu
tion
by
mix
ing
a 12
%
acid
sol
uti
on w
ith
a 2
0% s
olu
tion
. How
man
y ga
llon
s of
eac
h s
olu
tion
are
nee
ded?
3. G
EOM
ETRY
Th
e pe
rim
eter
of
a tr
ian
gle
is 2
4 in
ches
. Th
e lo
nge
st s
ide
is 4
in
ches
lon
ger
than
th
e sh
orte
st s
ide,
an
d th
e sh
orte
st s
ide
is t
hre
e-fo
urt
hs
the
len
gth
of
the
mid
dle
side
. Fin
d th
e le
ngt
h o
f ea
ch s
ide
of t
he
tria
ngl
e. 6
in.,
8 in
., 10
in.
6-2
Exam
ple
25 g
al o
f 12
% s
olu
tio
n a
nd
75
gal
of
20%
so
luti
on
010_
018_
ALG
1_A
_CR
M_C
06_C
R_6
6028
0.in
dd
1312
/21/
10
8:43
AM
Answers (Lesson 6-2)
A01_A21_ALG1_A_CRM_C06_AN_660280.indd A5A01_A21_ALG1_A_CRM_C06_AN_660280.indd A5 12/21/10 9:29 AM12/21/10 9:29 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
PDF Pass
Chapter 6 A6 Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DAT
E
P
ER
IOD
Cha
pte
r 6
14
Gle
ncoe
Alg
ebra
1
Use
su
bst
itu
tion
to
solv
e ea
ch s
yste
m o
f eq
uat
ion
s.
1. y
= 4
x 2.
y =
2x
x +
y =
5 (
1, 4
) x
+ 3
y =
-14
(-
2, -
4)
3. y
= 3
x 4.
x =
-4y
2x +
y =
15
(3, 9
) 3
x +
2y
= 2
0 (8
, -2)
5. y
= x
- 1
6.
x =
y -
7x
+ y
= 3
(2,
1)
x +
8y
= 2
(-
6, 1
)
7. y
= 4
x -
1
8. y
= 3
x +
8y
= 2
x -
5 (
-2,
-9)
5
x +
2y
= 5
(-
1, 5
)
9. 2
x -
3y
= 2
1 10
. y =
5x
- 8
y =
3 -
x (
6, -
3)
4x
+ 3
y =
33
(3, 7
)
11. x
+ 2
y =
13
12. x
+ 5
y =
43x
- 5
y =
6 (
7, 3
) 3
x +
15y
= -
1 n
o s
olu
tio
n
13. 3
x -
y =
4
14. x
+ 4
y =
82x
- 3
y =
-9
(3, 5
) 2
x -
5y
= 2
9 (1
2, -
1)
15. x
- 5
y =
10
16. 5
x -
2y
= 1
42x
- 1
0y =
20
infi
nit
ely
man
y 2
x -
y =
5 (
4, 3
)
17. 2
x +
5y
= 3
8 18
. x -
4y
= 2
7x
- 3
y =
-3
(9, 4
) 3
x +
y =
-23
(-
5, -
8)
19. 2
x +
2y
= 7
20
. 2.5
x +
y =
-2
x -
2y
= -
1 (2
, 3 −
2 )
3x
+ 2
y =
0 (
-2,
3)
6-2
Skill
s Pr
acti
ceS
ub
sti
tuti
on
010_
018_
ALG
1_A
_CR
M_C
06_C
R_6
6028
0.in
dd
1412
/21/
10
8:43
AM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DAT
E
P
ER
IOD
Lesson 6-2
Cha
pte
r 6
15
Gle
ncoe
Alg
ebra
1
Use
su
bst
itu
tion
to
solv
e ea
ch s
yste
m o
f eq
uat
ion
s.
1. y
= 6
x 2.
x =
3y
3. x
= 2
y +
72x
+ 3
y =
-20
(-
1, -
6)
3x
- 5
y =
12
(9, 3
) x
= y
+ 4
(1,
-3)
4. y
= 2
x -
2
5. y
= 2
x +
6
6. 3
x +
y =
12
y =
x +
2 (
4, 6
) 2
x -
y =
2 n
o s
olu
tio
n
y =
-x
- 2
(7,
-9)
7. x
+ 2
y =
13
(-3,
8)
8. x
- 2
y =
3 i
nfi
nit
ely
9. x
- 5
y =
36
(-4,
-8)
- 2
x -
3y
= -
18
4x
- 8
y =
12
man
y 2
x +
y =
-16
10. 2
x -
3y
= -
24
11. x
+ 1
4y =
84
12. 0
.3x
- 0
.2y
= 0
.5x
+ 6
y =
18
(-6,
4)
2x
- 7
y =
-7
(14,
5)
x -
2y
= -
5 (5
, 5)
13. 0
.5x
+ 4
y =
-1
14. 3
x -
2y
= 1
1 15
. 1 −
2 x
+ 2
y =
12
x
+ 2
.5y
= 3
.5 (
6, -
1)
x -
1 −
2 y
= 4
(5,
2)
x -
2y
= 6
(12
, 3)
16. 1
−
3 x -
y =
3
17. 4
x -
5y
= -
7 18
. x +
3y
= -
4
2
x +
y =
25
y =
5x
2x
+ 6
y =
5
(
12, 1
) (
1 −
3 ,
1 2 −
3 )
no
so
luti
on
19. E
MPL
OY
MEN
T K
enis
ha
sell
s at
hle
tic
shoe
s pa
rt-t
ime
at a
dep
artm
ent
stor
e. S
he
can
ea
rn e
ith
er $
500
per
mon
th p
lus
a 4%
com
mis
sion
on
her
tot
al s
ales
, or
$400
per
mon
th
plu
s a
5% c
omm
issi
on o
n t
otal
sal
es.
a. W
rite
a s
yste
m o
f eq
uat
ion
s to
rep
rese
nt
the
situ
atio
n.
y =
0.0
4x +
50
0 an
d y
= 0
.05x
+ 4
00
b.
Wh
at i
s th
e to
tal
pric
e of
th
e at
hle
tic
shoe
s K
enis
ha
nee
ds t
o se
ll t
o ea
rn t
he
sam
e in
com
e fr
om e
ach
pay
sca
le?
$10,
00
0
c. W
hic
h i
s th
e be
tter
off
er?
the
fi rs
t o
ffer
if s
he
exp
ects
to
sel
l les
s th
an
$10,
00
0 in
sh
oes
, an
d t
he
seco
nd
off
er if
sh
e ex
pec
ts t
o s
ell m
ore
th
an
$10,
00
0 in
sh
oes
20. M
OV
IE T
ICK
ETS
Tic
kets
to
a m
ovie
cos
t $7
.25
for
adu
lts
and
$5.5
0 fo
r st
ude
nts
. A
grou
p of
fri
ends
pu
rch
ased
8 t
icke
ts f
or $
52.7
5.
a.
Wri
te a
sys
tem
of
equ
atio
ns
to r
epre
sen
t th
e si
tuat
ion
.
b.
How
man
y ad
ult
tic
kets
an
d st
ude
nt
tick
ets
wer
e pu
rch
ased
? 5
adu
lt a
nd
3 s
tud
ent
6-2
Prac
tice
Su
bsti
tuti
on
x +
y =
8 a
nd
7.2
5x +
5.5
y =
52.
75
010_
018_
ALG
1_A
_CR
M_C
06_C
R_6
6028
0.in
dd
1512
/21/
10
8:43
AM
Answers (Lesson 6-2)
A01_A21_ALG1_A_CRM_C06_AN_660280.indd A6A01_A21_ALG1_A_CRM_C06_AN_660280.indd A6 12/21/10 8:49 AM12/21/10 8:49 AM
An
swer
s
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
PDF Pass
Chapter 6 A7 Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DAT
E
P
ER
IOD
Cha
pte
r 6
16
Gle
ncoe
Alg
ebra
1
1. B
USI
NES
S M
r. R
ando
lph
fin
ds t
hat
th
e su
pply
an
d de
man
d fo
r ga
soli
ne
at h
is
stat
ion
are
gen
eral
ly g
iven
by
the
foll
owin
g eq
uat
ion
s.x
-y
=-
2x
+y
= 1
0
Use
su
bsti
tuti
on t
o fi
nd
the
equ
ilib
riu
m
poin
t w
her
e th
e su
pply
an
d de
man
d li
nes
in
ters
ect.
2. G
EOM
ETRY
Th
e m
easu
res
of
com
plem
enta
ry a
ngl
es h
ave
a su
m o
f 90
deg
rees
. An
gle
A a
nd
angl
e B
are
co
mpl
emen
tary
, an
d th
eir
mea
sure
s h
ave
a di
ffer
ence
of
20°.
Wh
at a
re t
he
mea
sure
s of
th
e an
gles
?
AB
3. M
ON
EY H
arve
y h
as s
ome
$1 b
ills
an
d so
me
$5 b
ills
. In
all
, he
has
6 b
ills
wor
th
$22.
Let
x b
e th
e n
um
ber
of $
1 bi
lls
and
let
y be
th
e n
um
ber
of $
5 bi
lls.
Wri
te a
sy
stem
of
equ
atio
ns
to r
epre
sen
t th
e in
form
atio
n a
nd
use
su
bsti
tuti
on t
o de
term
ine
how
man
y bi
lls
of e
ach
de
nom
inat
ion
Har
vey
has
.
x +
y =
6, x
+ 5
y =
22;
he
has
fo
ur
$5 b
ills
and
tw
o $
1 b
ills.
4. P
OPU
LATI
ON
San
jay
is r
esea
rch
ing
popu
lati
on t
ren
ds i
n S
outh
Am
eric
a. H
e fo
un
d th
at t
he
popu
lati
on o
f E
cuad
or t
o in
crea
sed
by 1
,000
,000
an
d th
e po
pula
tion
of
Ch
ile
to i
ncr
ease
d by
60
0,00
0 fr
om 2
004
to 2
009.
Th
e ta
ble
disp
lays
th
e in
form
atio
n h
e fo
un
d.
Sour
ce:W
orld
Alm
anac
If t
he
popu
lati
on g
row
th f
or e
ach
cou
ntr
y co
nti
nu
es a
t th
e sa
me
rate
, in
wh
at y
ear
are
the
popu
lati
ons
of E
cuad
or a
nd
Ch
ile
pred
icte
d to
be
equ
al?
5. C
HEM
ISTR
Y S
hel
by a
nd
Cal
vin
are
do
ing
a ch
emis
try
expe
rim
ent.
Th
ey n
eed
5 ou
nce
s of
a s
olu
tion
th
at i
s 65
% a
cid
and
35%
dis
till
ed w
ater
. Th
ere
is n
o u
ndi
lute
d ac
id i
n t
he
chem
istr
y la
b, b
ut
they
do
hav
e tw
o fl
asks
of
dilu
ted
acid
: F
lask
A c
onta
ins
70%
aci
d an
d 30
%
dist
ille
d w
ater
. Fla
sk B
con
tain
s 20
%
acid
an
d 80
% d
isti
lled
wat
er.
a. W
rite
a s
yste
m o
f eq
uat
ion
s th
at
Sh
elby
an
d C
alvi
n c
ould
use
to
dete
rmin
e h
ow m
any
oun
ces
they
n
eed
to p
our
from
eac
h f
lask
to
mak
e th
eir
solu
tion
.S
amp
le a
nsw
er:
a +
b =
5;
0.7a
+ 0
.2b
= 0
.65(
5)
b.
Sol
ve y
our
syst
em o
f eq
uat
ion
s. H
ow
man
y ou
nce
s fr
om e
ach
fla
sk d
o S
hel
by a
nd
Cal
vin
nee
d? 4
.5 o
z fr
om
Fla
sk A
an
d 0
.5 o
z fr
om
F
lask
B.
Wor
d Pr
oble
m P
ract
ice
Su
bsti
tuti
on
6-2
Co
un
try
2004
P
op
ula
tio
n
5-Ye
ar
Po
pu
lati
on
C
han
ge
Ecu
ador
13,0
00,0
00+
1,00
0,00
0
Chi
le16
,000
,000
+60
0,00
0
(4, 6
)
35°
and
55°
2041
010_
018_
ALG
1_A
_CR
M_C
06_C
R_6
6028
0.in
dd
1612
/21/
10
8:43
AM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DAT
E
P
ER
IOD
Lesson 6-2
Cha
pte
r 6
17
Gle
ncoe
Alg
ebra
1
Enri
chm
ent
Inte
rsecti
on
of Tw
o P
ara
bo
las
Su
bsti
tuti
on c
an b
e u
sed
to f
ind
the
inte
rsec
tion
of
two
para
bola
s. R
epla
ce t
he
y-va
lue
in o
ne
of t
he
equ
atio
ns
wit
h t
he
y-va
lue
in t
erm
s of
x f
rom
th
e ot
her
equ
atio
n.
F
ind
th
e in
ters
ecti
on o
f th
e tw
o p
arab
olas
.
y =
x2 +
5x
+ 6
y
= x
2 + 4
x +
3
Gra
ph
th
e eq
uat
ion
s.
37.5
y
x-
1010
-5
5
12.5
25.0
50.0
Fro
m t
he
grap
h, n
otic
e th
at t
he
two
grap
hs
inte
rsec
t in
on
e po
int.
Use
su
bsti
tuti
on t
o so
lve
for
the
poin
t of
in
ters
ecti
on.
x2 +
5x
+ 6
= x
2 + 4
x +
3
5x +
6 =
4x
+ 3
S
ubtr
act
x2 fr
om e
ach
side
.
x
+ 6
= 3
S
ubtr
act
4x f
rom
eac
h si
de.
x
= -
3 S
ubtr
act
6 fr
om e
ach
side
.
Th
e gr
aph
s in
ters
ect
at x
= -
3.
Rep
lace
x w
ith
-3
in e
ith
er e
quat
ion
to
fin
d th
e y-
valu
e.
y =
x2 +
5x
+ 6
O
rigin
al e
quat
ion
y
= (-
3)2 +
5(-
3) +
6
Rep
lace
x w
ith -
3.
y
= 9
– 1
5 +
6 o
r 0
Sim
plify
.
Th
e po
int
of i
nte
rsec
tion
is
(-3,
0).
Exer
cise
sU
se s
ub
stit
uti
on t
o fi
nd
th
e p
oin
t of
in
ters
ecti
on o
f th
e gr
aph
s of
eac
h p
air
of e
qu
atio
ns.
1. y
= x
2 +
8x
+ 7
2.
y =
x2 +
6x
+ 8
3.
y =
x2 +
5x
+ 6
y =
x2 +
2x
+ 1
y
= x
2 + 4
x +
4
y =
x2 +
7x
+ 6
(-1,
0)
(-
2, 0
) (
0, 6
)
6-2
Exam
ple
010_
018_
ALG
1_A
_CR
M_C
06_C
R_6
6028
0.in
dd
1712
/21/
10
8:43
AM
Answers (Lesson 6-2)
A01_A21_ALG1_A_CRM_C06_AN_660280.indd A7A01_A21_ALG1_A_CRM_C06_AN_660280.indd A7 12/21/10 8:49 AM12/21/10 8:49 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
PDF Pass
Chapter 6 A8 Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DAT
E
P
ER
IOD
Cha
pte
r 6
18
Gle
ncoe
Alg
ebra
1
Stud
y G
uide
and
Inte
rven
tion
Elim
inati
on
Usin
g A
dd
itio
n a
nd
Su
btr
acti
on
Elim
inat
ion
Usi
ng
Ad
dit
ion
In
sys
tem
s of
equ
atio
ns
in w
hic
h t
he
coef
fici
ents
of
the
x or
y t
erm
s ar
e ad
diti
ve i
nve
rses
, sol
ve t
he
syst
em b
y ad
din
g th
e eq
uat
ion
s. B
ecau
se o
ne
of
the
vari
able
s is
eli
min
ated
, th
is m
eth
od i
s ca
lled
eli
min
atio
n.
U
se e
lim
inat
ion
to
solv
e th
e sy
stem
of
equ
atio
ns.
x -
3y
= 7
3x +
3y
= 9
Wri
te t
he
equ
atio
ns
in c
olu
mn
for
m a
nd
add
to e
lim
inat
e y.
x
- 3
y =
7
(+)
3x +
3y
=
9
4x
= 1
6S
olve
for
x.
