Chapter 3 Resource Masters - Wikispaces · PDF file©Glencoe/McGraw-Hill iv Glencoe...
Transcript of Chapter 3 Resource Masters - Wikispaces · PDF file©Glencoe/McGraw-Hill iv Glencoe...
Chapter 3Resource Masters
Consumable WorkbooksMany of the worksheets contained in the Chapter Resource Masters bookletsare available as consumable workbooks.
Study Guide and Intervention Workbook 0-07-828029-XSkills Practice Workbook 0-07-828023-0Practice Workbook 0-07-828024-9
ANSWERS FOR WORKBOOKS The answers for Chapter 3 of these workbookscan be found in the back of this Chapter Resource Masters booklet.
Copyright © by The McGraw-Hill Companies, Inc. All rights reserved.Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teacher, and families without charge; and be used solely in conjunction with Glencoe’s Algebra 2. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher.
Send all inquiries to:The McGraw-Hill Companies8787 Orion PlaceColumbus, OH 43240-4027
ISBN: 0-07-828006-0 Algebra 2Chapter 3 Resource Masters
2 3 4 5 6 7 8 9 10 066 11 10 09 08 07 06 05 04 03
Glencoe/McGraw-Hill
© Glencoe/McGraw-Hill iii Glencoe Algebra 2
Contents
Vocabulary Builder . . . . . . . . . . . . . . . . vii
Lesson 3-1Study Guide and Intervention . . . . . . . . 119–120Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 121Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 122Reading to Learn Mathematics . . . . . . . . . . 123Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 124
Lesson 3-2Study Guide and Intervention . . . . . . . . 125–126Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 127Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 128Reading to Learn Mathematics . . . . . . . . . . 129Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 130
Lesson 3-3Study Guide and Intervention . . . . . . . . 131–132Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 133Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 134Reading to Learn Mathematics . . . . . . . . . . 135Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 136
Lesson 3-4Study Guide and Intervention . . . . . . . . 137–138Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 139Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 140Reading to Learn Mathematics . . . . . . . . . . 141Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 142
Lesson 3-5Study Guide and Intervention . . . . . . . . 143–144Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 145Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 146Reading to Learn Mathematics . . . . . . . . . . 147Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 148
Chapter 3 AssessmentChapter 3 Test, Form 1 . . . . . . . . . . . . 149–150Chapter 3 Test, Form 2A . . . . . . . . . . . 151–152Chapter 3 Test, Form 2B . . . . . . . . . . . 153–154Chapter 3 Test, Form 2C . . . . . . . . . . . 155–156Chapter 3 Test, Form 2D . . . . . . . . . . . 157–158Chapter 3 Test, Form 3 . . . . . . . . . . . . 159–160Chapter 3 Open-Ended Assessment . . . . . . 161Chapter 3 Vocabulary Test/Review . . . . . . . 162Chapter 3 Quizzes 1 & 2 . . . . . . . . . . . . . . . 163Chapter 3 Quizzes 3 & 4 . . . . . . . . . . . . . . . 164Chapter 3 Mid-Chapter Test . . . . . . . . . . . . 165Chapter 3 Cumulative Review . . . . . . . . . . . 166Chapter 3 Standardized Test Practice . . 167–168
Standardized Test Practice Student Recording Sheet . . . . . . . . . . . . . . A1
ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A26
© Glencoe/McGraw-Hill iv Glencoe Algebra 2
Teacher’s Guide to Using theChapter 3 Resource Masters
The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 3 Resource Masters includes the core materials neededfor Chapter 3. 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 and printing in theAlgebra 2 TeacherWorks CD-ROM.
Vocabulary Builder Pages vii–viiiinclude a student study tool that presentsup to twenty of the key vocabulary termsfrom the chapter. Students are to recorddefinitions and/or examples for each term.You may suggest that students highlight orstar the terms with which they are notfamiliar.
WHEN TO USE Give these pages tostudents before beginning Lesson 3-1.Encourage them to add these pages to theirAlgebra 2 Study Notebook. Remind them to add definitions and examples as theycomplete each lesson.
Study Guide and InterventionEach lesson in Algebra 2 addresses twoobjectives. There is one Study Guide andIntervention master for each objective.
WHEN TO USE Use these masters asreteaching activities for students who needadditional reinforcement. These pages canalso be used in conjunction with the StudentEdition as an instructional tool for studentswho have been absent.
Skills Practice There is one master foreach lesson. These provide computationalpractice at a basic level.
WHEN TO USE These masters can be used with students who have weakermathematics backgrounds or needadditional reinforcement.
Practice There is one master for eachlesson. These problems more closely followthe structure of the Practice and Applysection of the Student Edition exercises.These exercises are of average difficulty.
WHEN TO USE These provide additionalpractice options or may be used ashomework for second day teaching of thelesson.
Reading to Learn MathematicsOne master is included for each lesson. Thefirst section of each master asks questionsabout the opening paragraph of the lessonin the Student Edition. Additionalquestions ask students to interpret thecontext of and relationships among termsin the lesson. Finally, students are asked tosummarize what they have learned usingvarious representation techniques.
WHEN TO USE This master can be usedas a study tool when presenting the lessonor as an informal reading assessment afterpresenting the lesson. It is also a helpfultool for ELL (English Language Learner)students.
Enrichment There is one extensionmaster for each lesson. These activities mayextend the concepts in the lesson, offer anhistorical or multicultural look at theconcepts, or widen students’ perspectives onthe mathematics they are learning. Theseare not written exclusively for honorsstudents, but are accessible for use with alllevels of students.
WHEN TO USE These may be used asextra credit, short-term projects, or asactivities for days when class periods areshortened.
© Glencoe/McGraw-Hill v Glencoe Algebra 2
Assessment OptionsThe assessment masters in the Chapter 3Resource Masters offer a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.
Chapter Assessment CHAPTER TESTS• Form 1 contains multiple-choice questions
and is intended for use with basic levelstudents.
• Forms 2A and 2B contain multiple-choicequestions aimed at the average levelstudent. These tests are similar in formatto offer comparable testing situations.
• Forms 2C and 2D are composed of free-response questions aimed at the averagelevel student. These tests are similar informat to offer comparable testingsituations. Grids with axes are providedfor questions assessing graphing skills.
• Form 3 is an advanced level test withfree-response questions. Grids withoutaxes are provided for questions assessinggraphing skills.
All of the above tests include a free-response Bonus question.
• The Open-Ended Assessment includesperformance assessment tasks that aresuitable for all students. A scoring rubricis included for evaluation guidelines.Sample answers are provided forassessment.
• A Vocabulary Test, suitable for allstudents, includes a list of the vocabularywords in the chapter and ten questionsassessing students’ knowledge of thoseterms. This can also be used in conjunc-tion with one of the chapter tests or as areview worksheet.
Intermediate Assessment• Four free-response quizzes are included
to offer assessment at appropriateintervals in the chapter.
• A Mid-Chapter Test provides an optionto assess the first half of the chapter. It iscomposed of both multiple-choice andfree-response questions.
Continuing Assessment• The Cumulative Review provides
students an opportunity to reinforce andretain skills as they proceed throughtheir study of Algebra 2. It can also beused as a test. This master includes free-response questions.
• The Standardized Test Practice offerscontinuing review of algebra concepts invarious formats, which may appear onthe standardized tests that they mayencounter. This practice includes multiple-choice, grid-in, and quantitative-comparison questions. Bubble-in andgrid-in answer sections are provided onthe master.
Answers• Page A1 is an answer sheet for the
Standardized Test Practice questionsthat appear in the Student Edition onpages 150–151. This improves students’familiarity with the answer formats theymay encounter in test taking.