4x
−
4 =
16
−
4
x
= 4
Su
bsti
tute
4 f
or x
in
eit
her
equ
atio
n a
nd
solv
e fo
r y.
4
- 3
y =
7 4
- 3
y -
4 =
7 -
4
-3y
= 3
-
3y
−
-3
=
3 −
-
3
y =
-1
Th
e so
luti
on i
s (4
, -1)
.
T
he
sum
of
two
nu
mb
ers
is 7
0 an
d t
hei
r d
iffe
ren
ce i
s 24
. Fin
d
the
nu
mb
ers.
Let
x r
epre
sen
t on
e n
um
ber
and
y re
pres
ent
the
oth
er n
um
ber.
x
+ y
= 7
0 (+
) x
- y
= 2
4
2x
= 9
4
2x
−
2 =
94
−
2
x
= 4
7S
ubs
titu
te 4
7 fo
r x
in e
ith
er e
quat
ion
.
47 +
y =
70
47 +
y -
47
= 7
0 -
47
y
= 2
3T
he
nu
mbe
rs a
re 4
7 an
d 23
.
Exer
cise
sU
se e
lim
inat
ion
to
solv
e ea
ch s
yste
m o
f eq
uat
ion
s.
1. x
+ y
= -
4 2.
2x
- 3
y =
14
3. 3
x -
y =
-9
x -
y =
2 (
-1,
-3)
x
+ 3
y =
-11
(1,
-4)
-
3x -
2y
= 0
(-
2, 3
)
4. -
3x -
4y
= -
1 5.
3x
+ y
= 4
6.
-2x
+ 2
y =
93x
- y
= -
4 (-
1, 1
) 2
x -
y =
6 (
2, -
2)
2x
- y
= -
6 (-
3 −
2 ,
3 )
7. 2
x +
2y
= -
2 8.
4x
- 2
y =
-1
9. x
- y
= 2
3x -
2y
= 1
2 (2
, -3)
-
4x +
4y
= -
2 (-
1, -
3 −
2 )
x
+ y
= -
3 (-
1 −
2 ,
- 5 −
2 )
10. 2
x -
3y
= 1
2 11
. -0.
2x +
y =
0.5
12
. 0.1
x +
0.3
y =
0.9
4
x +
3y
= 2
4 (6
, 0)
0.2
x +
2y
= 1
.6 (
1, 0
.7)
0.1
x -
0.3
y =
0.2
13. R
ema
is o
lder
th
an K
en. T
he
diff
eren
ce o
f th
eir
ages
is
12 a
nd
the
sum
of
thei
r ag
es i
s 50
. Fin
d th
e ag
e of
eac
h.
Rem
a is
31
and
Ken
is 1
9.
14. T
he
sum
of
the
digi
ts o
f a
two-
digi
t n
um
ber
is 1
2. T
he
diff
eren
ce o
f th
e di
gits
is
2. F
ind
the
nu
mbe
r if
th
e u
nit
s di
git
is l
arge
r th
an t
he
ten
s di
git.
57
6-3
Exam
ple
1Ex
amp
le 2
( 11
−
2 , 7 −
6 )
010_
018_
ALG
1_A
_CR
M_C
06_C
R_6
6028
0.in
dd
1812
/21/
10
8:43
AM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DAT
E
P
ER
IOD
Lesson 6-3
Cha
pte
r 6
19
Gle
ncoe
Alg
ebra
1
Stud
y G
uide
and
Inte
rven
tion
(co
nti
nu
ed)
Elim
inati
on
Usin
g A
dd
itio
n a
nd
Su
btr
acti
on
Elim
inat
ion
Usi
ng
Su
btr
acti
on
In
sys
tem
s of
equ
atio
ns
wh
ere
the
coef
fici
ents
of
the
x or
y t
erm
s ar
e th
e sa
me,
sol
ve t
he
syst
em b
y su
btra
ctin
g th
e eq
uat
ion
s.
U
se e
lim
inat
ion
to
solv
e th
e sy
stem
of
equ
atio
ns.
2x -
3y
= 1
15x
- 3
y =
14
2x
- 3
y =
11
Writ
e th
e eq
uatio
ns in
col
umn
form
and
sub
trac
t.
(-)
5x -
3y
= 1
4
-3x
=
-3
S
ubtr
act
the
two
equa
tions
. y is
elim
inat
ed.
-
3x
−
-3
= -
3 −
-
3 D
ivid
e ea
ch s
ide
by -
3.
x
= 1
S
impl
ify.
2(
1) -
3y
= 1
1 S
ubst
itute
1 fo
r x
in e
ither
equ
atio
n.
2
- 3
y =
11
Sim
plify
.
2 -
3y
- 2
= 1
1 -
2
Sub
trac
t 2
from
eac
h si
de.
-
3y =
9
Sim
plify
.
-
3y
−
-3
=
9 −
-
3 D
ivid
e ea
ch s
ide
by -
3.
y
= -
3 S
impl
ify.
Th
e so
luti
on i
s (1
, -3)
.
Exer
cise
sU
se e
lim
inat
ion
to
solv
e ea
ch s
yste
m o
f eq
uat
ion
s.
1. 6
x +
5y
= 4
2.
3m
- 4
n =
-14
3.
3a
+ b
= 1
6x -
7y
= -
20
3m
+ 2
n =
-2
a +
b =
3
(-
1, 2
) (
-2,
2)
(-
1, 4
)
4. -
3x -
4y
= -
23
5. x
- 3
y =
11
6. x
- 2
y =
6-
3x +
y =
2
2x
- 3
y =
16
x +
y =
3
(1,
5)
(5,
-2)
(
4, -
1)
7. 2
a -
3b
= -
13
8. 4
x +
2y
= 6
9.
5x
- y
= 6
2a +
2b
= 7
4
x +
4y
= 1
0 5
x +
2y
= 3
(
- 1 −
2 ,
4 )
( 1 −
2 ,
2 )
(1,
-1)
10. 6
x -
3y
= 1
2 11
. x +
2y
= 3
.5
12. 0
.2x
+ y
= 0
.74x
- 3
y =
24
x -
3y
= -
9 0
.2x
+ 2
y =
1.2
(
-6,
-16
) (
-1.
5, 2
.5)
(1,
0.5
)
13. T
he
sum
of
two
nu
mbe
rs i
s 70
. On
e n
um
ber
is t
en m
ore
than
tw
ice
the
oth
er n
um
ber.
Fin
d th
e n
um
bers
. 50
; 20
14. G
EOM
ETRY
Tw
o an
gles
are
su
pple
men
tary
. Th
e m
easu
re o
f on
e an
gle
is 1
0° m
ore
than
th
ree
tim
es t
he
oth
er. F
ind
the
mea
sure
of
each
an
gle.
42.
5°;
137.
5°
6-3
Exam
ple
019_
027_
ALG
1_A
_CR
M_C
06_C
R_6
6028
0.in
dd
1912
/21/
10
8:43
AM
Answers (Lesson 6-3)
A01_A21_ALG1_A_CRM_C06_AN_660280.indd A8A01_A21_ALG1_A_CRM_C06_AN_660280.indd A8 12/21/10 8:49 AM12/21/10 8:49 AM
An
swer
s
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
PDF Pass
Chapter 6 A9 Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DAT
E
P
ER
IOD
Cha
pte
r 6
20
Gle
ncoe
Alg
ebra
1
Skill
s Pr
acti
ceE
lim
inati
on
Usin
g A
dd
itio
n a
nd
Su
btr
acti
on
Use
eli
min
atio
n t
o so
lve
each
sys
tem
of
equ
atio
ns.
1. x
- y
= 1
2.
-x
+ y
= 1
x +
y =
3 (
2, 1
) x
+ y
= 1
1 (5
, 6)
3. x
+ 4
y =
11
4. -
x +
3y
= 6
x -
6y
= 1
1 (1
1, 0
) x
+ 3
y =
18
(6, 4
)
5. 3
x +
4y
= 1
9 6.
x +
4y
= -
83x
+ 6
y =
33
(-3,
7)
x -
4y
= -
8 (-
8, 0
)
7. 3
x +
4y
= 2
8.
3x
- y
= -
14x
- 4
y =
12
(2, -
1)
-3x
- y
= 5
(-
1, -
2)
9. 2
x -
3y
= 9
10
. x -
y =
4-
5x -
3y
= 3
0 (-
3, -
5)
2x
+ y
= -
4 (0
, -4)
11. 3
x -
y =
26
12. 5
x -
y =
-6
-2x
- y
= -
24 (
10, 4
) -
x +
y =
2 (
-1,
1)
13. 6
x -
2y
= 3
2 14
. 3x
+ 2
y =
-19
4x -
2y
= 1
8 (7
, 5)
-3x
- 5
y =
25
(-5,
-2)
15. 7
x +
4y
= 2
16
. 2x
- 5
y =
-28
7x +
2y
= 8
(2,
-3)
4
x +
5y
= 4
(-
4, 4
)
17. T
he
sum
of
two
nu
mbe
rs i
s 28
an
d th
eir
diff
eren
ce i
s 4.
Wh
at a
re t
he
nu
mbe
rs?
12, 1
6
18. F
ind
the
two
nu
mbe
rs w
hos
e su
m i
s 29
an
d w
hos
e di
ffer
ence
is
15.
7, 2
2
19. T
he
sum
of
two
nu
mbe
rs i
s 24
an
d th
eir
diff
eren
ce i
s 2.
Wh
at a
re t
he
nu
mbe
rs?
13, 1
1
20. F
ind
the
two
nu
mbe
rs w
hos
e su
m i
s 54
an
d w
hos
e di
ffer
ence
is
4. 2
5, 2
9
21. T
wo
tim
es a
nu
mbe
r ad
ded
to a
not
her
nu
mbe
r is
25.
Th
ree
tim
es t
he
firs
t n
um
ber
min
us
the
oth
er n
um
ber
is 2
0. F
ind
the
nu
mbe
rs.
9, 7
6-3
019_
027_
ALG
1_A
_CR
M_C
06_C
R_6
6028
0.in
dd
2012
/21/
10
8:43
AM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DAT
E
P
ER
IOD
Lesson 6-3
Cha
pte
r 6
21
Gle
ncoe
Alg
ebra
1
Prac
tice
E
lim
inati
on
Usin
g A
dd
itio
n a
nd
Su
btr
acti
on
Use
eli
min
atio
n t
o so
lve
each
sys
tem
of
equ
atio
ns.
1. x
- y
= 1
2.
p +
q =
-2
3. 4
x +
y =
23
x +
y =
-9
p -
q =
8
3x
- y
= 1
2
(
-4,
-5)
(
3, -
5)
(5,
3)
4. 2
x +
5y
= -
3 5.
3x
+ 2
y =
-1
6. 5
x +
3y
= 2
22x
+ 2
y =
6
4x
+ 2
y =
-6
5x
- 2
y =
2
(
6, -
3)
(-
5, 7
) (
2, 4
)
7. 5
x +
2y
= 7
8.
3x
- 9
y =
-12
9.
-4c
- 2
d =
-2
-2x
+ 2
y =
-14
3
x -
15y
= -
6 2
c -
2d
= -
14
(
3, -
4)
(-
7, -
1)
(-
2, 5
)
10. 2
x -
6y
= 6
11
. 7x
+ 2
y =
2
12. 4
.25x
- 1
.28y
= -
9.2
2x +
3y
= 2
4 7
x -
2y
= -
30
x +
1.2
8y =
17.
6
(
9, 2
) (
-2,
8)
(1.
6, 1
2.5)
13. 2
x +
4y
= 1
0 14
. 2.5
x +
y =
10.
7 15
. 6m
- 8
n =
3x
- 4
y =
-2.
5 2
.5x
+ 2
y =
12.
9 2
m -
8n
= -
3
(
2.5,
1.2
5)
(3.
4, 2
.2)
(1
1 −
2 ,
3 −
4 )
16. 4
a +
b =
2
17. -
1 −
3 x
- 4 −
3 y
= -
2 18
. 3 −
4 x
- 1 −
2 y
= 8
4
a +
3b
= 1
0
1 −
3 x
- 2 −
3 y
= 4
3 −
2 x
+ 1 −
2 y
= 1
9
(-
1 −
2 ,
4 )
(
10, -
1)
(12
, 2)
19. T
he
sum
of
two
nu
mbe
rs i
s 41
an
d th
eir
diff
eren
ce i
s 5.
Wh
at a
re t
he
nu
mbe
rs?
18, 2
3
20. F
our
tim
es o
ne
nu
mbe
r ad
ded
to a
not
her
nu
mbe
r is
36.
Th
ree
tim
es t
he
firs
t n
um
ber
min
us
the
oth
er n
um
ber
is 2
0. F
ind
the
nu
mbe
rs.
8, 4
21. O
ne
nu
mbe
r ad
ded
to t
hre
e ti
mes
an
oth
er n
um
ber
is 2
4. F
ive
tim
es t
he
firs
t n
um
ber
adde
d to
th
ree
tim
es t
he
oth
er n
um
ber
is 3
6. F
ind
the
nu
mbe
rs.
3, 7
22. L
AN
GU
AG
ES E
ngl
ish
is
spok
en a
s th
e fi
rst
or p
rim
ary
lan
guag
e in
78
mor
e co
un
trie
s th
an F
arsi
is
spok
en a
s th
e fi
rst
lan
guag
e. T
oget
her
, En
glis
h a
nd
Fars
i ar
e sp
oken
as
a fi
rst
lan
guag
e in
130
cou
ntr
ies.
In
how
man
y co
un
trie
s is
En
glis
h s
poke
n a
s th
e fi
rst
lan
guag
e? I
n h
ow m
any
cou
ntr
ies
is F
arsi
spo
ken
as
the
firs
t la
ngu
age?
E
ng
lish
: 10
4 co
un
trie
s, F
arsi
: 26
co
un
trie
s
23. D
ISC
OU
NTS
At
a sa
le o
n w
inte
r cl
oth
ing,
Cod
y bo
ugh
t tw
o pa
irs
of g
love
s an
d fo
ur
hat
s fo
r $4
3.00
. Tor
i bo
ugh
t tw
o pa
irs
of g
love
s an
d tw
o h
ats
for
$30.
00. W
hat
wer
e th
e pr
ices
fo
r th
e gl
oves
an
d h
ats?
6-3
glo
ves:
$8.
50, h
ats:
$6.
50.
019_
027_
ALG
1_A
_CR
M_C
06_C
R_6
6028
0.in
dd
2112
/21/
10
8:43
AM
Answers (Lesson 6-3)
A01_A21_ALG1_A_CRM_C06_AN_660280.indd A9A01_A21_ALG1_A_CRM_C06_AN_660280.indd A9 12/21/10 8:49 AM12/21/10 8:49 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
PDF Pass
Chapter 6 A10 Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DAT
E
P
ER
IOD
Cha
pte
r 6
22
Gle
ncoe
Alg
ebra
1
Wor
d Pr
oble
m P
ract
ice
Elim
inati
on
Usin
g A
dd
itio
n a
nd
Su
btr
acti
on
1. N
UM
BER
FU
N M
s. S
imm
s, t
he
sixt
h
grad
e m
ath
tea
cher
, gav
e h
er s
tude
nts
th
is c
hal
len
ge p
robl
em.
Tw
ice
a n
um
ber
adde
d to
an
oth
er n
um
ber
is 1
5. T
he
sum
of
the
two
nu
mbe
rs i
s 11
.L
oren
zo, a
n a
lgeb
ra s
tude
nt
wh
o w
as
Ms.
Sim
ms
aide
, rea
lize
d h
e co
uld
sol
ve
the
prob
lem
by
wri
tin
g th
e fo
llow
ing
syst
em o
f eq
uat
ion
s. 2
x+
y=
15
x+
y=
11
Use
th
e el
imin
atio
n m
eth
od t
o so
lve
the
syst
em a
nd
fin
d th
e tw
o n
um
bers
.
2. G
OV
ERN
MEN
T T
he
Tex
as S
tate
L
egis
latu
re i
s co
mpr
ised
of
stat
e se
nat
ors
and
stat
e re
pres
enta
tive
s. T
he
sum
of
the
nu
mbe
r of
sen
ator
s an
d re
pres
enta
tive
s is
181
. Th
ere
are
119
mor
e re
pres
enta
tive
s th
an s
enat
ors.
How
m
any
sen
ator
s an
d h
ow m
any
repr
esen
tati
ves
mak
e u
p th
e T
exas
Sta
te
Leg
isla
ture
?
3. R
ESEA
RC
H M
elis
sa w
onde
red
how
m
uch
it
wou
ld c
ost
to s
end
a le
tter
by
mai
l in
199
0, s
o sh
e as
ked
her
fat
her
. R
ath
er t
han
an
swer
dir
ectl
y, M
elis
sa’s
fa
ther
gav
e h
er t
he
foll
owin
g in
form
atio
n. I
t w
ould
hav
e co
st $
3.70
to
sen
d 13
pos
tcar
ds a
nd
7 le
tter
s, a
nd
it
wou
ld h
ave
cost
$2.
65 t
o se
nd
6 po
stca
rds
and
7 le
tter
s. U
se a
sys
tem
of
equ
atio
ns
and
elim
inat
ion
to
fin
d h
ow
mu
ch i
t co
st t
o se
nd
a le
tter
in
199
0.
4. S
POR
TS A
s of
201
0, t
he
New
Yor
k Ya
nke
es h
ad w
on m
ore
Wor
ld S
erie
s C
ham
pion
ship
s th
an a
ny
oth
er t
eam
. In
fa
ct, t
he
Yan
kees
had
won
3 f
ewer
th
an
3 ti
mes
th
e n
um
ber
of W
orld
Ser
ies
cham
pion
ship
s w
on b
y th
e se
con
d m
ost-
win
nin
g te
am, t
he
St.
Lou
is C
ardi
nal
s.
Th
e su
m o
f th
e tw
o te
ams’
Wor
ld S
erie
s ch
ampi
onsh
ips
is 3
7. H
ow m
any
tim
es
has
eac
h t
eam
won
th
e W
orld
Ser
ies?
T
he
Car
din
als
hav
e 10
win
s an
d
the
Yan
kees
hav
e 27
win
s.
5. B
ASK
ETB
ALL
In
200
5, t
he
aver
age
tick
et p
rice
s fo
r D
alla
s M
aver
icks
gam
es
and
Bos
ton
Cel
tics
gam
es a
re s
how
n i
n
the
tabl
e be
low
. Th
e ch
ange
in
pri
ce i
s fr
om t
he
2004
sea
son
to
the
2005
sea
son
.
Sour
ce: T
eam
Mar
ketin
g Re
port
a. A
ssu
me
that
tic
kets
con
tin
ue
to
chan
ge a
t th
e sa
me
rate
eac
h y
ear
afte
r 20
05. L
et x
be
the
nu
mbe
r of
ye
ars
afte
r 20
05, a
nd
y be
th
e pr
ice
of
an a
vera
ge t
icke
t. W
rite
a s
yste
m o
f eq
uat
ion
s to
rep
rese
nt
the
info
rmat
ion
in
th
e ta
ble.
y =
0.5
3x +
53.
60y =
-1.
08x +
55.
93
b.
In h
ow m
any
year
s w
ill
the
aver
age
tick
et p
rice
for
Dal
las
appr
oxim
atel
y eq
ual
th
at o
f B
osto
n?
x =
1.4
5; 1
or
2 yr
6-3
Team
Ave
rag
e T
icke
t P
rice
Ch
ang
e in
P
rice
Dal
las
$53.
60$0
.53
Bos
ton
$55.
93-
$1.0
8
(4, 7
)
31
sta
te s
enat
ors
an
d
150
stat
e re
pre
sen
tati
ves
$0.2
5
019_
027_
ALG
1_A
_CR
M_C
06_C
R_6
6028
0.in
dd
2212
/21/
10
8:43
AM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DAT
E
P
ER
IOD
Lesson 6-3
Cha
pte
r 6
23
Gle
ncoe
Alg
ebra
1
Sys
tem
s of
equ
atio
ns
can
in
volv
e m
ore
than
2 e
quat
ion
s an
d 2
vari
able
s. I
t is
pos
sibl
e to
so
lve
a sy
stem
of
3 eq
uat
ion
s an
d 3
vari
able
s u
sin
g el
imin
atio
n.
S
olve
th
e fo
llow
ing
syst
em.
x
+ y
+ z
= 6
3x
- y
+ z
= 8
x
- z
= 2
Ste
p 1
: U
se e
lim
inat
ion
to
get
rid
of t
he
y in
th
e fi
rst
two
equ
atio
ns.
x
+ y
+ z
= 6
3x
– y
+ z
= 8
4x
+ 2
z =
14
Ste
p 2
: U
se t
he
equ
atio
n y
ou f
oun
d in
ste
p 1
and
the
thir
d eq
uat
ion
to
elim
inat
e th
e z.
4x
+ 2
z =
14
x
– z
= 2
M
ult
iply
th
e se
con
d eq
uat
ion
by
2 so
th
at t
he
z’s
wil
l el
imin
ate.
4x
+ 2
z =
14
2x
– 2
z =
4
6x =
18
S
o, x
= 3
.
Ste
p 3
: R
epla
ce x
wit
h 3
in
th
e th
ird
orig
inal
equ
atio
n t
o de
term
ine
z.
3 –
z =
2, s
o z
= 1
.
Ste
p 4
: R
epla
ce x
wit
h 3
an
d z
wit
h 1
in
eit
her
of
the
firs
t tw
o or
igin
al
equ
atio
n t
o de
term
ine
the
valu
e of
y.
3
+ y
+ 1
= 6
or
4 +
y =
6.
S
o, y
= 2
.S
o, t
he
solu
tion
to
the
syst
em o
f eq
uat
ion
s is
(3,
2, 1
).
Exer
cise
sS
olve
eac
h s
yste
m o
f eq
uat
ion
s.
1. 3
x +
2y
+ z
= 4
2 2.
x
+ y
+ z
= -
3 3.
x
+ y
+ z
= 7
2
y +
z +
12
= 3
x
2x +
3y
+ 5
z =
-4
x
+ 2
y +
z =
10
x –
3y =
0
2y
– z
= 4
2 y +
z =
5
x =
9
x =
-5
x =
5
y =
3
y =
2
y =
3
z =
9
z =
0
z =
-1
Enri
chm
ent
So
lvin
g S
yste
ms o
f E
qu
ati
on
s i
n T
hre
e V
ari
ab
les
6-3
Exam
ple
019_
027_
ALG
1_A
_CR
M_C
06_C
R_6
6028
0.in
dd
2312
/21/
10
8:43
AM
Answers (Lesson 6-3)
A01_A21_ALG1_A_CRM_C06_AN_660280.indd A10A01_A21_ALG1_A_CRM_C06_AN_660280.indd A10 12/21/10 8:50 AM12/21/10 8:50 AM
An
swer
s
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
PDF Pass
Chapter 6 A11 Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DAT
E
P
ER
IOD
Cha
pte
r 6
24
Gle
ncoe
Alg
ebra
1
Stud
y G
uide
and
Inte
rven
tion
Elim
inati
on
Usin
g M
ult
iplicati
on
Elim
inat
ion
Usi
ng
Mu
ltip
licat
ion
Som
e sy
stem
s of
equ
atio
ns
can
not
be
solv
ed
sim
ply
by a
ddin
g or
su
btra
ctin
g th
e eq
uat
ion
s. I
n s
uch
cas
es, o
ne
or b
oth
equ
atio
ns
mu
st
firs
t be
mu
ltip
lied
by
a n
um
ber
befo
re t
he
syst
em c
an b
e so
lved
by
elim
inat
ion
.
U
se e
lim
inat
ion
to
solv
e th
e sy
stem
of
equ
atio
ns.
x +
10y
= 3
4x +
5y
= 5
If y
ou m
ult
iply
th
e se
con
d eq
uat
ion
by
-2,
yo
u c
an e
lim
inat
e th
e y
term
s.
x +
10y
= 3
(+
) -
8x -
10y
= -
10
-7x
=
-7
-
7x
−
-7
= -
7 −
-
7
x
= 1
Su
bsti
tute
1 f
or x
in
eit
her
equ
atio
n.
1
+ 1
0y =
3 1
+ 1
0y -
1 =
3 -
1
10y
= 2
10
y −
10
=
2 −
10
y
= 1 −
5
Th
e so
luti
on i
s (1
, 1 −
5 ) .
U
se e
lim
inat
ion
to
solv
e th
e sy
stem
of
equ
atio
ns.
3x -
2y
= -
72x
- 5
y =
10
If y
ou m
ult
iply
th
e fi
rst
equ
atio
n b
y 2
and
the
seco
nd
equ
atio
n b
y -
3, y
ou c
an
elim
inat
e th
e x
term
s.
6x -
4y
= -
14(+
) -
6x +
15y
=
-30
11
y =
-44
11
y −
11
=
- 44
−
11
y
= -
4S
ubs
titu
te -
4 fo
r y
in e
ith
er e
quat
ion
. 3
x -
2(-
4) =
-7
3x
+ 8
= -
7
3x +
8 -
8 =
-7
-8
3x
= -
15
3x
−
3
= -
15
−
3
x
= -
5T
he
solu
tion
is
(-5,
-4)
.
Exer
cise
sU
se e
lim
inat
ion
to
solv
e ea
ch s
yste
m o
f eq
uat
ion
s.
1. 2
x +
3y
= 6
2.
2m
+ 3
n =
4
3. 3
a -
b =
2x
+ 2
y =
5 (
-3,
4)
-m
+ 2
n =
5 (
-1,
2)
a +
2b
= 3
(1,
1)
4. 4
x +
5y
= 6
5.
4x
- 3
y =
22
6. 3
x -
4y
= -
46x
- 7
y =
-20
(-
1, 2
) 2
x -
y =
10
(4, -
2)
x +
3y
= -
10 (
-4,
-2)
7. 4
x -
y =
9
8. 4
a -
3b
= -
8 9.
2x
+ 2
y =
55x
+ 2
y =
8 (
2, -
1)
2a
+ 2
b =
3
(- 1 −
2 ,
2 )
4x
- 4
y =
10
10. 6
x -
4y
= -
8 11
. 4x
+ 2
y =
-5
12. 2
x +
y =
3.5
(0.9
, 1.7
)4x
+ 2
y =
-3
-
2x -
4y
= 1
-x
+ 2
y =
2.5
13. G
AR
DEN
ING
Th
e le
ngt
h o
f S
ally
’s g
arde
n i
s 4
met
ers
grea
ter
than
3 t
imes
th
e w
idth
. T
he
peri
met
er o
f h
er g
arde
n i
s 72
met
ers.
Wh
at a
re t
he
dim
ensi
ons
of S
ally
’s
gard
en?
28 m
by
8 m
6-4
Exam
ple
1Ex
amp
le 2
(- 3 −
2 ,
1 −
2 )
(-1,
1 −
2 )
( 5 −
2 ,
0 )
019_
027_
ALG
1_A
_CR
M_C
06_C
R_6
6028
0.in
dd
2412
/21/
10
8:43
AM
Lesson 6-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DAT
E
P
ER
IOD
Lesson 6-4
Cha
pte
r 6
25
Gle
ncoe
Alg
ebra
1
6-4
Stud
y G
uide
and
Inte
rven
tion
(co
nti
nu
ed)
Elim
inati
on
Usin
g M
ult
iplicati
on
Solv
e R
eal-
Wo
rld
Pro
ble
ms
Som
etim
es i
t is
nec
essa
ry t
o u
se m
ult
ipli
cati
on b
efor
e el
imin
atio
n i
n r
eal-
wor
ld p
robl
ems.
C
AN
OEI
NG
Du
rin
g a
can
oein
g tr
ip, i
t ta
kes
Ray
mon
d 4
hou
rs t
o p
add
le 1
2 m
iles
up
stre
am. I
t ta
kes
him
3 h
ours
to
mak
e th
e re
turn
tri
p p
add
lin
g d
own
stre
am. F
ind
th
e sp
eed
of
the
can
oe i
n s
till
wat
er.
Rea
d
You
are
ask
ed t
o fi
nd
the
spee
d of
th
e ca
noe
in
sti
ll w
ater
.
Sol
ve
Let
c =
th
e ra
te o
f th
e ca
noe
in
sti
ll w
ater
.L
et w
= t
he
rate
of
the
wat
er c
urr
ent.
rt
dr
· t
= d
Ag
ain
st t
he
Cu
rren
tc
– w
412
(c –
w)4
= 1
2
Wit
h t
he
Cu
rren
tc
+ w
312
(c +
w)3
= 1
2
So,
ou
r tw
o eq
uat
ion
s ar
e 4c
- 4
w=
12
and
3c+
3w
= 1
2.U
se e
lim
inat
ion
wit
h m
ult
ipli
cati
on t
o so
lve
the
syst
em. S
ince
th
e pr
oble
m a
sks
for
c,
elim
inat
e w
.4c
- 4
w=
12
⇒ M
ult
iply
by
3 ⇒
12
c-
12w
= 3
63c
+ 3
w=
12
⇒ M
ult
iply
by
4 ⇒
(+
) 1
2c+
12w
= 4
8
24c
= 8
4 w
is e
limin
ated
.
24c
− 24=
84 − 24D
ivid
e ea
ch s
ide
by 2
4.
c
= 3
.5
Sim
plify
.T
he
rate
of
the
can
oe i
n s
till
wat
er i
s 3.
5 m
iles
per
hou
r.
Exer
cise
s 1
. FLI
GH
T A
n a
irpl
ane
trav
elin
g w
ith
th
e w
ind
flie
s 45
0 m
iles
in
2 h
ours
. On
th
e re
turn
tr
ip, t
he
plan
e ta
kes
3 h
ours
to
trav
el t
he
sam
e di
stan
ce. F
ind
the
spee
d of
th
e ai
rpla
ne
if t
he
win
d is
sti
ll.
187.
5 m
iles
per
ho
ur
2. F
UN
DR
AIS
ING
Ben
ji a
nd
Joel
are
rai
sin
g m
oney
for
th
eir
clas
s tr
ip b
y se
llin
g gi
ft
wra
ppin
g pa
per.
Ben
ji r
aise
s $3
9 by
sel
lin
g 5
roll
s of
red
wra
ppin
g pa
per
and
2 ro
lls
of
foil
wra
ppin
g pa
per.
Joel
rai
ses
$57
by s
elli
ng
3 ro
lls
of r
ed w
rapp
ing
pape
r an
d 6
roll
s of
fo
il w
rapp
ing
pape
r. F
or h
ow m
uch
are
Ben
ji a
nd
Joel
sel
lin
g ea
ch r
oll
of r
ed a
nd
foil
w
rapp
ing
pape
r? r
ed:
$5;
foil:
$7
Exam
ple
019_
027_
ALG
1_A
_CR
M_C
06_C
R_6
6028
0.in
dd
2512
/21/
10
8:43
AM
Answers (Lesson 6-4)
A01_A21_ALG1_A_CRM_C06_AN_660280.indd A11A01_A21_ALG1_A_CRM_C06_AN_660280.indd A11 12/21/10 8:50 AM12/21/10 8:50 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
PDF Pass
Chapter 6 A12 Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DAT
E
P
ER
IOD
Cha
pte
r 6
26
Gle
ncoe
Alg
ebra
1
Skill
s Pr
acti
ceE
lim
inati
on
Usin
g M
ult
iplicati
on
Use
eli
min
atio
n t
o so
lve
each
sys
tem
of
equ
atio
ns.
1. x
+ y
= -
9 2.
3x
+ 2
y =
-9
5x -
2y
= 3
2 (2
, -11
) x
- y
= -
13 (
-7,
6)
3. 2
x +
5y
= 3
4.