• The answers for the lesson-by-lessonmasters are provided as reduced pageswith answers appearing in red.
• Full-size answer keys are provided forthe assessment masters in this booklet.
Reading to Learn MathematicsVocabulary Builder
NAME ______________________________________________ DATE ____________ PERIOD _____
33
Voca
bula
ry B
uild
erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 3.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 toyour Algebra Study Notebook to review vocabulary at the end of the chapter.
Vocabulary Term Found on Page Definition/Description/Example
bounded region
consistent system
constraints
kuhn·STRAYNTS
dependent system
elimination method
feasible region
FEE·zuh·buhl
inconsistent system
ihn·kuhn·SIHS·tuhnt
independent system
(continued on the next page)
© Glencoe/McGraw-Hill vii Glencoe Algebra 2
© Glencoe/McGraw-Hill viii Glencoe Algebra 2
Vocabulary Term Found on Page Definition/Description/Example
linear programming
ordered triple
substitution method
system of equations
system of inequalities
unbounded region
vertices
Reading to Learn MathematicsVocabulary Builder (continued)
NAME ______________________________________________ DATE ____________ PERIOD _____
33
Study Guide and InterventionSolving Systems of Equations by Graphing
NAME ______________________________________________ DATE ____________ PERIOD _____
3-13-1
© Glencoe/McGraw-Hill 119 Glencoe Algebra 2
Less
on
3-1
Graph Systems of Equations A system of equations is a set of two or moreequations containing the same variables. You can solve a system of linear equations bygraphing the equations on the same coordinate plane. If the lines intersect, the solution isthat intersection point.
Solve the system of equations by graphing. x � 2y � 4x � y � �2
Write each equation in slope-intercept form.
x � 2y � 4 → y � � 2
x � y � �2 → y � �x � 2
The graphs appear to intersect at (0, �2).
CHECK Substitute the coordinates into each equation.x � 2y � 4 x � y � �2
0 � 2(�2) � 4 0 � (�2) � �24 � 4 ✓ �2 � �2 ✓
The solution of the system is (0, �2).
Solve each system of equations by graphing.
1. y � � � 1 2. y � 2x � 2 3. y � � � 3
y � � 4 (6, �1) y � �x � 4 (2, 2) y � (4, 1)
4. 3x � y � 0 5. 2x � � �7 6. � y � 2
x � y � �2 (1, 3) � y � 1 (�4, 3) 2x � y � �1 (�2, �3)
x
y
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ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 120 Glencoe Algebra 2
Classify Systems of Equations The following chart summarizes the possibilities forgraphs of two linear equations in two variables.
Graphs of Equations Slopes of Lines Classification of System Number of Solutions
Lines intersect Different slopes Consistent and independent One
Lines coincide (same line)Same slope, same
Consistent and dependent Infinitely many y-intercept
Lines are parallelSame slope, different
Inconsistent None y-intercepts
Graph the system of equations x � 3y � 6and describe it as consistent and independent, 2x � y � �3consistent and dependent, or inconsistent.
Write each equation in slope-intercept form.
x � 3y � 6 → y � x � 2
2x � y � �3 → y � 2x � 3
The graphs intersect at (�3, �3). Since there is one solution, the system is consistent and independent.
Graph the system of equations and describe it as consistent and independent,consistent and dependent, or inconsistent.
1. 3x � y � �2 2. x � 2y � 5 3. 2x � 3y � 06x � 2y � 10 3x � 15 � �6y consistent 4x � 6y � 3inconsistent and dependent inconsistent
4. 2x � y � 3 5. 4x � y � �2 6. 3x � y � 2
x � 2y � 4 consistent 2x � � �1 consistent x � y � 6 consistent
and independent and dependent and independent
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Study Guide and Intervention (continued)
Solving Systems of Equations by Graphing
NAME ______________________________________________ DATE ____________ PERIOD _____
3-13-1
ExercisesExercises
ExampleExample
Skills PracticeSolving Systems of Equations By Graphing
NAME ______________________________________________ DATE ____________ PERIOD _____
3-13-1
© Glencoe/McGraw-Hill 121 Glencoe Algebra 2
Less
on
3-1
Solve each system of equations by graphing.
1. x � 2 2. y � �3x � 6 3. y � 4 � 3x
y � 0 (2, 0) y � 2x � 4 (2, 0) y � � x � 1 (2, �2)
4. y � 4 � x 5. y � �2x � 2 6. y � x
y � x � 2 (3, 1) y � x � 5 (3, �4) y � �3x � 4 (1, 1)
7. x � y � 3 8. x � y � 4 9. 3x � 2y � 4x � y � 1 (2, 1) 2x � 5y � 8 (4, 0) 2x � y � 1 (�2, �5)
Graph each system of equations and describe it as consistent and independent,consistent and dependent, or inconsistent.
10. y � �3x 11. y � x � 5 12. 2x � 5y � 10y � �3x � 2 �2x � 2y � �10 3x � y � 15
inconsistent consistent and consistent and dependent independent
2
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1�2
© Glencoe/McGraw-Hill 122 Glencoe Algebra 2
Solve each system of equations by graphing.
1. x � 2y � 0 2. x � 2y � 4 3. 2x � y � 3
y � 2x � 3 (2, 1) 2x � 3y � 1 (2, 1) y � x � (3, �3)
4. y � x � 3 5. 2x � y � 6 6. 5x � y � 4y � 1 (�2, 1) x � 2y � �2 (2, �2) �2x � 6y � 4 (1, 1)
Graph each system of equations and describe it as consistent and independent,consistent and dependent, or inconsistent.
7. 2x � y � 4 8. y � �x � 2 9. 2y � 8 � x
x � y � 2 x � y � �4 y � x � 4
consistent and indep. inconsistent consistent and dep.
SOFTWARE For Exercises 10–12, use the following information.Location Mapping needs new software. Software A costs $13,000 plus $500 per additionalsite license. Software B costs $2500 plus $1200 per additional site license.
10. Write two equations that represent the cost of each software. A: y � 13,000 � 500x, B: y � 2500 � 1200x
11. Graph the equations. Estimate the break-even point of the software costs.15 additional licenses
12. If Location Mapping plans to buy 10 additional site licenses, which software will cost less? B
Software Costs
Additional Licenses
Tota
l Co
st (
$)
40 8 12 14 16 18 202 6 10
24,000
20,000
16,000
12,000
8,000
4,000
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Practice (Average)
Solving Systems of Equations By Graphing
NAME ______________________________________________ DATE ____________ PERIOD _____
3-13-1
Reading to Learn MathematicsSolving Systems of Equations by Graphing
NAME ______________________________________________ DATE ____________ PERIOD _____
3-13-1
© Glencoe/McGraw-Hill 123 Glencoe Algebra 2
Less
on
3-1
Pre-Activity How can a system of equations be used to predict sales?
Read the introduction to Lesson 3-1 at the top of page 110 in your textbook.
• Which are growing faster, in-store sales or online sales? online sales
• In what year will the in-store and online sales be the same? 2005
Reading the Lesson
1. The Study Tip on page 110 of your textbook says that when you solve a system ofequations by graphing and find a point of intersection of the two lines, you must alwayscheck the ordered pair in both of the original equations. Why is it not good enough tocheck the ordered pair in just one of the equations?Sample answer: To be a solution of the system, the ordered pair mustmake both of the equations true.
2. Under each system graphed below, write all of the following words that apply: consistent,inconsistent, dependent, and independent.
inconsistent consistent; consistent; dependent independent
Helping You Remember
3. Look up the words consistent and inconsistent in a dictionary. How can the meaning ofthese words help you distinguish between consistent and inconsistent systems ofequations?
Sample answer: One meaning of consistent is “being in agreement,” or“compatible,” while one meaning of inconsistent is “not being inagreement” or “incompatible.” When a system is consistent, theequations are compatible because both can be true at the same time (forthe same values of x and y). When a system is inconsistent, theequations are incompatible because they can never be true at the same time.
x
y
Ox
y
Ox
y
O
© Glencoe/McGraw-Hill 124 Glencoe Algebra 2
InvestmentsThe following graph shows the value of two different investments over time.Line A represents an initial investment of $30,000 with a bank paying passbook savings interest. Line B represents an initial investment of $5,000 in a profitable mutual fund with dividends reinvested and capital gains accepted in shares. By deriving the linear equation y � mx � b for A and B, you can predict the value of these investments for years to come.
1. The y-intercept, b, is the initial investment. Find b for each of the following.a. line A 30,000 b. line B 5000
2. The slope of the line, m, is the rate of return. Find m for each of the following.a. line A 0.588 b. line B 2.73
3. What are the equations of each of the following lines?a. line A y � 0.588x � 30 b. line B y � 2.73x � 5
4. What will be the value of the mutual fund after 11 years of investment? $35,030
5. What will be the value of the bank account after 11 years of investment? $36,468
6. When will the mutual fund and the bank account have equal value?after 11.67 years of investment
7. Which investment has the greatest payoff after 11 years of investment? the mutual fund
A
B25
20
30
35
10
5
0
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1 2 3 4 5 6 7 8 9
Amou
nt In
vest
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hous
ands
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Enrichment
NAME ______________________________________________ DATE______________ PERIOD _____
3-13-1
Study Guide and InterventionSolving Systems of Equations Algebraically
NAME ______________________________________________ DATE ____________ PERIOD _____
3-23-2
© Glencoe/McGraw-Hill 125 Glencoe Algebra 2
Less
on
3-2
Substitution To solve a system of linear equations by substitution, first solve for onevariable in terms of the other in one of the equations. Then substitute this expression intothe other equation and simplify.
Use substitution to solve the system of equations. 2x � y � 9x � 3y � �6
Solve the first equation for y in terms of x.2x � y � 9 First equation
�y � �2x � 9 Subtract 2x from both sides.
y � 2x � 9 Multiply both sides by �1.
Substitute the expression 2x � 9 for y into the second equation and solve for x.x � 3y � �6 Second equation
x � 3(2x � 9) � �6 Substitute 2x � 9 for y.
x � 6x � 27 � �6 Distributive Property
7x � 27 � �6 Simplify.
7x � 21 Add 27 to each side.
x � 3 Divide each side by 7.
Now, substitute the value 3 for x in either original equation and solve for y.2x � y � 9 First equation
2(3) � y � 9 Replace x with 3.
6 � y � 9 Simplify.
�y � 3 Subtract 6 from each side.
y � �3 Multiply each side by �1.
The solution of the system is (3, �3).
Solve each system of linear equations by using substitution.
1. 3x � y � 7 2. 2x � y � 5 3. 2x � 3y � �34x � 2y � 16 3x � 3y � 3 x � 2y � 2
(�1, 10) (2, 1) (�12, 7)
4. 2x � y � 7 5. 4x � 3y � 4 6. 5x � y � 66x � 3y � 14 2x � y � �8 3 � x � 0
no solution (�2, �4) (3, �9)
7. x � 8y � �2 8. 2x � y � �4 9. x � y � �2x � 3y � 20 4x � y � 1 2x � 3y � 2
(14, �2) �� , 3� (�8, �6)
10. x � 4y � 4 11. x � 3y � 2 12. 2x � 2y � 42x � 12y � 13 4x � 12 y � 8 x � 2y � 0
�5, � infinitely many � , �2�
4�
1�
1�
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 126 Glencoe Algebra 2
Elimination To solve a system of linear equations by elimination, add or subtract theequations to eliminate one of the variables. You may first need to multiply one or both of theequations by a constant so that one of the variables has the same (or opposite) coefficient inone equation as it has in the other.
Use the elimination method to solve the system of equations.2x � 4y � �263x � y � �24
Study Guide and Intervention (continued)
Solving Systems of Equations Algebraically
NAME ______________________________________________ DATE ____________ PERIOD _____
3-23-2
Example 1Example 1
Example 2Example 2
Multiply the second equation by 4. Then subtractthe equations to eliminate the y variable.2x � 4y � �26 2x � 4y � �263x � y � �24 Multiply by 4. 12x � 4y � �96
�10x � 70x � �7
Replace x with �7 and solve for y.2x � 4y � �26
2(�7) �4y � �26�14 � 4y � �26
�4y � �12y � 3
The solution is (�7, 3).
Multiply the first equation by 3 and the secondequation by 2. Then add the equations toeliminate the y variable.3x � 2y � 4 Multiply by 3. 9x � 6y � 125x � 3y � �25 Multiply by 2. 10x � 6y � �50
19x � �38x � �2
Replace x with �2 and solve for y.3x � 2y � 4
3(�2) � 2y � 4�6 � 2y � 4
�2y � 10y � �5
The solution is (�2, �5).
Use the elimination method to solve the system of equations.3x � 2y � 45x � 3y � �25
ExercisesExercises
Solve each system of equations by using elimination.
1. 2x � y � 7 2. x � 2y � 4 3. 3x � 4y � �10 4. 3x � y � 123x � y � 8 �x � 6y � 12 x � 4y � 2 5x � 2y � 20
(3, �1) (12, 4) (�2, �1) (4, 0)
5. 4x � y � 6 6. 5x � 2y � 12 7. 2x � y � 8 8. 7x � 2y � �1
2x � � 4 �6x � 2y � �14 3x � y � 12 4x � 3y � �13
no solution (2, 1) infinitely many (�1, 3)
9. 3x � 8y � �6 10. 5x � 4y � 12 11. �4x � y � �12 12. 5m � 2n � �8x � y � 9 7x � 6y � 40 4x � 2y � 6 4m � 3n � 2
(6, �3) (4, �2) � , �2� (�4, 6)5�
3�2
y�2
Skills PracticeSolving Systems of Equations Algebraically
NAME ______________________________________________ DATE ____________ PERIOD _____
3-23-2
© Glencoe/McGraw-Hill 127 Glencoe Algebra 2
Less
on
3-2
Solve each system of equations by using substitution.
1. m � n � 20 2. x � 3y � �3 3. w � z � 1m � n � �4 (8, 12) 4x � 3y � 6 (3, �2) 2w � 3z � 12 (3, 2)
4. 3r � s � 5 5. 2b � 3c � �4 6. x � y � 12r � s � 5 (2, �1) b � c � 3 (13, �10) 2x � 3y � 12 (3, 2)
Solve each system of equations by using elimination.
7. 2p � q � 5 8. 2j � k � 3 9. 3c � 2d � 23p � q � 5 (2, �1) 3j � k � 2 (1, �1) 3c � 4d � 50 (6, 8)
10. 2f � 3g � 9 11. �2x � y � �1 12. 2x � y � 12f � g � 2 (3, 1) x � 2y � 3 (1, 1) 2x � y � 6 no solution
Solve each system of equations by using either substitution or elimination.