2x
+ y
= 3
-x
+ 3
y =
-7
(4, -
1)
-4x
- 4
y =
-8
(1, 1
)
5. 4
x -
2y
= -
14
6. 2
x +
y =
03x
- y
= -
8 (-
1, 5
) 5
x +
3y
= 2
(-
2, 4
)
7. 5
x +
3y
= -
10
8. 2
x +
3y
= 1
43x
+ 5
y =
-6
(-2,
0)
3x
- 4
y =
4 (
4, 2
)
9. 2
x -
3y
= 2
1 10
. 3x
+ 2
y =
-26
5x -
2y
= 2
5 (3
, -5)
4
x -
5y
= -
4 (-
6, -
4)
11. 3
x -
6y
= -
3 12
. 5x
+ 2
y =
-3
2x +
4y
= 3
0 (7
, 4)
3x
+ 3
y =
9 (
-3,
6)
13. T
wo
tim
es a
nu
mbe
r pl
us
thre
e ti
mes
an
oth
er n
um
ber
equ
als
13. T
he
sum
of
the
two
nu
mbe
rs i
s 7.
Wh
at a
re t
he
nu
mbe
rs?
8, -
1
14. F
our
tim
es a
nu
mbe
r m
inu
s tw
ice
anot
her
nu
mbe
r is
-16
. Th
e su
m o
f th
e tw
o n
um
bers
is
-1.
Fin
d th
e n
um
bers
. -
3, 2
15. F
UN
DR
AIS
ING
Tri
sha
and
Byr
on a
re w
ash
ing
and
vacu
um
ing
cars
to
rais
e m
oney
for
a
clas
s tr
ip. T
rish
a ra
ised
$38
was
hin
g 5
cars
an
d va
cuu
min
g 4
cars
. Byr
on r
aise
d $2
8 by
w
ash
ing
4 ca
rs a
nd
vacu
um
ing
2 ca
rs. F
ind
the
amou
nt
they
ch
arge
d to
was
h a
car
an
d va
cuu
m a
car
.
6-4 was
h:
$6, v
acu
um
: $2
019_
027_
ALG
1_A
_CR
M_C
06_C
R_6
6028
0.in
dd
2612
/21/
10
8:43
AM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DAT
E
P
ER
IOD
Lesson 6-4
Cha
pte
r 6
27
Gle
ncoe
Alg
ebra
1
Prac
tice
Elim
inati
on
Usin
g M
ult
iplicati
on
Use
eli
min
atio
n t
o so
lve
each
sys
tem
of
equ
atio
ns.
1. 2
x -
y =
-1
2. 5
x -
2y
= -
10
3. 7
x +
4y
= -
43x
- 2
y =
1
3x
+ 6
y =
66
5x
+ 8
y =
28
(
-3,
-5)
(
2, 1
0)
(-
4, 6
)
4. 2
x -
4y
= -
22
5. 3
x +
2y
= -
9 6.
4x
- 2
y =
32
3x +
3y
= 3
0 5
x -
3y
= 4
-
3x -
5y
= -
11
(
3, 7
) (
-1,
-3)
(
7, -
2)
7. 3
x +
4y
= 2
7 8.
0.5
x +
0.5
y =
-2
9. 2
x -
3 −
4 y
= -
7
5
x -
3y
= 1
6 x
- 0
.25y
= 6
x
+ 1 −
2 y
= 0
(
5, 3
) (
4, -
8)
(-
2, 4
)
10. 6
x -
3y
= 2
1 11
. 3x
+ 2
y =
11
12. -
3x +
2y
= -
152x
+ 2
y =
22
2x
+ 6
y =
-2
2x
- 4
y =
26
(
6, 5
) (
5, -
2)
(1,
-6)
13. E
igh
t ti
mes
a n
um
ber
plu
s fi
ve t
imes
an
oth
er n
um
ber
is -
13. T
he
sum
of
the
two
nu
mbe
rs i
s 1.
Wh
at a
re t
he
nu
mbe
rs?
-6,
7
14. T
wo
tim
es a
nu
mbe
r pl
us
thre
e ti
mes
an
oth
er n
um
ber
equ
als
4. T
hre
e ti
mes
th
e fi
rst
nu
mbe
r pl
us
fou
r ti
mes
th
e ot
her
nu
mbe
r is
7. F
ind
the
nu
mbe
rs.
5, -
2
15. F
INA
NC
E G
un
ther
in
vest
ed $
10,0
00 i
n t
wo
mu
tual
fu
nds
. On
e of
th
e fu
nds
ros
e 6%
in
on
e ye
ar, a
nd
the
oth
er r
ose
9% i
n o
ne
year
. If
Gu
nth
er’s
in
vest
men
t ro
se a
tot
al o
f $6
84
in o
ne
year
, how
mu
ch d
id h
e in
vest
in
eac
h m
utu
al f
un
d?$7
200
in t
he
6% f
un
d a
nd
$28
00
in t
he
9% f
un
d
16. C
AN
OEI
NG
Lau
ra a
nd
Bre
nt
padd
led
a ca
noe
6 m
iles
ups
trea
m i
n f
our
hou
rs. T
he
retu
rn t
rip
took
th
ree
hou
rs. F
ind
the
rate
at
wh
ich
Lau
ra a
nd
Bre
nt
padd
led
the
can
oe
in s
till
wat
er.
1.75
mi/h
17. N
UM
BER
TH
EORY
Th
e su
m o
f th
e di
gits
of
a tw
o-di
git
nu
mbe
r is
11.
If
the
digi
ts a
re
reve
rsed
, th
e n
ew n
um
ber
is 4
5 m
ore
than
th
e or
igin
al n
um
ber.
Fin
d th
e n
um
ber.
38
6-4
019_
027_
ALG
1_A
_CR
M_C
06_C
R_6
6028
0.in
dd
2712
/21/
10
8:43
AM
Answers (Lesson 6-4)
A01_A21_ALG1_A_CRM_C06_AN_660280.indd A12A01_A21_ALG1_A_CRM_C06_AN_660280.indd A12 12/21/10 8:50 AM12/21/10 8:50 AM
An
swer
s
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
PDF Pass
Chapter 6 A13 Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DAT
E
P
ER
IOD
Cha
pte
r 6
28
Gle
ncoe
Alg
ebra
1
1. S
OC
CER
Su
ppos
e a
you
th s
occe
r fi
eld
has
a p
erim
eter
of
320
yard
s an
d it
s le
ngt
h m
easu
res
40 y
ards
mor
e th
an i
ts
wid
th. M
s. H
ugh
ey a
sks
her
pla
yers
to
dete
rmin
e th
e le
ngt
h a
nd
wid
th o
f th
eir
fiel
d. S
he
give
s th
em t
he
foll
owin
g sy
stem
of
equ
atio
ns
to r
epre
sen
t th
e si
tuat
ion
. Use
eli
min
atio
n t
o so
lve
the
syst
em t
o fi
nd
the
len
gth
an
d w
idth
of
the
fiel
d.
2L +
2W
= 3
20 L
– W
= 4
0
w
idth
= 6
0 yd
; le
ng
th =
10
0 yd
2. S
POR
TS T
he
Fan
Cos
t In
dex
(FC
I)
trac
ks t
he
aver
age
cost
s fo
r at
ten
din
g sp
orti
ng
even
ts, i
ncl
udi
ng
tick
ets,
dri
nks
, fo
od, p
arki
ng,
pro
gram
s, a
nd
sou
ven
irs.
A
ccor
din
g to
th
e F
CI,
a f
amil
y of
fou
r w
ould
spe
nd
a to
tal
of $
592.
30 t
o at
ten
d tw
o M
ajor
Lea
gue
Bas
ebal
l (M
LB
) ga
mes
an
d on
e N
atio
nal
Bas
ketb
all A
ssoc
iati
on
(NB
A)
gam
e. T
he
fam
ily
wou
ld s
pen
d $6
91.3
1 to
att
end
one
ML
B a
nd
two
NB
A
gam
es. W
rite
an
d so
lve
a sy
stem
of
equ
atio
ns
to f
ind
the
fam
ily’
s co
sts
for
each
kin
d of
gam
e ac
cord
ing
to t
he
FC
I.N
BA
: $2
63.4
4; M
LB
: $1
64.4
3
3. A
RT
Mr.
San
tos,
th
e cu
rato
r of
th
e ch
ildr
en’s
mu
seu
m, r
ecen
tly
mad
e tw
o pu
rch
ases
of
clay
an
d w
ood
for
a vi
siti
ng
arti
st t
o sc
ulp
t. U
se t
he
tabl
e to
fin
d th
e co
st o
f ea
ch p
rodu
ct p
er k
ilog
ram
.
Cla
y (k
g)
Wo
od
(kg
)To
tal C
ost
54
$35.
50
3.5
6$5
0.45
C
lay
was
$0.
70 p
er k
ilog
ram
an
d
wo
od
was
$8.
00
per
kilo
gra
m.
4. T
RA
VEL
An
ton
io f
lies
fro
m H
oust
on
to P
hil
adel
phia
, a d
ista
nce
of
abou
t 13
40 m
iles
. His
pla
ne
trav
els
wit
h t
he
win
d an
d ta
kes
2 h
ours
an
d 20
min
ute
s.
At
the
sam
e ti
me,
Pau
l is
on
a p
lan
e fr
om
Ph
ilad
elph
ia t
o H
oust
on. S
ince
his
pla
ne
is h
eadi
ng
agai
nst
th
e w
ind,
Pau
l’s f
ligh
t ta
kes
2 h
ours
an
d 50
min
ute
s. W
hat
was
th
e sp
eed
of t
he
win
d in
mil
es p
er h
our?
abo
ut
50.6
7 m
ph
5. B
USI
NES
S S
upp
ose
you
sta
rt a
bu
sin
ess
asse
mbl
ing
and
sell
ing
mot
oriz
ed
scoo
ters
. It
cost
s yo
u $
1500
for
too
ls a
nd
equ
ipm
ent
to g
et s
tart
ed, a
nd
the
mat
eria
ls c
ost
$200
for
eac
h s
coot
er. Y
our
scoo
ters
sel
l fo
r $3
00 e
ach
.
a. W
rite
an
d so
lve
a sy
stem
of
equ
atio
ns
repr
esen
tin
g th
e to
tal
cost
s an
d re
ven
ue
of y
our
busi
nes
s.
y =
20
0x +
150
0 (
cost
);
y =
30
0x
(rev
enu
e);
solu
tio
n:
(15,
450
0)
b.
Des
crib
e w
hat
th
e so
luti
on m
ean
s in
te
rms
of t
he
situ
atio
n. T
he
solu
tio
n
rep
rese
nts
th
e p
oin
t at
wh
ich
co
st e
qu
als
reve
nu
e. T
his
is t
he
bre
ak-e
ven
po
int
for
the
busi
nes
s o
wn
er.
c. G
ive
an e
xam
ple
of a
rea
son
able
n
um
ber
of s
coot
ers
you
cou
ld a
ssem
ble
and
sell
in
ord
er t
o m
ake
a pr
ofit
, an
d fi
nd
the
prof
it y
ou w
ould
mak
e fo
r th
at n
um
ber
of s
coot
ers.
Sam
ple
an
swer
: 20
sco
ote
rs:
It c
ost
s $5
500
to a
ssem
ble
th
ese
sco
ote
rs, w
hic
h s
ell f
or
$60
00,
le
avin
g a
$50
0 p
rofi t
.
Wor
d Pr
oble
m P
ract
ice
Elim
inati
on
Usin
g M
ult
iplicati
on
6-4
028_
036_
ALG
1_A
_CR
M_C
06_C
R_6
6028
0.in
dd
2812
/21/
10
8:43
AM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DAT
E
P
ER
IOD
Lesson 6-4
Cha
pte
r 6
29
Gle
ncoe
Alg
ebra
1
Enri
chm
ent
Geo
rge W
ash
ing
ton
Carv
er
an
d P
erc
y J
ulian
In
199
0, G
eorg
e W
ash
ingt
on C
arve
r an
d P
ercy
Ju
lian
bec
ame
the
firs
t A
fric
an A
mer
ican
s el
ecte
d to
th
e N
atio
nal
In
ven
tors
Hal
l of
Fam
e. C
arve
r (1
864–
1943
) w
as a
n a
gric
ult
ura
l sc
ien
tist
kn
own
wor
ldw
ide
for
deve
lopi
ng
hu
ndr
eds
of u
ses
for
the
pean
ut
and
the
swee
t po
tato
. His
wor
k re
vita
lize
d th
e ec
onom
y of
th
e so
uth
ern
Un
ited
Sta
tes
beca
use
it
was
no
lon
ger
depe
nde
nt
sole
ly u
pon
cot
ton
. Ju
lian
(18
98–1
975)
was
a r
esea
rch
ch
emis
t w
ho
beca
me
fam
ous
for
inve
nti
ng
a m
eth
od o
f m
akin
g a
syn
thet
ic c
orti
son
e fr
om s
oybe
ans.
His
di
scov
ery
has
had
man
y m
edic
al a
ppli
cati
ons,
par
ticu
larl
y in
th
e tr
eatm
ent
of a
rth
riti
s.T
her
e ar
e do
zen
s of
oth
er A
fric
an A
mer
ican
in
ven
tors
wh
ose
acco
mpl
ish
men
ts a
re n
ot a
s w
ell
know
n. T
hei
r in
ven
tion
s ra
nge
fro
m c
omm
on h
ouse
hol
d it
ems
like
th
e ir
onin
g bo
ard
to
com
plex
dev
ices
th
at h
ave
revo
luti
oniz
ed m
anu
fact
uri
ng.
Th
e ex
erci
ses
that
fol
low
wil
l h
elp
you
ide
nti
fy ju
st a
few
of
thes
e in
ven
tors
an
d th
eir
inve
nti
ons.
Mat
ch t
he
inve
nto
rs w
ith
th
eir
inve
nti
ons
by
mat
chin
g ea
ch s
yste
m w
ith
its
so
luti
on. (
Not
all
th
e so
luti
ons
wil
l b
e u
sed
.)
1. S
ara
Boo
ne
x +
y =
2 E
A
. (1
, 4)
auto
mat
ic t
raff
ic s
ign
al
x -
y =
10
2. S
arah
Goo
de
x =
2 -
y D
B
. (4
, -2)
eg
gbea
ter
2y
+ x
= 9
3. F
rede
rick
M.
y =
2x
+ 6
G
C.
(-2,
3)
fire
ext
ingu
ish
er
Jon
es
y =
-x
- 3
4. J
. L. L
ove
2x +
3y
= 8
F
D.
(-5,
7)
fold
ing
cabi
net
bed
2x -
y =
-8
5. T
. J. M
arsh
all
y -
3x
= 9
C
E.
(6, -
4)
iron
ing
boar
d
2y +
x =
4
6. J
an M
atze
lige
r y
+ 4
= 2
x J
F.
(-2,
4)
pen
cil
shar
pen
er
6x
- 3
y =
12
7. G
arre
tt A
. 3x
- 2
y =
-5
A
G.
(-3,
0)
port
able
x-r
ay m
ach
ine
Mor
gan
3y
- 4
x =
8
8. N
orbe
rt R
illi
eux
3x -
y =
12
I H
. (2
, -3)
pl
ayer
pia
no
y -
3x
= 1
5
I.
n
o so
luti
on
evap
orat
ing
pan
for
re
fin
ing
suga
r
J.
in
fin
itel
y la
stin
g (s
hap
ing)
m
any
mac
hin
e fo
r
so
luti
ons
man
ufa
ctu
rin
g sh
oes
6-4
028_
036_
ALG
1_A
_CR
M_C
06_C
R_6
6028
0.in
dd
2912
/21/
10
8:43
AM
Answers (Lesson 6-4)
A01_A21_ALG1_A_CRM_C06_AN_660280.indd A13A01_A21_ALG1_A_CRM_C06_AN_660280.indd A13 12/21/10 8:50 AM12/21/10 8:50 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
PDF Pass
Chapter 6 A14 Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DAT
E
P
ER
IOD
Cha
pte
r 6
30
Gle
ncoe
Alg
ebra
1
Stud
y G
uide
and
Inte
rven
tion
Ap
ply
ing
Syste
ms o
f Lin
ear
Eq
uati
on
s
Det
erm
ine
The
Bes
t M
eth
od
Yo
u h
ave
lear
ned
fiv
e m
eth
ods
for
solv
ing
syst
ems
of
lin
ear
equ
atio
ns:
gra
phin
g, s
ubs
titu
tion
, eli
min
atio
n u
sin
g ad
diti
on, e
lim
inat
ion
usi
ng
subt
ract
ion
, an
d el
imin
atio
n u
sin
g m
ult
ipli
cati
on. F
or a
n e
xact
sol
uti
on, a
n a
lgeb
raic
m
eth
od i
s be
st.