13. �r � t � 5 14. 2x � y � �5 15. x � 3y � �12�2r � t � 4 (1, 6) 4x � y � 2 �� , 4� 2x � y � 11 (3, 5)
16. 2p � 3q � 6 17. 6w � 8z � 16 18. c � d � 6�2p � 3q � �6 (3, 0) 3w � 4z � 8 c � d � 0 (3, 3)
infinitely many
19. 2u � 4v � �6 20. 3a � b � �1 21. 2x � y � 6u � 2v � 3 no solution �3a � b � 5 (�1, 2) 3x � 2y � 16 (4, �2)
22. 3y � z � �6 23. c � 2d � �2 24. 3r � 2s � 1�3y � z � 6 (�2, 0) �2c � 5d � 3 (�4, 1) 2r � 3s � 9 (�3, �5)
25. The sum of two numbers is 12. The difference of the same two numbers is �4.Find the numbers. 4, 8
26. Twice a number minus a second number is �1. Twice the second number added to three times the first number is 9. Find the two numbers. 1, 3
1�
© Glencoe/McGraw-Hill 128 Glencoe Algebra 2
Solve each system of equations by using substitution.
1. 2x � y � 4 2. x � 3y � 9 3. g � 3h � 8 no3x � 2y � 1 (7, �10) x � 2y � �1 (3, �2) g � h � 9 solution
4. 2a � 4b � 6 infinitely 5. 2m � n � 6 6. 4x � 3y � �6�a � 2b � �3 many 5m � 6n � 1 (5, �4) �x � 2y � 7 (�3, �2)
7. u � 2v � 8. x � 3y � 16 9. w � 3z � 1
�u � 2v � 5 no solution 4x � y � 9 (1, �5) 3w � 5z � �4 �� , �
Solve each system of equations by using elimination.
10. 2r � s � 5 11. 2m � n � �1 12. 6x � 3y � 63r � s � 20 (5, �5) 3m � 2n � 30 (4, 9) 8x � 5y � 12 (�1, 4)
13. 3j � k � 10 14. 2x � y � �4 no 15. 2g � h � 64j � k � 16 (6, 8) �4x � 2y � 6 solution 3g � 2h � 16 (4, �2)
16. 2t � 4v � 6 infinitely 17. 3x � 2y � 12 18. x � 3y � 11�t � 2v � �3 many
2x � y � 14 (6, 3) 8x � 5y � 17 (4, 3)
Solve each system of equations by using either substitution or elimination.
19. 8x � 3y � �5 20. 8q � 15r � �40 21. 3x � 4y � 12 infinitely10x � 6y � �13 � , �3� 4q � 2r � 56 (10, 8) x � y � many
22. 4b � 2d � 5 no 23. s � 3y � 4 24. 4m � 2p � 0�2b � d � 1 solution s � 1 (1, 1) �3m � 9p � 5 � , �
25. 5g � 4k � 10 26. 0.5x � 2y � 5 27. h � z � 3 no�3g � 5k � 7 (6, �5) x � 2y � �8 (�2, 3) �3h � 3z � 6 solution
SPORTS For Exercises 28 and 29, use the following information.Last year the volleyball team paid $5 per pair for socks and $17 per pair for shorts on atotal purchase of $315. This year they spent $342 to buy the same number of pairs of socksand shorts because the socks now cost $6 a pair and the shorts cost $18.
28. Write a system of two equations that represents the number of pairs of socks and shortsbought each year. 5x � 17y � 315, 6x � 18y � 342
29. How many pairs of socks and shorts did the team buy each year?socks: 12, shorts: 15
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Practice (Average)
Solving Systems of Equations Algebraically
NAME ______________________________________________ DATE ____________ PERIOD _____
3-23-2
Reading to Learn MathematicsSolving Systems of Equations Algebraically
NAME ______________________________________________ DATE ____________ PERIOD _____
3-23-2
© Glencoe/McGraw-Hill 129 Glencoe Algebra 2
Less
on
3-2
Pre-Activity How can systems of equations be used to make consumerdecisions?
Read the introduction to Lesson 3-2 at the top of page 116 in your textbook.
• How many more minutes of long distance time did Yolanda use inFebruary than in January? 13 minutes
• How much more were the February charges than the January charges?$1.04
• Using your answers for the questions above, how can you find the rateper minute? Find $1.04 � 13.
Reading the Lesson
1. Suppose that you are asked to solve the system of equations 4x � 5y � 7at the right by the substitution method. 3x � y � �9
The first step is to solve one of the equations for one variable in terms of the other. To make your work as easy as possible,which equation would you solve for which variable? Explain.Sample answer: Solve the second equation for y because in thatequation the variable y has a coefficient of 1.
2. Suppose that you are asked to solve the system of equations 2x � 3y � �2at the right by the elimination method. 7x � y � 39
To make your work as easy as possible, which variable would you eliminate? Describe how you would do this.Sample answer: Eliminate the variable y; multiply the second equationby 3 and then add the result to the first equation.
Helping You Remember
3. The substitution method and elimination method for solving systems both have severalsteps, and it may be difficult to remember them. You may be able to remember themmore easily if you notice what the methods have in common. What step is the same inboth methods?Sample answer: After finding the value of one of the variables, you findthe value of the other variable by substituting the value you have foundin one of the original equations.
© Glencoe/McGraw-Hill 130 Glencoe Algebra 2
Using CoordinatesFrom one observation point, the line of sight to a downed plane is given by y � x � 1. This equation describes the distance from the observation point to the plane in a straight line. From another observation point, the line of sight is given by x � 3y � 21. What are the coordinates of the point at which the crash occurred?
Solve the system of equations � y � x � 1 .x � 3y � 21
x � 3y � 21x � 3(x � 1) � 21 Substitute x � 1 for y.
x � 3x � 3 � 214x � 24x � 6
x � 3y � 216 � 3y � 21 Substitute 6 for x.
3y � 15y � 5
The coordinates of the crash are (6, 5).
Solve the following.
1. The lines of sight to a forest fire are as follows.
From Ranger Station A: 3x � y � 9From Ranger Station B: 2x � 3y � 13Find the coordinates of the fire.
2. An airplane is traveling along the line x � y � �1 when it sees another airplane traveling along the line 5x � 3y � 19. If they continue along the same lines, at what point will their flight paths cross?
3. Two mine shafts are dug along the paths of the following equations.
x � y � 14002x � y � 1300
If the shafts meet at a depth of 200 feet, what are the coordinates of the point at which they meet?
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
3-23-2
Study Guide and InterventionSolving Systems of Inequalities by Graphing
NAME ______________________________________________ DATE ____________ PERIOD _____
3-33-3
© Glencoe/McGraw-Hill 131 Glencoe Algebra 2
Less
on
3-3
Graph Systems of Inequalities To solve a system of inequalities, graph the inequalitiesin the same coordinate plane. The solution set is represented by the intersection of the graphs.
Solve the system of inequalities by graphing.y � 2x � 1 and y � � 2
The solution of y � 2x � 1 is Regions 1 and 2.
The solution of y � � 2 is Regions 1 and 3.
The intersection of these regions is Region 1, which is the solution set of the system of inequalities.
Solve each system of inequalities by graphing.
1. x � y � 2 2. 3x � 2y � �1 3. y � 1x � 2y � 1 x � 4y � �12 x � 2
4. y � � 3 5. y � � 2 6. y � � � 1
y � 2x y � �2x � 1 y � 3x � 1
7. x � y � 4 8. x � 3y � 3 9. x � 2y � 62x � y � 2 x � 2y � 4 x � 4y � �4
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Region 3Region 1
Region 2
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 132 Glencoe Algebra 2
Find Vertices of a Polygonal Region Sometimes the graph of a system ofinequalities forms a bounded region. You can find the vertices of the region by a combinationof the methods used earlier in this chapter: graphing, substitution, and/or elimination.