A
t a
bas
ebal
l ga
me,
Hen
ry b
ough
t 3
hot
dog
s an
d a
bag
of
chip
s fo
r $1
4. S
cott
bou
ght
2 h
otd
ogs
and
a b
ag o
f ch
ips
for
$10.
Eac
h o
f th
e b
oys
pai
d t
he
sam
e p
rice
for
th
eir
hot
dog
s, a
nd
th
e sa
me
pri
ce f
or t
hei
r ch
ips.
Th
e fo
llow
ing
syst
em o
f eq
uat
ion
s ca
n b
e u
sed
to
rep
rese
nt
the
situ
atio
n. D
eter
min
e th
e b
est
met
hod
to
solv
e th
e sy
stem
of
equ
atio
ns.
Th
en s
olve
th
e sy
stem
.
3x +
y =
14
2x +
y =
10
•
Sin
ce n
eith
er t
he
coef
fici
ents
of
x n
or t
he
coef
fici
ents
of
y ar
e ad
diti
ve i
nve
rses
, you
ca
nn
ot u
se e
lim
inat
ion
usi
ng
addi
tion
.
•
Sin
ce t
he
coef
fici
ent
of y
in
bot
h e
quat
ion
s is
1, y
ou c
an u
se e
lim
inat
ion
usi
ng
subt
ract
ion
. You
cou
ld a
lso
use
th
e su
bsti
tuti
on m
eth
od o
r el
imin
atio
n u
sin
g m
ult
ipli
cati
on
Th
e fo
llow
ing
solu
tion
use
s el
imin
atio
n b
y su
btra
ctio
n t
o so
lve
this
sys
tem
.
3x +
y =
14
Wri
te t
he
equ
atio
ns
in c
olu
mn
for
m a
nd
subt
ract
. (-
) 2x
+ (-
) y
= (-
)10
x
= 4
T
he
vari
able
y i
s el
imin
ated
.
3(4)
+ y
= 1
4 S
ubs
titu
te t
he
valu
e fo
r x
back
in
to t
he
firs
t eq
uat
ion
. y
= 2
S
olve
for
y.
Th
is m
ean
s th
at h
ot d
ogs
cost
$4
each
an
d a
bag
of c
hip
s co
sts
$2.
Exer
cise
sD
eter
min
e th
e b
est
met
hod
to
solv
e ea
ch s
yste
m o
f eq
uat
ion
s. T
hen
sol
ve
the
syst
em.
1. 5
x +
3y
= 1
6 2.
3x
- 5
y =
73x
- 5
y =
-4
2x
+ 5
y =
13
elim
inat
ion
(×
); (
2, 2
) e
limin
atio
n (
+);
(4,
1)
3. y
+ 3
x =
24
4. -
11x
- 1
0y =
17
5x -
y =
8
5x
– 7y
= 5
0el
imin
atio
n (
+);
(4,
12)
e
limin
atio
n (
×);
(3,
-5)
6-5
Exam
ple
028_
036_
ALG
1_A
_CR
M_C
06_C
R_6
6028
0.in
dd
3012
/21/
10
8:43
AM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DAT
E
P
ER
IOD
Lesson 6-5
Cha
pte
r 6
31
Gle
ncoe
Alg
ebra
1
Ap
ply
Sys
tem
s O
f Li
nea
r Eq
uat
ion
s W
hen
app
lyin
g sy
stem
s of
lin
ear
equ
atio
ns
to
prob
lem
sit
uat
ion
s, i
t is
im
port
ant
to a
nal
yze
each
sol
uti
on i
n t
he
con
text
of
the
situ
atio
n.
B
USI
NES
S A
T-s
hir
t p
rin
tin
g co
mp
any
sell
s T-
shir
ts f
or $
15 e
ach
. Th
e co
mp
any
has
a f
ixed
cos
t fo
r th
e m
ach
ine
use
d t
o p
rin
t th
e T-
shir
ts a
nd
an
ad
dit
ion
al c
ost
per
T-
shir
t. U
se t
he
tab
le t
o es
tim
ate
the
nu
mb
er o
f T-
shir
ts t
he
com
pan
y m
ust
sel
l in
ord
er f
or t
he
inco
me
to e
qu
al e
xpen
ses.
Un
der
stan
d Y
ou k
now
th
e in
itia
l in
com
e an
d th
e in
itia
l ex
pen
se a
nd
the
rate
s of
ch
ange
of
each
qu
anti
ty w
ith
eac
h T
-sh
irt
sold
.
Pla
n
W
rite
an
equ
atio
n t
o re
pres
ent
the
inco
me
and
the
expe
nse
s. T
hen
sol
ve t
o fi
nd
how
man
y T-
shir
ts n
eed
to b
e so
ld f
or b
oth
val
ues
to
be e
qual
.
Sol
ve
L
et x
= t
he
nu
mbe
r of
T-s
hir
ts s
old
and
let
y =
th
e to
tal
amou
nt.
in
com
e y
=
0 +
15
x
ex
pen
ses
y
= 3
000
+
5x
Yo
u c
an u
se s
ubs
titu
tion
to
solv
e th
is s
yste
m.
y
= 1
5x
The
fi rs
t eq
uatio
n.
15
x =
300
0 +
5x
Sub
stitu
te t
he v
alue
for
y in
to t
he s
econ
d eq
uatio
n.
10
x =
300
0 S
ubtr
act
10x
from
eac
h si
de a
nd s
impl
ify.
x
= 3
00
Div
ide
each
sid
e by
10
and
sim
plify
.
T
his
mea
ns
that
if
300
T-sh
irts
are
sol
d, t
he
inco
me
and
expe
nse
s of
th
e T-
shir
t co
mpa
ny
are
equ
al.
Ch
eck
Doe
s th
is s
olu
tion
mak
e se
nse
in
th
e co
nte
xt o
f th
e pr
oble
m?
Aft
er s
elli
ng
300
T-sh
irts
, th
e in
com
e w
ould
be
abou
t 30
0 ×
$15
or
$450
0. T
he
cost
s w
ould
be
abou
t $3
000
+ 3
00 ×
$5
or $
4500
.
Exer
cise
sR
efer
to
the
exam
ple
ab
ove.
If
the
cost
s of
th
e T-
shir
t co
mp
any
chan
ge t
o th
e gi
ven
val
ues
an
d t
he
sell
ing
pri
ce r
emai
ns
the
sam
e, d
eter
min
e th
e n
um
ber
of
T-sh
irts
th
e co
mp
any
mu
st s
ell
in o
rder
for
in
com
e to
eq
ual
exp
ense
s.
1. p
rin
tin
g m
ach
ine:
$50
00.0
0;
2. p
rin
tin
g m
ach
ine:
$21
00.0
0;
T-sh
irt:
$10
.00
each
10
00
T-s
hir
t: $
8.00
eac
h
300
3. p
rin
tin
g m
ach
ine:
$88
00.0
0;
4. p
rin
tin
g m
ach
ine:
$12
00.0
0;T-
shir
t: $
4.00
eac
h
800
T-s
hir
t: $
12.0
0 ea
ch
400
Stud
y G
uide
and
Inte
rven
tion
(co
nti
nu
ed)
Ap
ply
ing
Syste
ms o
f Lin
ear
Eq
uati
on
s
6-5
T-sh
irt
Pri
nti
ng
Co
st
prin
ting
mac
hine
$300
0.00
blan
k T-
shir
t$5
.00
Exam
ple
tota
l am
ount
initi
al
amou
ntra
te o
f ch
ange
tim
es
num
ber
of T
-shi
rts
sold
028_
036_
ALG
1_A
_CR
M_C
06_C
R_6
6028
0.in
dd
3112
/21/
10
8:43
AM
Answers (Lesson 6-5)
A01_A21_ALG1_A_CRM_C06_AN_660280.indd A14A01_A21_ALG1_A_CRM_C06_AN_660280.indd A14 12/21/10 8:50 AM12/21/10 8:50 AM
An
swer
s
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
PDF Pass
Chapter 6 A15 Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DAT
E
P
ER
IOD
Cha
pte
r 6
32
Gle
ncoe
Alg
ebra
1
Det
erm
ine
the
bes
t m
eth
od t
o so
lve
each
sys
tem
of
equ
atio
ns.
T
hen
sol
ve t
he
syst
em.
1. 5
x +
3y
= 1
6 2.
3x
– 5y
= 7
3x –
5y
= -
4 2
x +
5y
= 1
3el
imin
atio
n (
×);
(2,
2)
elim
inat
ion
(+
); (
4, 1
)
3. y
= 3
x -
24
4. -
11x
– 10
y =
17
5x -
y =
8
5x
– 7y
= 5
0su
bst
ituti
on
; (-
8, -
48)
elim
inat
ion
(×
); (
3, -
5)
5. 4
x +
y =
24
6. 6
x –
y =
-14
55x
- y
= 1
2
x =
4 –
2y
elim
inat
ion
(+
); (
4, 8
) s
ub
stitu
tio
n;
(-22
, 13)
7. V
EGET
AB
LE S
TAN
D
A r
oads
ide
vege
tabl
e st
and
sell
s pu
mpk
ins
for
$5 e
ach
an
d sq
uas
hes
for
$3
each
. On
e da
y th
ey s
old
6 m
ore
squ
ash
th
an p
um
pkin
s, a
nd
thei
r sa
les
tota
led
$98.
Wri
te a
nd
solv
e a
syst
em o
f eq
uat
ion
s to
fin
d h
ow m
any
pum
pkin
s an
d sq
uas
h t
hey
sol
d?
y =
6 +
x a
nd
5x +
3y =
98;
10
pu
mp
kin
s, 1
6 sq
uas
hes
8. I
NC
OM
E R
amir
o ea
rns
$20
per
hou
r du
rin
g th
e w
eek
and
$30
per
hou
r fo
r ov
erti
me
on t
he
wee
ken
ds. O
ne
wee
k R
amir
o ea
rned
a t
otal
of
$650
. He
wor
ked
5 ti
mes
as
man
y h
ours
du
rin
g th
e w
eek
as h
e di
d on
th
e w
eeke
nd.
Wri
te a
nd
solv
e a
syst
em o
f eq
uat
ion
s to
det
erm
ine
how
man
y h
ours
of
over
tim
e R
amir
o w
orke
d on
th
e w
eeke
nd.
20
x +
30y
= 6
50 a
nd
x =
5y;
5 h
ou
rs
9. B
ASK
ETB
ALL
A
nya
mak
es 1
4 ba
sket
s du
rin
g h
er g
ame.
Som
e of
th
ese
bask
ets
wer
e w
orth
2-p
oin
ts a
nd
oth
ers
wer
e w
orth
3-p
oin
ts. I
n t
otal
, sh
e sc
ored
30
poin
ts. W
rite
an
d so
lve
a sy
stem
of
equ
atio
ns
to f
ind
how
2-p
oin
ts b
aske
ts s
he
mad
e.
x +
y =
14
and
2x +
3y =
30;
12
6-5
Skill
s Pr
acti
ceA
pp
lyin
g S
yste
ms o
f Lin
ear
Eq
uati
on
s
028_
036_
ALG
1_A
_CR
M_C
06_C
R_6
6028
0.in
dd
3212
/21/
10
8:43
AM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DAT
E
P
ER
IOD
Lesson 6-5
Cha
pte
r 6
33
Gle
ncoe
Alg
ebra
1
Det
erm
ine
the
bes
t m
eth
od t
o so
lve
each
sys
tem
of
equ
atio
ns.
Th
en s
olve
th
e sy
stem
.
1. 1
.5x
– 1.
9y =
-29
2.
1.2
x –
0.8y
= -
6 3.
18x
–16
y =
-31
2x
– 0.
9y =
4.5
4
.8x
+ 2
.4y
= 6
0 7
8x –
16y
= 4
08su
bst
ituti
on
;
elim
inat
ion
(×
);
elim
inat
ion
(-
);
(63,
65)
(
5, 1
5)
(12
, 33)
4. 1
4x +
7y
= 2
17
5. x
= 3
.6y
+ 0
.7
6. 5
.3x
– 4y
= 4
3.5
14x
+ 3
y =
189
2
x +
0.2
y =
38.
4 x
+ 7
y =
78
elim
inat
ion
(-
);
su
bst
ituti
on
;
su
bst
ituti
on
; (1
2, 7
) (
18.7
, 5)
(15
, 9)
7. B
OO
KS
A l
ibra
ry c
onta
ins
2000
boo
ks. T
her
e ar
e 3
tim
es a
s m
any
non
-fic
tion
boo
ks a
s fi
ctio
n b
ooks
. Wri
te a
nd
solv
e a
syst
em o
f eq
uat
ion
s to
det
erm
ine
the
nu
mbe
r of
non
-fi
ctio
n a
nd
fict
ion
boo
ks.
x +
y =
20
00
and
x =
3y;
150
0 n
on
-fi c
tio
n,
500
fi ct
ion
8. S
CH
OO
L C
LUB
S T
he
ches
s cl
ub
has
16
mem
bers
an
d ga
ins
a n
ew m
embe
r ev
ery
mon
th. T
he
film
clu
b h
as 4
mem
bers
an
d ga
ins
4 n
ew m
embe
rs e
very
mon
th. W
rite
an
d so
lve
a sy
stem
of
equ
atio
ns
to f
ind
wh
en t
he
nu
mbe
r of
mem
bers
in
bot
h c
lubs
wil
l be
eq
ual
. y =
16
+ x
an
d y
= 4
+ 4
x;
4 m
on
ths
9. T
ia a
nd
Ken
eac
h s
old
snac
k ba
rs a
nd
mag
azin
e su
bscr
ipti
ons
for
a sc
hoo
l fu
nd-
rais
er, a
s sh
own
in
th
e ta
ble.
T
ia e
arn
ed $
132
and
Ken
ear
ned
$19
0.
a. D
efin
e va
riab
le a
nd
form
ula
te a
sys
tem
of
lin
ear
equ
atio
n f
rom
th
is s
itu
atio
n.
L
et x
= t
he
cost
per
sn
ack
bar
an
d le
t y =
th
e co
st p
er m
agaz
ine
sub
scri
pti
on
s; 1
6x +
4y =
132
an
d 2
0x +
6y =
190
.
b.
Wh
at w
as t
he
pric
e pe
r sn
ack
bar?
Det
erm
ine
the
reas
onab
len
ess
of y
our
solu
tion
.
$2;
Sam
ple
an
swer
: $2
is a
rea
son
able
pri
ce f
or
a sn
ack
bar
, an
d t
he
solu
tio
n c
hec
ks if
th
e m
agaz
ine
sub
scri
pti
on
s ar
e $2
5.
Prac
tice
Ap
ply
ing
Syste
ms o
f Lin
ear
Eq
uati
on
s
6-5
Item
Nu
mb
er S
old
Tia
Ken
snac
k ba
rs16
20
mag
azin
e su
bscr
iptio
ns4
6
028_
036_
ALG
1_A
_CR
M_C
06_C
R_6
6028
0.in
dd
3312
/21/
10
8:43
AM
Answers (Lesson 6-5)
A01_A21_ALG1_A_CRM_C06_AN_660280.indd A15A01_A21_ALG1_A_CRM_C06_AN_660280.indd A15 12/21/10 8:50 AM12/21/10 8:50 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
PDF Pass
Chapter 6 A16 Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DAT
E
P
ER
IOD
Cha
pte
r 6
34
Gle
ncoe
Alg
ebra
1
Wor
d Pr
oble
m P
ract
ice
Ap
ply
ing
Syste
ms o
f Lin
ear
Eq
uati
on
s
1. M
ON
EY V
eron
ica
has
bee
n s
avin
g di
mes
an
d qu
arte
rs. S
he
has
94
coin
s in
all
, an
d th
e to
tal
valu
e is
$19
.30.
How
man
y di
mes
an
d h
ow m
any
quar
ters
doe
s sh
e h
ave?
28
dim
es;
66 q
uar
ters
2. C
HEM
ISTR
Y H
ow m
any
lite
rs o
f 15
%
acid
an
d 33
% a
cid
shou
ld b
e m
ixed
to
mak
e 40
lit
ers
of 2
1% a
cid
solu
tion
?
Co
nce
ntr
atio
no
f S
olu
tio
nA
mo
un
t o
f S
olu
tio
n (
L)
Am
ou
nt
of A
cid
15%
x
33%
y
21%
40
26
2 −
3 L
of
15%
; 13
1
−
3 L o
f 33
%
3. B
UIL
DIN
GS
Th
e S
ears
Tow
er i
n C
hic
ago
is t
he
tall
est
buil
din
g in
Nor
th A
mer
ica.
T
he
tota
l h
eigh
t of
th
e to
wer
t a
nd
the
ante
nn
a th
at s
tan
ds o
n t
op o
f it
a i
s 17
29 f
eet.
Th
e di
ffer
ence
in
hei
ghts
be
twee
n t
he
buil
din
g an
d th
e an
ten
na
is
279
feet
. How
tal
l is
th
e S
ears
Tow
er?
1450
ft
4. P
RO
DU
CE
Rog
er a
nd
Tre
vor
wen
t sh
oppi
ng
for
prod
uce
on
th
e sa
me
day.
T
hey
eac
h b
ough
t so
me
appl
es a
nd
som
e po
tato
es. T
he
amou
nt
they
bou
ght
and
the
tota
l pr
ice
they
pai
d ar
e li
sted
in
th
e ta
ble
belo
w.
Wh
at w
as t
he
pric
e of
app
les
and
pota
toes
per
pou
nd?
ap
ple
s: $
1.49
p
er lb
; p
ota
toes
: $0
.99
per
lb
5. S
HO
PPIN
G T
wo
stor
es a
re h
avin
g a
sale
on
T-s
hir
ts t
hat
nor
mal
ly s
ell
for
$20.
S
tore
S i
s ad
vert
isin
g an
s p
erce
nt
disc
oun
t, a
nd
Sto
re T
is
adve
rtis
ing
at
doll
ar d
isco
un
t. R
ose
spen
ds $
63 f
or
thre
e T-
shir
ts f
rom
Sto
re S
an
d on
e fr
om
Sto
re T
. Man
ny
spen
ds $
140
on f
ive
T-sh
irts
fro
m S
tore
S a
nd
fou
r fr
om
Sto
re T
. Fin
d th
e di
scou
nt
at e
ach
sto
re.
S
tore
S:
20%
; S
tore
T:
$5
6. T
RA
NSP
OR
TATI
ON
A S
peed
y R
iver
ba
rge
bou
nd
for
New
Orl
ean
s le
aves
B
aton
Rou
ge, L
ouis
ian
a, a
t 9:
00 A
.M. a
nd
trav
els
at a
spe
ed o
f 10
mil
es p
er h
our.
A
Rai
l Tra
nsp
ort
frei
ght
trai
n a
lso
bou
nd
for
New
Orl
ean
s le
aves
Bat
on R
ouge
at
1:30
P.M
. th
e sa
me
day.
Th
e tr
ain
tra
vels
at
25
mil
es p
er h
our,
and
the
rive
r ba
rge
trav
els
at 1
0 m
iles
per
hou
r. B
oth
th
e ba
rge
and
the
trai
n w
ill
trav
el 1
00 m
iles
to
rea
ch N
ew O
rlea
ns.
a. H
ow f
ar w
ill
the
trai
n t
rave
l be
fore
ca
tch
ing
up
to t
he
barg
e?
75 m
i
b.
Wh
ich
sh
ipm
ent
wil
l re
ach
New
O
rlea
ns
firs
t? A
t w
hat
tim
e?
Th
e tr
ain
will
arr
ive
fi rs
t. It
will
ar
rive
in N
ew O
rlea
ns
at
5:30
P.M
. of
the
sam
e d
ay.
c. I
f bo
th s
hip
men
ts t
ake
an h
our
to
un
load
bef
ore
hea
din
g ba
ck t
o B
aton
R
ouge
, wh
at i
s th
e ea
rlie
st t
ime
that
ei
ther
on
e of
th
e co
mpa
nie
s ca
n b
egin
to
loa
d gr
ain
to
ship
in
Bat
on
Rou
ge?
at
10:3
0 P.M
. th
e sa
me
day
6-5
Ap
ple
s(I
b)
Po
tato
es(I
b)
Tota
l Co
st
($)
Rog
er8
718
.85
Trev
or2
1012
.88
028_
036_
ALG
1_A
_CR
M_C
06_C
R_6
6028
0.in
dd
3412
/21/
10
8:43
AM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DAT
E
P
ER
IOD
Lesson 6-5
Cha
pte
r 6
35
Gle
ncoe
Alg
ebra
1
Cra
mer’
s R
ule
Cra
mer
’s R
ule
is
a m
eth
od f
or s
olvi
ng
a sy
stem
of
equ
atio
ns.
To
use
Cra
mer
’s R
ule
, set
up
a m
atri
x to
rep
rese
nt
the
equ
atio
ns.
A m
atri
x is
a w
ay o
f or
gan
izin
g da
ta.
S
olve
th
e fo
llow
ing
syst
em o
f eq
uat
ion
s u
sin
g C
ram
er’s
Ru
le.
2x +
3y
= 1
3
x +
y =
5
Ste
p 1
: S
et u
p a
mat
rix
repr
esen
tin
g th
e co
effi
cien
ts o
f x
and
y.
A
=
x y
Ste
p 2
: F
ind
the
dete
rmin
ant
of m
atri
x A
.
If a
mat
rix
A =
⎪ a c b
d
⎥ , th
en t
he
dete
rmin
ant,
det
(A) =
ad
– b
c.
det(
A) =
2(1
) –
1(3)
= -
1
Ste
p 3
: R
epla
ce t
he
firs
t co
lum
n i
n A
wit
h 1
3 an
d 5
and
fin
d th
e de
term
inan
t of
th
e n
ew
mat
rix.
A1 =
⎪ 13
5 3 1 ⎥ ;
det
(A1)
= 1
3(1)
– 5
(3)
= -
2
Ste
p 4
: T
o fi
nd
the
valu
e of
x i
n t
he
solu
tion
to
the
syst
em o
f eq
uat
ion
s,
dete
rmin
e th
e va
lue
of
det(
A1)
−
det(
A) .
det(
A1)
−
det(
A) =
-2
−
-1 o
r 2
Ste
p 5
: R
epea
t th
e pr
oces
s to
fin
d th
e va
lue
of y
. Th
is t
ime,
rep
lace
th
e se
con
d co
lum
n w
ith
13
and
5 an
d fi
nd
the
dete
rmin
ant.
A2 =
⎪ 2 1 13
5 ⎥ ;
det
(A2)
= 2
(5)
– 1(
13) =
-3
and
det(
A2)
−
det(
A)
= -
3 −
-
1 or
3.
So,
th
e so
luti
on t
o th
e sy
stem
of
equ
atio
ns
is (
2, 3
).
Exer
cise
s U
se C
ram
er’s
Ru
le t
o so
lve
each
sys
tem
of
equ
atio
ns.
1. 2
x +
y =
1
2. x
+ y
= 4
3.
x –
y =
43x
+ 5
y =
5
2x
– 3y
= -
2 3
x –
5y =
8(0
, 1)
(2,
2)
(6,
2)
4. 4
x –
y =
3
5. 3
x –
2y =
7
6. 6
x –
5y =
1x
+ y
= 7
2
x +
y =
14
3x
+ 2
y =
5(2
, 5)
(5,
4)
(1,
1)
Enri
chm
ent
6-5
Exam
ple
⎪ 2 1 3 1 ⎥
028_
036_
ALG
1_A
_CR
M_C
06_C
R_6
6028
0.in
dd
3512
/21/
10
8:43
AM
Answers (Lesson 6-5)
A01_A21_ALG1_A_CRM_C06_AN_660280.indd A16A01_A21_ALG1_A_CRM_C06_AN_660280.indd A16 12/21/10 8:50 AM12/21/10 8:50 AM
An
swer
s
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
PDF Pass
Chapter 6 A17 Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DAT
E
P
ER
IOD
Cha
pte
r 6
36
Gle
ncoe
Alg
ebra
1
Stud
y G
uide
and
Inte
rven
tion
Syste
ms o
f In
eq
ualiti
es
Syst
ems
of
Ineq
ual
itie
s T
he
solu
tion
of
a sy
stem
of
ineq
ual
itie
s is
th
e se
t of
all
or
dere
d pa
irs
that
sat
isfy
bot
h i
neq
ual
itie
s. I
f yo
u g
raph
th
e in
equ
alit
ies
in t
he
sam
e co
ordi
nat
e pl
ane,
th
e so
luti
on i
s th
e re
gion
wh
ere
the
grap
hs
over
lap.
S
olve
th
e sy
stem
of
ineq
ual
itie
s
by
grap
hin
g.y
> x
+ 2
y ≤
-2x
- 1
Th
e so
luti
on i
ncl
ude
s th
e or
dere
d pa
irs
in t
he
inte
rsec
tion
of
the
grap
hs.
Th
is r
egio
n i
s sh
aded
at
the
righ
t. T
he
grap
hs
of y
= x
+ 2
an
d y
= -
2x -
1 a
re b
oun
dari
es o
f th
is r
egio
n. T
he
grap
h o
f y
= x
+ 2
is
dash
ed a
nd
is n
ot i
ncl
ude
d in
th
e gr
aph
of
y >
x +
2.
S
olve
th
e sy
stem
of
ineq
ual
itie
s
by
grap
hin
g.x
+ y
> 4
x +
y <
-1
Th
e gr
aph
s of
x +
y =
4 a
nd
x +
y =
-1
are
para
llel
. Bec
ause
th
e tw
o re
gion
s h
ave
no
poin
ts i
n c
omm
on, t
he
syst
em o
f in
equ
alit
ies
has
no
solu
tion
.
Exer
cise
sS
olve
eac
h s
yste
m o
f in
equ
alit
ies
by
grap
hin
g.
1. y
> -
1 2.
y >
-2x
+ 2
3.
y <
x +
1x
< 0
y
≤ x
+ 1
3
x +
4y
≥ 1
2
x
y
O
y =
-1
x =
0
x
y
O
y =
-2x
+ 2
y =
x +
1
x
y
O
3x +
4y
= 1
2
y =
x +
1
4. 2
x +
y ≥
1
5. y
≤ 2
x +
3
6. 5
x -
2y
< 6
x -
y ≥
-2
y ≥
-1
+ 2
x y
> -
x +
1
x
y
O
2x +
y =
1
x -
y =
-2
x
y
O
y =
2x
+ 3
y =
-1
+ 2
x
x
y
O
y =
-x
+ 1
5x -
2y
= 6
x
y O
x +
y =
1
x +
y =
4
x
y O
y =
x +
2
y =
-2x
- 1
6-6
Exam
ple
1
Exam
ple
2
028_
036_
ALG
1_A
_CR
M_C
06_C
R_6
6028
0.in
dd
3612
/21/
10
8:43
AM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DAT
E
P
ER
IOD
Lesson 6-6
Cha
pte
r 6
37
Gle
ncoe
Alg
ebra
1
Stud
y G
uide
and
Inte
rven
tion
(co
nti
nu
ed)
Syste
ms o
f In
eq
ualiti
es
Ap
ply
Sys
tem
s o
f In
equ
alit
ies
In r
eal-
wor
ld p
robl
ems,
som
etim
es o
nly
wh
ole
nu
mbe
rs m
ake
sen
se f
or t
he
solu
tion
, an
d of
ten
on
ly p
osit
ive
valu
es o
f x
and
y m
ake
sen
se.
BU
SIN
ESS
AA
A G
em C
omp
any
pro
du
ces
n
eck
lace
s an
d b
race
lets
. In
a 4
0-h
our
wee
k, t
he
com
pan
y h
as 4
00 g
ems
to u
se. A
nec
kla
ce r
equ
ires
40
gem
s an
d a
b
race
let
req
uir
es 1
0 ge
ms.
It
tak
es 2
hou
rs t
o p
rod
uce
a
nec
kla
ce a
nd
a b
race
let
req
uir
es o
ne
hou
r. H
ow m
any
of
each
typ
e ca
n b
e p
rod
uce
d i
n a
wee
k?
Let
n =
th
e n
um
ber
of n
eckl
aces
th
at w
ill
be p
rodu
ced
and
b =
th
e n
um
ber
of b
race
lets
th
at w
ill
be p
rodu
ced.
Nei
ther
n o
r b
can
be
a n
egat
ive
nu
mbe
r, so
th
e fo
llow
ing
syst
em o
f in
equ
alit
ies
repr
esen
ts
the
con
diti
ons
of t
he
prob
lem
s.
n ≥
0b
≥ 0
b +
2n
≤ 4
010
b +
40n
≤ 4
00T
he
solu
tion
is
the
set
orde
red
pair
s in
th
e in
ters
ecti
on o
f th
e gr
aph
s. T
his
reg
ion
is
shad
ed
at t
he
righ
t. O
nly
wh
ole-
nu
mbe
r so
luti
ons,
su
ch a
s (5
, 20)
mak
e se
nse
in
th
is p
robl
em.
Exer
cise
sF
or e
ach
exe
rcis
e, g
rap
h t
he
solu
tion
set
. Lis
t th
ree
pos
sib
le s
olu
tion
s to
th
e p
rob
lem
.
1. H
EALT
H M
r. F
low
ers
is o
n a
res
tric
ted
2.
REC
REA
TIO
N M
aria
had
$15
0 in
gif
t di
et t
hat
all
ows
him
to
hav
e be
twee
n
cer
tifi
cate
s to
use
at
a re
cord
sto
re. S
he
1600
an
d 20
00 C
alor
ies
per
day.
His
b
ough
t fe
wer
th
an 2
0 re
cord
ings
. Eac
h
dail
y fa
t in
take
is
rest
rict
ed t
o be
twee
n
tap
e co
st $
5.95
an
d ea
ch C
D c
ost
$8.9
5.
45 a
nd
55 g
ram
s. W
hat
dai
ly C
alor
ie
How
man
y of
eac
h t
ype
of r
ecor
din
g m
igh
t an
d fa
t in
take
s ar
e ac
cept
able
? s
he
hav
e bo
ugh
t?
60 50 40 30 20 10
Cal
ori
es
Fat Grams
1000
2000
3000
0
S
amp
le a
nsw
ers:
160
0 C
alo
ries
,
Sam
ple
an
swer
s: 1
0 ta
pes
, 9 C
Ds;
45
fat
gra
ms;
180
0 C
alo
ries
, 50
fat
0
tap
es, 1
6 C
Ds;
14
tap
es, 5
CD
sg
ram
s; 2
00
0 C
alo
ries
, 55
fat
gra
ms
30 25 20 15 10 5
Tap
es
Compact Discs
510
1520
2530
0
5.95
t + 8
.95c
= 1
50
t + c
= 2
0
Nec
klac
es
Bracelets
1020
3040
500
50 40 30 20 10
10b
+ 4
0n =
400
b +
2n
= 4
0
6-6
Exam
ple
037_
044_
ALG
1_A
_CR
M_C
06_C
R_6
6028
0.in
dd
3712
/21/
10
8:43
AM
Answers (Lesson 6-6)
A01_A21_ALG1_A_CRM_C06_AN_660280.indd A17A01_A21_ALG1_A_CRM_C06_AN_660280.indd A17 12/21/10 8:50 AM12/21/10 8:50 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
PDF 2nd
Chapter 6 A18 Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DAT
E
P
ER
IOD
Cha
pte
r 6
38
Gle
ncoe
Alg
ebra
1
Skill
s Pr
acti
ceS
yste
ms o
f In
eq
ualiti
es
Sol
ve e
ach
sys
tem
of
ineq
ual
itie
s b
y gr
aph
ing.
1. x
> -
1 2.
y >
2
3. y
> x
+ 3
y ≤
-3
x <
-2
y ≤
-1
x
y
O
x =
-1
y =
-3
x
y
O
y =
2
x =
-2
x
y
O
y =
x +
3
y =
-1
4. x
< 2
5.
x +
y ≤
-1
6. y
- x
> 4
y -
x ≤
2
x +
y ≥
3 Ø
x
+ y
> 2
x
y
O
y -
x =
2x
= 2
y
xO
y =
-x
+ 3
y =
-x
- 1
x
y
O
x +
y =
2
y -
x =
4
7. y
> x
+ 1
8.
y ≥
-x
+ 2
9.
y <
2x
+ 4
y ≥
-x
+ 1
y
< 2
x -
2
y ≥
x +
1
x
y
O
y =
-x
+ 1
y =
x +
1
x
y
O
y =
2x
- 2
y =
-x
+ 2
x
y
O
y =
x +
1
y =
2x
+ 4
Wri
te a
sys
tem
of
ineq
ual
itie
s fo
r ea
ch g
rap
h.