Find the coordinates of the vertices of the figure formed by 5x � 4y � 20, y � 2x � 3, and x � 3y � 4.
Graph the boundary of each inequality. The intersections of the boundary lines are the vertices of a triangle.The vertex (4, 0) can be determined from the graph. To find thecoordinates of the second and third vertices, solve the two systems of equations
y � 2x � 3 and y � 2x � 35x � 4y � 20 x � 3y � 4
x
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Study Guide and Intervention (continued)
Solving Systems of Inequalities by Graphing
NAME ______________________________________________ DATE ____________ PERIOD _____
3-33-3
For the first system of equations, rewrite the firstequation in standard form as 2x � y � �3. Thenmultiply that equation by 4 and add to the secondequation.2x � y � �3 Multiply by 4. 8x � 4y � �125x � 4y � 20 (�) 5x � 4y � 20
13x � 8
x �
Then substitute x � in one of the original equations and solve for y.
2� � � y � �3
� y � �3
y �
The coordinates of the second vertex are � , 4 �.3�13
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For the second system ofequations, use substitution.Substitute 2x � 3 for y in thesecond equation to getx � 3(2x � 3) � 4
x � 6x � 9 � 4�5x � 13
x � �
Then substitute x � � in the
first equation to solve for y.
y � 2�� � � 3
y � � � 3
y � �
The coordinates of the third
vertex are ��2 , �2 �.1�5
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ExercisesExercises
Thus, the coordinates of the three vertices are (4, 0), � , 4 �, and ��2 , �2 �.
Find the coordinates of the vertices of the figure formed by each system ofinequalities.
1. y � �3x � 7 2. x � �3 3. y � � x � 3
y � x (2, 1), (�4, �2), y � � x � 3 (�3, 4), y � x � 1 (�2, 4), (2, 2),
y � �2(3, �2)
y � x � 1(�3, �4), (3, 2)
y � 3x � 10 ��3 , � �4�
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ExampleExample
Skills PracticeSolving Systems of Inequalities by Graphing
NAME ______________________________________________ DATE ____________ PERIOD _____
3-33-3
© Glencoe/McGraw-Hill 133 Glencoe Algebra 2
Less
on
3-3
Solve each system of inequalities by graphing.
1. x � 1 2. x � �3 3. x � 2y � �1 y � �3 x � 4 no solution
4. y � x 5. y � �4x 6. x � y � �1y � �x y � 3x � 2 3x � y � 4
7. y � 3 8. y � �2x � 3 9. x � y � 4x � 2y � 12 y � x � 2 2x � y � 4
Find the coordinates of the vertices of the figure formed by each system ofinequalities.
10. y � 0 11. y � 3 � x 12. x � �2x � 0 y � 3 y � x � 2y � �x � 1 x � �5 x � y � 2 (�2, 4), (0, 0), (0, �1), (�1, 0) (0, 3), (�5, 3), (�5, 8) (�2, �4), (2, 0)
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© Glencoe/McGraw-Hill 134 Glencoe Algebra 2
Solve each system of inequalities by graphing.
1. y � 1 � �x 2. x � �2 3. y � 2x � 3
y � 1 2y � 3x � 6 y � � x � 2
4. x � y � �2 5. y � 1 6. 3y � 4x3x � y � �2 y � x � 1 2x � 3y � �6
Find the coordinates of the vertices of the figure formed by each system ofinequalities.
7. y � 1 � x 8. x � y � 2 9. y � 2x � 2y � x � 1 x � y � 2 2x � 3y � 6x � 3 x � �2 y � 4
(1, 0), (3, 2), (3, �2) (�2, 4), (�2, �4), (2, 0) (�3, 4), � , 1�, (3, 4)
DRAMA For Exercises 10 and 11, use the following information.The drama club is selling tickets to its play. An adult ticket costs $15 and a student ticket costs $11. The auditorium willseat 300 ticket-holders. The drama club wants to collect atleast $3630 from ticket sales.
10. Write and graph a system of four inequalities that describe how many of each type of ticket the club must sell to meets its goal.x 0, y 0, x � y � 300, 15x � 11y 3630
11. List three different combinations of tickets sold thatsatisfy the inequalities. Sample answer: 250 adult and 50 student, 200 adult and 100 student, 145 adult and 148 student
Play Tickets
Adult Tickets
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1000 200 300 400
400
350
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250
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Practice (Average)
Solving Systems of Inequalities by Graphing
NAME ______________________________________________ DATE ____________ PERIOD _____
3-33-3
Reading to Learn MathematicsSolving Systems of Inequalities by Graphing
NAME ______________________________________________ DATE ____________ PERIOD _____
3-33-3
© Glencoe/McGraw-Hill 135 Glencoe Algebra 2
Less
on
3-3
Pre-Activity How can you determine whether your blood pressure is in a normal range?
Read the introduction to Lesson 3-3 at the top of page 123 in your textbook.
Satish is 37 years old. He has a blood pressure reading of 135/99. Is hisblood pressure within the normal range? Explain.Sample answer: No; his systolic pressure is normal, but hisdiastolic pressure is too high. It should be between 60 and 90.
Reading the Lesson
1. Without actually drawing the graph, describe the boundary lines x � 3for the system of inequalities shown at the right. y � 5Two dashed vertical lines (x � 3 and x � �3) and two solid horizontallines (y � �5 and y � 5)
2. Think about how the graph would look for the system given above. What will be theshape of the shaded region? (It is not necessary to draw the graph. See if you canimagine it without drawing anything. If this is difficult to do, make a rough sketch tohelp you answer the question.)a rectangle
3. Which system of inequalities matches the graph shown at the right? BA. x � y � �2 B. x � y � �2
x � y � 2 x � y � 2
C. x � y � �2 D. x � y � �2x � y � 2 x � y � 2
Helping You Remember
4. To graph a system of inequalities, you must graph two or more boundary lines. Whenyou graph each of these lines, how can the inequality symbols help you rememberwhether to use a dashed or solid line?Use a dashed line if the inequality symbol is � or �, because thesesymbols do not include equality and the dashed line reminds you thatthe line itself is not included in the graph. Use a solid line if the symbolis or �, because these symbols include equality and tell you that theline itself is included in the graph.
x
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© Glencoe/McGraw-Hill 136 Glencoe Algebra 2
Tracing StrategyTry to trace over each of the figures below without tracing the same segment twice.
The figure at the left cannot be traced, but the one at the right can. The rule is that a figure is traceable if it has no more than two points where an odd number of segments meet. The figure at the left has three segments meeting at each of the four corners. However, the figure at the right has only two points, L and Q, where an odd number of segments meet.
Determine if each figure can be traced without tracing the same segment twice. If it can, then name the starting point and name the segments in the order they should be traced.
1. 2.
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Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
3-33-3
Study Guide and InterventionLinear Programming
NAME ______________________________________________ DATE ____________ PERIOD _____
3-43-4
© Glencoe/McGraw-Hill 137 Glencoe Algebra 2
Less
on
3-4
Maximum and Minimum Values When a system of linear inequalities produces abounded polygonal region, the maximum or minimum value of a related function will occurat a vertex of the region.
Graph the system of inequalities. Name the coordinates of thevertices of the feasible region. Find the maximum and minimum values of thefunction f(x, y) � 3x � 2y for this polygonal region.
y � 4y � �x � 6
y x �
y � 6x � 4First find the vertices of the bounded region. Graph the inequalities.The polygon formed is a quadrilateral with vertices at (0, 4), (2, 4), (5, 1), and (�1, �2). Use the table to find themaximum and minimum values of f(x, y) � 3x � 2y.
The maximum value is 17 at (5, 1). The minimum value is �7 at (�1, �2).
Graph each system of inequalities. Name the coordinates of the vertices of thefeasible region. Find the maximum and minimum values of the given function forthis region.
1. y � 2 2. y � �2 3. x � y � 21 � x � 5 y � 2x � 4 4y � x � 8y � x � 3 x � 2y � �1 y � 2x � 5f(x, y) � 3x � 2y f(x, y) � 4x � y f(x, y) � 4x � 3y
vertices: (1, 2), (1, 4), vertices: (�5, �2), vertices (0, 2), (4, 3),(5, 8), (5, 2); max: 11; (3, 2), (1, �2); � , � �; max: 25; min: 6 min; �5 max: 10; min: �18
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(x, y ) 3x � 2y f (x, y )
(0, 4) 3(0) � 2(4) 8
(2, 4) 3(2) � 2(4) 14
(5, 1) 3(5) � 2(1) 17
(�1, �2) 3(�1) � 2(�2) �7
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1�2
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 138 Glencoe Algebra 2
Real-World Problems When solving linear programming problems, use thefollowing procedure.
1. Define variables.2. Write a system of inequalities.3. Graph the system of inequalities.4. Find the coordinates of the vertices of the feasible region.5. Write an expression to be maximized or minimized.6. Substitute the coordinates of the vertices in the expression.7. Select the greatest or least result to answer the problem.
A painter has exactly 32 units of yellow dye and 54 units of greendye. He plans to mix as many gallons as possible of color A and color B. Eachgallon of color A requires 4 units of yellow dye and 1 unit of green dye. Eachgallon of color B requires 1 unit of yellow dye and 6 units of green dye. Find themaximum number of gallons he can mix.
Step 1 Define the variables.x � the number of gallons of color A madey � the number of gallons of color B made
Step 2 Write a system of inequalities.Since the number of gallons made cannot benegative, x � 0 and y � 0.There are 32 units of yellow dye; each gallon of color A requires 4 units, and each gallon of color B requires 1 unit.So 4x � y � 32.Similarly for the green dye, x � 6y � 54.Steps 3 and 4 Graph the system of inequalities and find the coordinates of the vertices of the feasible region.The vertices of the feasible region are (0, 0), (0, 9), (6, 8), and (8, 0).Steps 5–7 Find the maximum number of gallons,x � y, that he can make.The maximum number of gallons the painter can make is 14,6 gallons of color A and 8 gallons of color B.
1. FOOD A delicatessen has 8 pounds of plain sausage and 10 pounds of garlic-flavoredsausage. The deli wants to make as much bratwurst as possible. Each pound of
bratwurst requires pound of plain sausage and pound of garlic-flavored sausage.
Find the maximum number of pounds of bratwurst that can be made. 10 lb
2. MANUFACTURING Machine A can produce 30 steering wheels per hour at a cost of $16per hour. Machine B can produce 40 steering wheels per hour at a cost of $22 per hour.At least 360 steering wheels must be made in each 8-hour shift. What is the least costinvolved in making 360 steering wheels in one shift? $194
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(x, y) x � y f (x, y)
(0, 0) 0 � 0 0
(0, 9) 0 � 9 9
(6, 8) 6 � 8 14
(8, 0) 8 � 0 8
Color A (gallons)
Co
lor
B (
gal
lon
s)
5 10 15 20 25 30 35 40 45 50 550
40
35
30
25
20
15
10
5
(6, 8)
(8, 0)(0, 9)
Study Guide and Intervention (continued)
Linear Programming
NAME ______________________________________________ DATE______________ PERIOD _____
3-43-4
ExampleExample
ExercisesExercises
Skills PracticeLinear Programming
NAME ______________________________________________ DATE ____________ PERIOD _____
3-43-4
© Glencoe/McGraw-Hill 139 Glencoe Algebra 2
Less
on
3-4
Graph each system of inequalities. Name the coordinates of the vertices of thefeasible region. Find the maximum and minimum values of the given function forthis region.
1. x � 2 2. x � 1 3. x � 0x � 5 y � 6 y � 0y � 1 y � x � 2 y � 7 � xy � 4 f(x, y) � x � y f(x, y) � 3x � yf(x, y) � x � y
max.: 9, min.: 3 max.: 2, min.: �5 max.: 21, min.: 0
4. x � �1 5. y � 2x 6. y � �x � 2x � y � 6 y � 6 � x y � 3x � 2f(x, y) � x � 2y y � 6 y � x � 4
f(x, y) � 4x � 3y f(x, y) � �3x � 5y
max.: 13, no min. no max., min.: 20 max.: 22, min.: �2
7. MANUFACTURING A backpack manufacturer produces an internal frame pack and anexternal frame pack. Let x represent the number of internal frame packs produced inone hour and let y represent the number of external frame packs produced in one hour.Then the inequalities x � 3y � 18, 2x � y � 16, x � 0, and y � 0 describe the constraintsfor manufacturing both packs. Use the profit function f(x) � 50x � 80y and theconstraints given to determine the maximum profit for manufacturing both backpacksfor the given constraints. $620
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Ox
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© Glencoe/McGraw-Hill 140 Glencoe Algebra 2
Graph each system of inequalities. Name the coordinates of the vertices of thefeasible region. Find the maximum and minimum values of the given function forthis region.
1. 2x � 4 � y 2. 3x � y � 7 3. x � 0�2x � 4 � y 2x � y � 3 y � 0y � 2 y � x � 3 y � 6f(x, y) � �2x � y f(x, y) � x � 4y y � �3x � 15
f(x, y) � 3x � y
max.: 8, min.: �4 max.: 12, min.: �16 max.: 15, min.: 0
4. x � 0 5. y � 3x � 6 6. 2x � 3y � 6y � 0 4y � 3x � 3 2x � y � 24x � y � �7 x � �2 x � 0f(x, y) � �x � 4y f(x, y) � �x � 3y y � 0
f(x, y) � x � 4y � 3
max.: 28, min.: 0 max.: , no min. no max., min.:
PRODUCTION For Exercises 7–9, use the following information.A glass blower can form 8 simple vases or 2 elaborate vases in an hour. In a work shift of nomore than 8 hours, the worker must form at least 40 vases.
7. Let s represent the hours forming simple vases and e the hours forming elaborate vases.Write a system of inequalities involving the time spent on each type of vase.s 0, e 0, s � e � 8, 8s � 2e 40
8. If the glass blower makes a profit of $30 per hour worked on the simple vases and $35per hour worked on the elaborate vases, write a function for the total profit on the vases.f(s, e) � 30s � 35e
9. Find the number of hours the worker should spend on each type of vase to maximizeprofit. What is that profit? 4 h on each; $260
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Practice (Average)
Linear Programming
NAME ______________________________________________ DATE ____________ PERIOD _____
3-43-4
Reading to Learn MathematicsLinear Programming
NAME ______________________________________________ DATE ____________ PERIOD _____
3-43-4
© Glencoe/McGraw-Hill 141 Glencoe Algebra 2
Less
on
3-4
Pre-Activity How is linear programming used in scheduling work?
Read the introduction to Lesson 3-4 at the top of page 129 in your textbook.
Name two or more facts that indicate that you will need to use inequalitiesto model this situation.Sample answer: The buoy tender can carry up to 8 new buoys.There seems to be a limit of 24 hours on the time the crew hasat sea. The crew will want to repair or replace the maximumnumber of buoys possible.
Reading the Lesson
1. Complete each sentence.
a. When you find the feasible region for a linear programming problem, you are solving
a system of linear called . The points
in the feasible region are of the system.
b. The corner points of a polygonal region are the of the feasible region.