10.
x
y
O
11
.
x
y
O
12
.
x
y
O
y
≤ x
+ 2
, y ≥
x -
3
y >
-x, y
> x
y ≥
x +
1, y
< 1
6-6
037_
044_
ALG
1_A
_CR
M_C
06_C
R_6
6028
0.in
dd
3812
/24/
10
6:11
PM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DAT
E
P
ER
IOD
Lesson 6-6
Cha
pte
r 6
39
Gle
ncoe
Alg
ebra
1
Prac
tice
Syste
ms o
f In
eq
ualiti
es
$4 S
ton
es
Turq
uo
ise S
ton
es
$6 Stones
13
24
56
706 5 4 3 2 1
Gym
(h
ou
rs)
Die
go
’s R
ou
tin
e
Walking (miles)
13
24
56
78
0
16 14 12 10 8 6 4 2
6-6
Sol
ve e
ach
sys
tem
of
ineq
ual
itie
s b
y gr
aph
ing.
1. y
> x
- 2
2.
y ≥
x +
2
3. x
+ y
≥ 1
y ≤
x
y >
2x
+ 3
x
+ 2
y >
1
x
y
Oy
= x
y =
x -
2
x
y
O
y =
2x
+ 3
y =
x +
2
x
y
Ox
+ 2
y =
1
x +
y =
1
4. y
< 2
x -
1
5. y
> x
- 4
6.
2x
- y
≥ 2
y >
2 -
x
2x
+ y
≤ 2
x
- 2
y ≥
2
x
y
O
y =
2 -
x
y =
2x
- 1
x
y
O
2x +
y =
2
y =
x -
4
x
y
O
2x -
y =
2
x -
2y
= 2
7. F
ITN
ESS
Die
go s
tart
ed a
n e
xerc
ise
prog
ram
in
wh
ich
eac
h
wee
k h
e w
orks
ou
t at
th
e gy
m b
etw
een
4.5
an
d 6
hou
rs a
nd
wal
ks b
etw
een
9 a
nd
12 m
iles
.
a.
Mak
e a
grap
h t
o sh
ow t
he
nu
mbe
r of
hou
rs D
iego
wor
ks
out
at t
he
gym
an
d th
e n
um
ber
of m
iles
he
wal
ks p
er
wee
k.
b.
Lis
t th
ree
poss
ible
com
bin
atio
ns
of w
orki
ng
out
and
wal
kin
g th
at m
eet
Die
go’s
goa
ls.
8. S
OU
VEN
IRS
Em
ily
wan
ts t
o bu
y tu
rqu
oise
sto
nes
on
her
tr
ip t
o N
ew M
exic
o to
giv
e to
at
leas
t 4
of h
er f
rien
ds. T
he
gift
sh
op s
ells
sto
nes
for
eit
her
$4
or $
6 pe
r st
one.
Em
ily
has
n
o m
ore
than
$30
to
spen
d.
a. M
ake
a gr
aph
sh
owin
g th
e n
um
bers
of
each
pri
ce o
f st
one
Em
ily
can
pu
rch
ase.
b.
Lis
t th
ree
poss
ible
sol
uti
ons.
Sam
ple
an
swer
s:
gym
5 h
, wal
k 9
mi;
gym
6 h
, wal
k 10
mi,
gym
5.
5 h
, wal
k 11
mi
Sam
ple
an
swer
: o
ne
$4 s
ton
e an
d f
ou
r $6
sto
nes
; th
ree
$4
sto
nes
an
d t
hre
e $6
sto
nes
; fi
ve $
4 st
on
es a
nd
on
e $6
sto
ne
037_
044_
ALG
1_A
_CR
M_C
06_C
R_6
6028
0.in
dd
3912
/21/
10
8:43
AM
Answers (Lesson 6-6)
A01_A21_ALG1_A_CRM_C06_AN_660280.indd A18A01_A21_ALG1_A_CRM_C06_AN_660280.indd A18 12/24/10 6:23 PM12/24/10 6:23 PM
An
swer
s
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
PDF Pass
Chapter 6 A19 Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DAT
E
P
ER
IOD
Cha
pte
r 6
40
Gle
ncoe
Alg
ebra
1
Wor
d Pr
oble
m P
ract
ice
Syste
ms o
f In
eq
ualiti
es
1. P
ETS
Ren
ée’s
Pet
Sto
re n
ever
has
mor
e th
an a
com
bin
ed t
otal
of
20 c
ats
and
dogs
an
d n
ever
mor
e th
an 8
cat
s. T
his
is
repr
esen
ted
by t
he
ineq
ual
itie
s x
≤ 8
an
d x
+ y
≤ 2
0. S
olve
th
e sy
stem
of
ineq
ual
itie
s by
gra
phin
g.
2. W
AG
ES T
he
min
imu
m w
age
for
one
grou
p of
wor
kers
in
Tex
as i
s $7
.25
per
hou
r ef
fect
ive
Sep
t. 1
, 200
8. T
he
grap
h
belo
w s
how
s th
e po
ssib
le w
eekl
y w
ages
fo
r a
pers
on w
ho
mak
es a
t le
ast
min
imu
m w
ages
an
d w
orks
at
mos
t 40
hou
rs. W
rite
th
e sy
stem
of
ineq
ual
itie
s fo
r th
e gr
aph
.
3. F
UN
D R
AIS
ING
Th
e C
amp
Cou
rage
C
lub
plan
s to
sel
l ti
ns
of p
opco
rn a
nd
pean
uts
as
a fu
ndr
aise
r. T
he
Clu
b m
embe
rs h
ave
$900
to
spen
d on
pro
duct
s to
sel
l an
d w
ant
to o
rder
up
to 2
00 t
ins
in a
ll. T
hey
als
o w
ant
to o
rder
at
leas
t as
m
any
tin
s of
pop
corn
as
tin
s of
pea
nu
ts.
Eac
h t
in o
f po
pcor
n c
osts
$3
and
each
tin
of
pea
nu
ts c
osts
$4.
Wri
te a
sys
tem
of
equ
atio
ns
to r
epre
sen
t th
e co
ndi
tion
s of
th
is p
robl
em.
x +
y ≤
20
0; x
≥ y
;
3x +
4y ≤
90
0S
tud
ents
may
als
o in
clu
de
x ≥
0
and
y ≥
0.
4. B
USI
NES
S F
or m
axim
um
eff
icie
ncy
, a
fact
ory
mu
st h
ave
at l
east
100
wor
kers
, bu
t n
o m
ore
than
200
wor
kers
on
a s
hif
t.
Th
e fa
ctor
y al
so m
ust
man
ufa
ctu
re a
t le
ast
30 u
nit
s pe
r w
orke
r.
a. L
et x
be
the
nu
mbe
r of
wor
kers
an
d le
t y
be
the
nu
mbe
r of
un
its.
Wri
te
fou
r in
equ
alit
ies
expr
essi
ng
the
con
diti
ons
in t
he
prob
lem
giv
en
abov
e.
x ≤
20
0;
x ≥
10
0; y
≥ 3
00
0; y
≥ 3
0x
b.
Gra
ph t
he
syst
ems
of i
neq
ual
itie
s.
c. L
ist
at l
east
th
ree
poss
ible
sol
uti
ons.
S
amp
le a
nsw
er:
(110
, 341
0),
(150
, 510
0), (
180,
630
0)
O
Dogs
Cat
s
46 28101214161820y
64
210
1416
1820
812
x
x
y
(40,
290
)
255075100
Wages ($)
125
150
175
200
225
250
275
300
Ho
urs
510
1520
2530
3540
4550
OUnits (1000)
23 14567
Wo
rker
s
y
150
100
5025
035
020
030
040
0x
x ≤
40,
y ≥
7.2
5x
6-6
037_
044_
ALG
1_A
_CR
M_C
06_C
R_6
6028
0.in
dd
4012
/21/
10
8:43
AM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DAT
E
P
ER
IOD
Lesson 6-6
Cha
pte
r 6
41
Gle
ncoe
Alg
ebra
1
Enri
chm
ent
Descri
bin
g R
eg
ion
s
Th
e sh
aded
reg
ion
in
side
th
e tr
ian
gle
can
be
desc
ribe
d
wit
h a
sys
tem
of
thre
e in
equ
alit
ies.
y <
-x
+ 1
y >
1 −
3 x
- 3
y >
-9x
- 3
1
Wri
te s
yste
ms
of i
neq
ual
itie
s to
des
crib
e ea
ch r
egio
n. Y
ou m
ay
firs
t n
eed
to
div
ide
a re
gion
in
to t
rian
gles
or
qu
adri
late
rals
.
1.
x
y
O
2.
x
y
O
y
≤ 5 −
2 x
+ 1
2
y ≥
-x -
2
y ≤
x +
2
y
≤ -
5 −
2 x
+ 19
−
2
y ≥
x -
2
y ≤
-x +
2
y
≤ 2
y
≥ -
3
y ≤
5
y ≥
-5
3.
x
y
O
6-6
x
y
O
top
: y
≤ 3 −
2 x
+ 7
, y ≤
- 3 −
2 x
+ 7
, y ≥
4
mid
dle
: y ≤
4, y
≥ 0
, y ≥
- 4 −
3 x
- 4
,
y ≥
4 −
3 x
- 4
bo
tto
m le
ft:
y ≤
4x +
12,
y ≥
1 −
2 x
- 2
,y ≤
0, x
≤ 0
bo
tto
m r
igh
t: y
≤ -
4x +
12,
y ≥
- 1 −
2 x
- 2
, y ≤
0, x
≥ 0
037_
044_
ALG
1_A
_CR
M_C
06_C
R_6
6028
0.in
dd
4112
/21/
10
8:43
AM
Answers (Lesson 6-6)
A01_A21_ALG1_A_CRM_C06_AN_660280.indd A19A01_A21_ALG1_A_CRM_C06_AN_660280.indd A19 12/21/10 8:50 AM12/21/10 8:50 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
PDF Pass
Chapter 6 A20 Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DAT
E
P
ER
IOD
Cha
pte
r 6
42
G
lenc
oe A
lgeb
ra 1
Gra
phin
g Ca
lcul
ator
Act
ivit
yS
had
ing
Ab
so
lute
Valu
e I
neq
ualiti
es
Abs
olu
te v
alu
e in
equ
alit
ies
of t
he
form
y ≤
a|
x -
h
| +
k o
r y
≥ a|
x -
h|
+ k
ca
n b
e gr
aph
ed u
sin
g th
e S
HA
DE
com
man
d or
by
usi
ng
the
shad
ing
feat
ure
in
th
e Y
= sc
reen
.
G
rap
h y
≤
2|
x -
3|
+
2.
Met
hod
1
SH
AD
E c
omm
and
En
ter
the
bou
nda
ry e
quat
ion
y =
2 |
x -
3|
+
2
into
Y1.
Fro
m t
he
hom
e sc
reen
, use
SH
AD
E t
o sh
ade
the
regi
on u
nde
r th
e bo
un
dary
eq
uat
ion
bec
ause
th
e in
equ
alit
y u
ses
<. U
se Y
min
as
the
low
er
bou
nda
ry a
nd
the
bou
nda
ry e
quat
ion
, Y1,
as
the
upp
er b
oun
dary
.K
eyst
roke
s:
2nd
[D
RA
W]
7 V
AR
S E
NT
ER
4
,
VA
RS
E
NT
ER
E
NT
ER
) E
NT
ER
.
Met
hod
2 Y
=
On
th
e Y
= sc
reen
, mov
e th
e cu
rsor
to
the
far
left
an
d re
peat
edly
pr
ess
EN
TE
R t
o to
ggle
th
rou
gh t
he
grap
h c
hoi
ces.
Be
sure
to
sele
ct
the
opti
on t
o sh
ade
belo
w
. Th
en e
nte
r th
e bo
un
dary
equ
atio
n a
nd
grap
h.
Th
e sa
me
tech
niq
ues
can
be
use
d to
sol
ve a
sys
tem
of
abso
lute
va
lue
ineq
ual
itie
s.
Exer
cise
sS
olve
eac
h s
yste
m o
f in
equ
alit
ies
by
grap
hin
g.
1. y
≤
2 |
x -
4|
+ 1
2.
y <
1 −
2 |x
- 3|
+ 2
3.
y ≤
4|
x -
4|
+ 3
4.
y ≤
1 −
4 |x
+ 1
| +
1
y
> |
x +
3|
+ 6
y
≤ |
x +
1|
+ 4
y
≥ 3
|x -
4|
- 2
y
≥ 2
|x +
3|
+ 5
S
olve
th
e sy
stem
y ≥
|x
+ 5
| -
4
and
y ≤
3|
x -
6|
+ 1.
En
ter
y =
|x
+ 5|
- 4
in
Y1
and
set
the
grap
h t
o sh
ade
abov
e.
The
n en
ter
y =
3 |
x -
6|
+ 1
in Y
2 an
d se
t th
e gr
aph
to s
hade
bel
ow.
Gra
ph t
he in
equa
litie
s. N
otic
e th
at a
dif
fere
nt p
atte
rn is
use
d fo
r th
e se
cond
ineq
ualit
y. T
he r
egio
n w
here
the
tw
o pa
tter
ns o
verl
ap is
the
so
luti
on t
o th
e sy
stem
.
[-10, 10]
scl:
1 b
y [
-10, 10]
scl:
1
[-10, 10]
scl:
1 b
y [
-10, 10]
scl:
1
[-20, 20]
scl:
2 b
y [
-20, 20]
scl:
2
[-3
0, 3
0]
scl:
5 b
y [
-10
, 5
0]
scl:
5[-
10
, 10
] s
cl:
1 b
y [
-10
, 10
] s
cl:
1[-
5, 15
] s
cl:
1 b
y [
-5
, 15
] s
cl:
1[-
10
, 10
] s
cl:
1 b
y [
-10
, 10
] s
cl:
1
6-6
Exam
ple
1
Exam
ple
2
037_
044_
ALG
1_A
_CR
M_C
06_C
R_6
6028
0.in
dd
4212
/21/
10
8:43
AM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DAT
E
P
ER
IOD
Lesson 6-6
Cha
pte
r 6
43
G
lenc
oe A
lgeb
ra 1
6-6
Spre
adsh
eet
Act
ivit
yS
yste
ms o
f In
eq
ualiti
es
Exer
cise
sU
se t
he
spre
adsh
eet
to d
eter
min
e w
het
her
Top
Sp
ort
can
mak
e th
e fo
llow
ing
com
bin
atio
ns
of s
hoe
s.
1. 1
000
Ru
nn
ers,
110
0 F
lyer
s yes
2. 1
200
Ru
nn
ers,
100
0 F
lyer
s yes
3. 2
000
Ru
nn
ers,
500
Fly
ers
no
4.
300
Ru
nn
ers,
150
0 F
lyer
s yes
4. 9
50 R
un
ner
s, 1
400
Fly
ers
yes
5.
180
0 R
un
ner
s, 1
300
Fly
ers
no
6. I
f Top
Spo
rt c
an e
ith
er b
uy
anot
her
cu
ttin
g m
ach
ine
or h
ire
mor
e as
sem
bler
s, w
hic
h w
ould
mak
e m
ore
com
bin
atio
ns
of s
hoe
pro
duct
ion
po
ssib
le?
Exp
lain
. H
ire m
ore
assem
ble
rs. E
ach
ord
ere
d p
air
th
at
was n
ot
a s
olu
tio
n o
f th
e
syste
m w
as t
rue f
or
the fi
rst
ineq
uality
. C
han
gin
g t
he s
eco
nd
in
eq
uality
m
ay m
ake s
om
e o
f th
ose o
rdere
d p
air
s s
olu
tio
ns o
f th
e s
yste
m.