2. A polygonal region always takes up only a limited part of the coordinate plane. One wayto think of this is to imagine a circle or rectangle that the region would fit inside. In thecase of a polygonal region, you can always find a circle or rectangle that is large enoughto contain all the points of the polygonal region. What word is used to describe a regionthat can be enclosed in this way? What word is used to describe a region that is too largeto be enclosed in this way? bounded; unbounded
3. How do you find the corner points of the polygonal region in a linear programmingproblem? You solve a system of two linear equations.
4. What are some everyday meanings of the word feasible that remind you of themathematical meaning of the term feasible region?Sample answer: possible or achievable
Helping You Remember
5. Look up the word constraint in a dictionary. If more than one definition is given, choosethe one that seems closest to the idea of a constraint in a linear programming problem.How can this definition help you to remember the meaning of constraint as it is used inthis lesson? Sample answer: A constraint is a restriction or limitation. Theconstraints in a linear programming problem are restrictions on thevariables that translate into inequality statements.
vertices
solutions
constraintsinequalities
© Glencoe/McGraw-Hill 142 Glencoe Algebra 2
Computer Circuits and LogicComputers operate according to the laws of logic. The circuits of a computer can be described using logic.
1. With switch A open, no current flows. The value 0 is assigned to an open switch.
2. With switch A closed, current flows. The value 1 is assigned to a closed switch.
3. With switches A and B open, no current flows. This circuit can be described by the conjunction, A B.
4. In this circuit, current flows if either A or B is closed. This circuit can be described by the disjunction, A � B.
Truth tables are used to describe the flow of current in a circuit.The table at the left describes the circuit in diagram 4. According to the table, the only time current does not flow through the circuit is when both switches A and B are open.
Draw a circuit diagram for each of the following.
1. (A B) � C 2. (A � B) C
3. (A � B) (C � D) 4. (A B) � (C D)
5. Construct a truth table for the following circuit.
B
C
A
A
A
A
B
A B
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
3-43-4
A B A � B
0 0 00 1 11 0 11 1 1
Study Guide and InterventionSolving Systems of Equations in Three Variables
NAME ______________________________________________ DATE ____________ PERIOD _____
3-53-5
© Glencoe/McGraw-Hill 143 Glencoe Algebra 2
Less
on
3-5
Systems in Three Variables Use the methods used for solving systems of linearequations in two variables to solve systems of equations in three variables. A system ofthree equations in three variables can have a unique solution, infinitely many solutions, orno solution. A solution is an ordered triple.
Solve this system of equations. 3x � y � z � �62x � y � 2z � 84x � y � 3z � �21
Step 1 Use elimination to make a system of two equations in two variables.3x � y � z � �6 First equation 2x � y � 2z � 8 Second equation
(�) 2x � y � 2z � 8 Second equation (�) 4x � y � 3z � �21 Third equation
5x � z � 2 Add to eliminate y. 6x � z � �13 Add to eliminate y.
Step 2 Solve the system of two equations.5x � z � 2
(�) 6x � z � �1311x � �11 Add to eliminate z.
x � �1 Divide both sides by 11.
Substitute �1 for x in one of the equations with two variables and solve for z.5x � z � 2 Equation with two variables
5(�1) � z � 2 Replace x with �1.
�5 � z � 2 Multiply.
z � 7 Add 5 to both sides.
The result so far is x � �1 and z � 7.
Step 3 Substitute �1 for x and 7 for z in one of the original equations with three variables.3x � y � z � �6 Original equation with three variables
3(�1) � y � 7 � �6 Replace x with �1 and z with 7.
�3 � y � 7 � �6 Multiply.
y � 4 Simplify.
The solution is (�1, 4, 7).
Solve each system of equations.
1. 2x � 3y � z � 0 2. 2x � y � 4z � 11 3. x � 2y � z � 8x � 2y � 4z � 14 x � 2y � 6z � �11 2x � y � z � 03x � y � 8z � 17 3x � 2y �10z � 11 3x � 6y � 3z � 24
(4, �3, �1) �2, �5, � infinitely manysolutions
4. 3x � y � z � 5 5. 2x � 4y � z � 10 6. x � 6y � 4z � 23x � 2y � z � 11 4x � 8y � 2z � 16 2x � 4y � 8z � 166x � 3y � 2z � �12 3x � y � z � 12 x � 2y � 5
� , 2, �5� no solution �6, , � �1�
1�
2�
1�
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 144 Glencoe Algebra 2
Real-World Problems
The Laredo Sports Shop sold 10 balls, 3 bats, and 2 bases for $99 onMonday. On Tuesday they sold 4 balls, 8 bats, and 2 bases for $78. On Wednesdaythey sold 2 balls, 3 bats, and 1 base for $33.60. What are the prices of 1 ball, 1 bat,and 1 base?
First define the variables.x � price of 1 bally � price of 1 batz � price of 1 base
Translate the information in the problem into three equations.
10x � 3y � 2z � 994x � 8y � 2z � 782x � 3y �z � 33.60
Study Guide and Intervention (continued)
Solving Systems of Equations in Three Variables
NAME ______________________________________________ DATE ____________ PERIOD _____
3-53-5
ExampleExample
Subtract the second equation from the firstequation to eliminate z.
10x � 3y � 2z � 99(�) 4x � 8y � 2z � 78
6x � 5y � 21
Multiply the third equation by 2 andsubtract from the second equation.
4x � 8y � 2z � 78(�) 4x � 6y � 2z � 67.20
2y � 10.80y � 5.40
Substitute 5.40 for y in the equation 6x � 5y � 21.
6x �5(5.40) � 216x � 48x � 8
Substitute 8 for x and 5.40 for y in one ofthe original equations to solve for z.
10x � 3y � 2z � 9910(8) � 3(5.40) � 2z � 99
80 � 16.20 � 2z � 992z � 2.80z � 1.40
So a ball costs $8, a bat $5.40, and a base $1.40.
1. FITNESS TRAINING Carly is training for a triathlon. In her training routine each week,she runs 7 times as far as she swims, and she bikes 3 times as far as she runs. One weekshe trained a total of 232 miles. How far did she run that week? 56 miles
2. ENTERTAINMENT At the arcade, Ryan, Sara, and Tim played video racing games, pinball,and air hockey. Ryan spent $6 for 6 racing games, 2 pinball games, and 1 game of airhockey. Sara spent $12 for 3 racing games, 4 pinball games, and 5 games of air hockey. Timspent $12.25 for 2 racing games, 7 pinball games, and 4 games of air hockey. How much dideach of the games cost? Racing game: $0.50; pinball: $0.75; air hockey: $1.50
3. FOOD A natural food store makes its own brand of trail mix out of dried apples, raisins,and peanuts. One pound of the mixture costs $3.18. It contains twice as much peanutsby weight as apples. One pound of dried apples costs $4.48, a pound of raisins $2.40, anda pound of peanuts $3.44. How many ounces of each ingredient are contained in 1 poundof the trail mix? 3 oz of apples, 7 oz of raisins, 6 oz of peanuts
ExercisesExercises
Skills PracticeSolving Systems of Equations in Three Variables
NAME ______________________________________________ DATE ____________ PERIOD _____
3-53-5
© Glencoe/McGraw-Hill 145 Glencoe Algebra 2
Less
on
3-5
Solve each system of equations.