T
opS
por
t S
hoe
Com
pan
y h
as a
tot
al o
f 96
00 m
inu
tes
of m
ach
ine
tim
e ea
ch w
eek
to
cut
the
mat
eria
ls f
or t
he
two
typ
es o
f at
hle
tic
shoe
s th
ey m
ake.
Th
ere
are
a to
tal
of 2
8,00
0 m
inu
tes
of w
ork
er
tim
e p
er w
eek
for
ass
emb
ly. I
t ta
kes
3 m
inu
tes
to c
ut
and
12
min
ute
s to
ass
emb
le a
pai
r of
Ru
nn
ers
and
2 m
inu
tes
to c
ut
and
10
min
ute
s to
as
sem
ble
a p
air
of F
lyer
s. I
s it
pos
sib
le f
or t
he
com
pan
y to
mak
e 12
00 p
airs
of
Ru
nn
ers
and
140
0 p
airs
of
Fly
ers
in a
wee
k?
Ste
p 1
R
epre
sent
the
sit
uati
on u
sing
a s
yste
m o
f in
equa
litie
s. L
et r
rep
rese
nt
the
nu
mbe
r of
Ru
nn
ers
and
f re
pres
ent
the
nu
mbe
r of
Fly
ers.
3r +
2f
≤
96
00
12
r +
10
f ≤ 28
,000
Ste
p 2
C
olu
mn
s A
an
d B
con
tain
th
e va
lues
of
r an
d f.
Col
um
ns
C a
nd
D
con
tain
th
e fo
rmu
las
for
the
ineq
ual
itie
s. T
he
form
ula
s w
ill
retu
rn
TR
UE
or
FA
LS
E. I
f bo
th i
neq
ual
itie
s ar
e tr
ue
for
an o
rder
ed p
air,
then
th
e or
dere
d pa
ir i
s a
solu
tion
to
the
syst
em o
f in
equ
alit
ies.
Sin
ce o
ne
of t
he
ineq
ual
itie
s is
fal
se f
or (
1200
, 140
0), t
he
orde
red
pair
is n
ot p
art
of t
he
solu
tion
set
. The
com
pany
ca
nn
ot m
ake
1200
pai
rs o
f R
un
ner
s an
d 14
00 p
airs
of
Fly
ers
in a
wee
k.Th
e fo
rmul
a in
cel
l D2
is12
*A2
+ 1
0*B
2 <
= 2
8,00
0.
+≤
+≤
Exam
ple
037_
044_
ALG
1_A
_CR
M_C
06_C
R_6
6028
0.in
dd
4312
/21/
10
8:43
AM
Answers (Lesson 6-6)
A01_A21_ALG1_A_CRM_C06_AN_660280.indd A20A01_A21_ALG1_A_CRM_C06_AN_660280.indd A20 12/21/10 8:50 AM12/21/10 8:50 AM
A01_A21_ALG1_A_CRM_C06_AN_660280.indd A21A01_A21_ALG1_A_CRM_C06_AN_660280.indd A21 12/21/10 8:50 AM12/21/10 8:50 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
PDF Pass
Chapter 6 A22 Glencoe Algebra 1
Chapter 6 Assessment Answer KeyQuiz 1 (Lessons 6-1 and 6-2) Quiz 3 (Lessons 6-5) Mid-Chapter Test
Page 47 Page 48 Page 49
Quiz 2 (Lessons 6-3 and 6-4)
Page 47
Quiz 4 (Lessons 6-6)
Page 48
1.
2.
3.
4.
5.
y
x
x - 2y = 3
x - 2y = -2
y
x
y = -x + 5
y = x32
1.
2.
3.
4.
5.
(1, 5)
infi nitely many solutions
A
one solution; (2, 3)
no solution
(
5 1 −
2 , -1 1 −
2 )
(-1 1 − 2 , 2)
(2, -3)
(2, -1)
D
7.
8.
9.
10.
11.
12.
y
x
x + 3y = 9
x - 3y = -3
(-9, -5)
(1, -4)
(
3 3 −
5 , 3 −
5 )
(-1, -4)
34 games
one solution; (3, 2)
y
x
x + 3y = 9
x - 3y = -3
1. B
2. A
3. F
4. C
5. G
6. D
1.
2.
3.
x
y
y
x
D
1.
2.
3.
4.
5.
substitution; (5, 2)
elimination (×); (2, 3)
elimination (-); (-2, 2)
elimination
(+); (
- 1 −
4 , 1
)
D
A22_A34_ALG1_A_CRM_C06_AN_660280.indd A22A22_A34_ALG1_A_CRM_C06_AN_660280.indd A22 12/21/10 8:49 AM12/21/10 8:49 AM
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
An
swer
s
PDF Pass
Chapter 6 A23 Glencoe Algebra 1
Chapter 6 Assessment Answer KeyVocabulary Test Form 1Page 50 Page 51 Page 52
1.
2.
3.
4.
5.
6.
Sample answer:
A system of equations
that has exactly one
solution is called
independent.
system of equations
consistent
dependent
eliminationSample answer:
Substitution is a
method of solving a
system of equations
where one variable is
solved in terms of the
other variable.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11. D
H
B
G
C
G
A
H
C
F
G
J
C
A
J
AB 12.
13.
14.
15.
16.
17.
B: substitution; (0, 0)
A22_A34_ALG1_A_CRM_C06_AN_660280.indd A23A22_A34_ALG1_A_CRM_C06_AN_660280.indd A23 12/21/10 8:49 AM12/21/10 8:49 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
PDF Pass
Chapter 6 A24 Glencoe Algebra 1
Chapter 6 Assessment Answer KeyForm 2A Form 2BPage 53 Page 54 Page 55 Page 56
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11. J
B
H
A
H
C
F
A
J
B
A
12.
13.
14.
15.
16.
17. J
J
C
F
A
(
-7, -3 1 −
2 )
B:
D
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11. H
C
F
D
H
A
J
H
B
B
C
12.
13.
14.
15.
16.
17.
C
G
A
J
C
F
Manuel is 15 years old; his sister is 7 years old.
B:
A22_A34_ALG1_A_CRM_C06_AN_660280.indd A24A22_A34_ALG1_A_CRM_C06_AN_660280.indd A24 12/21/10 8:49 AM12/21/10 8:49 AM
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
An
swer
s
PDF Pass
Chapter 6 A25 Glencoe Algebra 1
Chapter 6 Assessment Answer KeyForm 2CPage 57 Page 58
y
xO
2x - y = -3
6x - 3y = -9
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
(7, -1)
(-3, 2)
(4, -1)
(-8, 0)
(3, 5)
(1, 3)
y
xO
y = -x + 4
y = x - 4
one solution
no solution
one solution; (4, 0)
infi nitely many solutions
elimination (-); (2, -7)
substitution; (3, 2)
14 dimes; 19 nickels
186 student tickets;
135 adult tickets
23 and -6
16.
x
y
17.
x
y
18.
x = test score, y = family
income; x ≥ 75, y ≤ 40,000 ;
sample answer: (80, 38,000)
Mavis is 15 years old;
her brother is 10 years
old B:
A22_A34_ALG1_A_CRM_C06_AN_660280.indd A25A22_A34_ALG1_A_CRM_C06_AN_660280.indd A25 12/21/10 8:49 AM12/21/10 8:49 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
PDF Pass
Chapter 6 A26 Glencoe Algebra 1
Chapter 6 Assessment Answer KeyForm 2DPage 59 Page 60
1.
2. 3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
(-2, 5)
(-1, 4)
(2, 2)
(5, 3)
infi nitely many solutions
(2, 1)
(2, 4)
y
x
4x - 2y = 10
2x - y = 5
y
x
y = x - 3
y = -x + 3
one solution
one solution; (3, 0)
infi nitely many solutions
18 and -2
substitution; (-2, -5)
elimination (-);
(-1, 1)
16 dimes;
7 quarters
34 hours
16.
x
y
17.
x
y
18.
x = credit score, y = cost
of car; x ≥ 600, y ≥
5000; sample answer:
(650, 8000)
(-1, -3) B:
A22_A34_ALG1_A_CRM_C06_AN_660280.indd A26A22_A34_ALG1_A_CRM_C06_AN_660280.indd A26 12/21/10 8:49 AM12/21/10 8:49 AM
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
An
swer
s
PDF 2nd
Chapter 6 A27 Glencoe Algebra 1
Chapter 6 Assessment Answer KeyForm 3Page 61 Page 62
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
(-32, 15)
(-2, -6)
(-14, 7)
no solution
(1, 2)
no solution
(4, 1)
y
x
y = x 13
y + x + 4 = 0
one solution; (-1, -3)
no solution
y
x
3y = -x + 9
x + 3y = 3
substitution; (63, 84)
6, 10
3 1 −
2 , 8 1 −
2
10 lb of $2.45 mix; 20 lb of $2.30 mix
(3, - 3 −
7 )
16.
17.
x
y
18.
x
y
19.
B:
5 mph
x = women’s shoes
sold; y = men’s shoes
sold; x ≥ 10, y ≥ 5,
2.5x + 3y ≥ 60; sample
answer: (15, 10)
10 square unitselimination (×);
(-
1 −
2 , 1 −
2 )
A22_A34_ALG1_A_CRM_C06_AN_660280.indd A27A22_A34_ALG1_A_CRM_C06_AN_660280.indd A27 12/24/10 7:28 AM12/24/10 7:28 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
PDF Pass
Chapter 6 A28 Glencoe Algebra 1
Chapter 6 Assessment Answer Key Page 63, Extended-Response Test Scoring Rubric
Score General Description Specifi c Criteria
4 Superior
A correct solution that
is supported by well-
developed, accurate
explanations
• Shows thorough understanding of the concepts of solving systems of equations.
• Uses appropriate strategies to solve problems.
• Computations are correct.
• Written explanations are exemplary.
• raphs are accurate and appropriate.
• Goes beyond requirements of some or all problems.
3 Satisfactory
A generally correct solution,
but may contain minor fl aws in
reasoning or computation
• Shows an understanding of the concepts of solving systems of equations.
• Uses appropriate strategies to solve problems.
• Computations are mostly correct.
• Written explanations are effective.
• Graphs are mostly accurate and appropriate.
• Satisfi es all requirements of problems.
2 Nearly Satisfactory
A partially correct
interpretation and/or solution
to the problem
• Shows an understanding of most of the concepts of solving systems of equations.
• May not use appropriate strategies to solve problems.
• Computations are mostly correct.
• Written explanations are satisfactory.
• Graphs are mostly accurate.
• Satisfi es the requirements of most of the problems.
1 Nearly Unsatisfactory
A correct solution with no
supporting evidence or
explanation
• Final computation is correct.
• No written explanations or work is shown to substantiate
the fi nal computation.
• Graphs may be accurate but lack detail or explanation.
• Satisfi es minimal requirements of some of the problems.
0 Unsatisfactory
An incorrect solution indicating
no mathematical
understanding of the concept
or task, or no solution is given
• Shows little or no understanding of most of the concepts of
solving systems of equations.• Does not use appropriate strategies to solve problems.
• Computations are incorrect.
• Written explanations are unsatisfactory.
• Graphs are inaccurate or inappropriate.
• Does not satisfy requirements of problems.
• No answer may be given.
A22_A34_ALG1_A_CRM_C06_AN_660280.indd A28A22_A34_ALG1_A_CRM_C06_AN_660280.indd A28 12/21/10 8:49 AM12/21/10 8:49 AM
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
An
swer
s
PDF Pass
Chapter 6 A29 Glencoe Algebra 1
Chapter 6 Assessment Answer Key Page 63, Extended-Response Test Sample Answers
1a. Sample answer: (5000, 5000); $600
1b. (15,000, -5000); To represent a possible investment both x and y must be positive. Thus, there is no interpretation for a negative value for y.
2a. The student should recognize that the total cost of a one-day rental of a compact car from the car rental company is modeled by the equation y = A + Bx, where y is the total cost and x is the number of miles driven during the day. Likewise, the total cost of a one-day rental of a compact car from the leading competitor is modeled by the equation y = 15 + 0.25x. The solution to the system is the number of miles driven during the day that makes the cost of renting a compact car from the two companies for one day equal, and the corresponding total cost.
2b. The value of A determines if the one-day fee for the rental of the compact car from the car rental company will be less than, greater than, or equal to the one-day fee from their leading competitor. The value of B determines whether the number of miles driven will keep the total cost less than, greater than, or equal to the total cost of renting from the leading competitor, or change which company has the higher total cost for the rental.
3a. x + y = S2.5x + 0.75y = PBoth S and P must be positive. The student may recognize that the maximum profit is 2.5 times the weekly sales, and the minimum profit is 0.75 times the weekly sales. i.e., 0.75S ≤ P ≤ 2.5S.
3b. Sample answer:
The graph of this system of equations shows the ordered pair that corresponds to weekly sales of 250 and a weekly profit of $450. In order to have a profit of $450 from magazine and book sales, the store would have to sell 150 books and 100 magazines.
4a. a + b = 21a - 5 b = 3
4b. 17 adults; 4 babies
Mag
azin
es
100
0
200
400
600
Books100 200 300 400
y
x
2.5x + .75y = 450
x + y = 250
In addition to the scoring rubric found on page A28, the following sample answers may be used as guidance in evaluating extended-response assessment items.
A22_A34_ALG1_A_CRM_C06_AN_660280.indd A29A22_A34_ALG1_A_CRM_C06_AN_660280.indd A29 12/21/10 8:49 AM12/21/10 8:49 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
PDF Pass
Chapter 6 A30 Glencoe Algebra 1
Chapter 6 Assessment Answer KeyStandardized Test PracticePage 64 Page 65
1. A B C D
2.
3. A B C D
4. F G H J
5. A B C D
6. F G H J
7. A B C D
8.
9. A B C D
10.
F G H J
F G H J
F G H J
11. A B C D
12. F G H J
13. A B C D
14. F G H J
15. A B C D
16. F G H J
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
2 5
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
8 17. 18.
A22_A34_ALG1_A_CRM_C06_AN_660280.indd A30A22_A34_ALG1_A_CRM_C06_AN_660280.indd A30 12/21/10 8:49 AM12/21/10 8:49 AM
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
An
swer
s
PDF Pass
Chapter 6 A31 Glencoe Algebra 1
Chapter 6 Assessment Answer KeyStandardized Test PracticePage 66
19.
20.
21.
22.
23.
24.
25.
26.
27.
28a.
28b. -12, 4
Sample answer: Let
x = fi rst number
and y = second number;
3x - y = -40
x + 2y = -4
y
xOx - 2y = 4
2x - y = 4
-1-2-3-4 0 1 2 43
{r | r < 3}
d = 55t ; 275 miles
yes; common difference is 3
-2
3
2
10
-1-2
decrease; 25%
- 8 −
7
Let n = the number; 2n - 7 > 13 or 2n - 7 ≤ -5; {n | n > 10 or n ≤ 1}
y = 3x + 5
A22_A34_ALG1_A_CRM_C06_AN_660280.indd A31A22_A34_ALG1_A_CRM_C06_AN_660280.indd A31 12/21/10 8:49 AM12/21/10 8:49 AM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
PDF Pass
Chapter 6 A32 Glencoe Algebra 1
Chapter 6 Assessment Answer KeyUnit 2 TestPage 67 Page 68
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
y = -3x - 6
3x + 2y = -10
y = 5x - 2
y = 2x + 9
d = 40t
-41
y
xO
no solution
0
40
0
80
120
160
1 2 3 4
Dis
tan
ce
(m
ile
s)
Time (hours)
positive; sample
answer: y = 130 −
3 x + 20
−
3
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23. y
xO
-3-2-1 0 1 2 4 53
no solution
{w | w < -6 or w > -4}
elimination (×); (3, -5)
elimination (-); (10, -4)
elimination (+); (3, -2)
substitution; (1, 1)
{v | v < -3 or v > 11}
{t | -1 < t < 2}
{a | a ≤ 8}
{x | x > -5}
-7-6-5-4-8 -3-2-1 0
about $2.25
{
b | 2 −
3 ≤ b ≤ 4
}
A22_A34_ALG1_A_CRM_C06_AN_660280.indd A32A22_A34_ALG1_A_CRM_C06_AN_660280.indd A32 12/21/10 8:49 AM12/21/10 8:49 AM