1. 2a � c � �10 (5, �5, �20) 2. x � y � z � 3 (0, 2, 1)b � c � 15 13x � 2z � 2a � 2b � c � �5 �x � 5z � �5
3. 2x � 5y � 2z � 6 (�3, 2, 1) 4. x � 4y � z � 1 no solution5x � 7y � �29 3x � y � 8z � 0z � 1 x � 4y � z � 10
5. �2z � �6 (2, �1, 3) 6. 3x � 2y � 2z � �2 (�2, 1, 3)2x � 3y � z � �2 x � 6y � 2z � �2x � 2y � 3z � 9 x � 2y � 0
7. �x � 5z � �5 (0, 0, 1) 8. �3r � 2t � 1 (1, �6, 2)y � 3x � 0 4r � s � 2t � �613x � 2z � 2 r � s � 4t � 3
9. x � y � 3z � 3 no solution 10. 5m � 3n � p � 4 (�2, 3, 5)�2x � 2y � 6z � 6 3m � 2n � 0y � 5z � �3 2m � n � 3p � 8
11. 2x � 2y � 2z � �2 infinitely many 12. x � 2y � z � 4 (1, 2, 1)2x � 3y � 2z � 4 3x � y � 2z � 3x � y � z � �1 �x � 3y � z � 6
13. 3x � 2y � z � 1 (5, 7, 0) 14. 3x � 5y � 2z � �12 infinitely many�x � y � z � 2 x � 4y � 2z � 85x � 2y � 10z � 39 �3x � 5y � 2z � 12
15. 2x � y � 3z � �2 (�1, 3, �1) 16. 2x � 4y � 3z � 0 (3, 0, �2)x � y � z � �3 x � 2y � 5z � 133x � 2y � 3z � �12 5x � 3y � 2z � 19
17. �2x � y � 2z � 2 (1, �2, 3) 18. x � 2y � 2z � �1 infinitely many3x � 3y � z � 0 x � 2y � z � 6x � y � z � 2 �3x � 6y � 6z � 3
19. The sum of three numbers is 18. The sum of the first and second numbers is 15,and the first number is 3 times the third number. Find the numbers. 9, 6, 3
© Glencoe/McGraw-Hill 146 Glencoe Algebra 2
Solve each system of equations.
1. 2x � y � 2z � 15 2. x � 4y � 3z � �27 3. a � b � 3�x � y � z � 3 2x � 2y � 3z � 22 �b � c � 33x � y � 2z � 18 4z � �16 a � 2c � 10(3, 1, 5) (1, 4, �4) (2, 1, 4)
4. 3m � 2n � 4p � 15 5. 2g � 3h � 8j � 10 6. 2x � y � z � �8m � n � p � 3 g � 4h � 1 4x � y � 2z � �3m � 4n � 5p � 0 �2g � 3h � 8j � 5 �3x � y � 2z � 5(3, 3, 3) no solution (�2, �3, 1)
7. 2x � 5y � z � 5 8. 2x � 3y � 4z � 2 9. p � 4r � �73x � 2y � z � 17 5x � 2y � 3z � 0 p � 3q � �84x � 3y � 2z � 17 x � 5y � 2z � �4 q � r � 1(5, 1, 0) (2, 2, �2) (1, 3, �2)
10. 4x � 4y � 2z � 8 11. d � 3e � f � 0 12. 4x � y � 5z � �93x � 5y � 3z � 0 �d � 2e � f � �1 x � 4y � 2z � �2 2x � 2y � z � 4 4d � e � f � 1 2x � 3y � 2z � 21infinitely many (1, �1, 2) (2, 3, �4)
13. 5x � 9y � z � 20 14. 2x � y � 3z � �3 15. 3x � 3y � z � 102x � y � z � �21 3x � 2y � 4z � 5 5x � 2y � 2z � 75x � 2y � 2z � �21 �6x � 3y � 9z � 9 3x � 2y � 3z � �9(�7, 6, 1) infinitely many (1, 3, �2)
16. 2u � v � w � 2 17. x � 5y � 3z � �18 18. x � 2y � z � �1�3u � 2v � 3w � 7 3x � 2y � 5z � 22 �x � 2y � z � 6 �u � v � 2w � 7 �2x � 3y � 8z � 28 �4y � 2z � 1(0, �1, 3) (1, �2, 3) no solution
19. 2x � 2y � 4z � �2 20. x � y � 9z � �27 21. 2x � 5y � 3z � 73x � 3y � 6z � �3 2x � 4y � z � �1 �4x � 10y � 2z � 6�2x � 3y � z � 7 3x � 6y � 3z � 27 6x � 15y � z � �19(4, 5, 0) (2, 2, �3) (1, 2, �5)
22. The sum of three numbers is 6. The third number is the sum of the first and secondnumbers. The first number is one more than the third number. Find the numbers.4, �1, 3
23. The sum of three numbers is �4. The second number decreased by the third is equal tothe first. The sum of the first and second numbers is �5. Find the numbers. �3, �2, 1
24. SPORTS Alexandria High School scored 37 points in a football game. Six points areawarded for each touchdown. After each touchdown, the team can earn one point for theextra kick or two points for a 2-point conversion. The team scored one fewer 2-pointconversions than extra kicks. The team scored 10 times during the game. How manytouchdowns were made during the game? 5
Practice (Average)
Solving Systems of Equations in Three Variables
NAME ______________________________________________ DATE ____________ PERIOD _____
3-53-5
Reading to Learn MathematicsSolving Systems of Equations in Three Variables
NAME ______________________________________________ DATE ____________ PERIOD _____
3-53-5
© Glencoe/McGraw-Hill 147 Glencoe Algebra 2
Less
on
3-5
Pre-Activity How can you determine the number and type of medals U.S.Olympians won?
Read the introduction to Lesson 3-5 at the top of page 138 in your textbook.
At the 1996 Summer Olympics in Atlanta, Georgia, the United States won101 medals. The U.S. team won 12 more gold medals than silver and 7fewer bronze medals than silver. Using the same variables as those in theintroduction, write a system of equations that describes the medals won forthe 1996 Summer Olympics.g � s � b � 101; g � s � 12; b � s � 7
Reading the Lesson
1. The planes for the equations in a system of three linear equations in three variablesdetermine the number of solutions. Match each graph description below with thedescription of the number of solutions of the system. (Some of the items on the right maybe used more than once, and not all possible types of graphs are listed.)
a. three parallel planes I. one solution
b. three planes that intersect in a line II. no solutions
c. three planes that intersect in one point III. infinite solutions
d. one plane that represents all three equations
2. Suppose that three classmates, Monique, Josh, and Lilly, are studying for a quiz on thislesson. They work together on solving a system of equations in three variables, x, y, andz, following the algebraic method shown in your textbook. They first find that z � 3, thenthat y � �2, and finally that x � �1. The students agree on these values, but disagreeon how to write the solution. Here are their answers:
Monique: (3, �2, �1) Josh: (�2, �1, 3) Lilly: (�1, �2, 3)
a. How do you think each student decided on the order of the numbers in the orderedtriple? Sample answer: Monique arranged the values in the order inwhich she found them. Josh arranged them from smallest to largest.Lilly arranged them in alphabetical order of the variables.
b. Which student is correct? Lilly
Helping You Remember
3. How can you remember that obtaining the equation 0 � 0 indicates a system withinfinitely many solutions, while obtaining an equation such as 0 � 8 indicates a systemwith no solutions? 0 � 0 is always true, while 0 � 8 is never true.
III
I
III
II
© Glencoe/McGraw-Hill 148 Glencoe Algebra 2
BilliardsThe figure at the right shows a billiard table.The object is to use a cue stick to strike the ball at point C so that the ball will hit the sides (or cushions) of the table at least once before hitting the ball located at point A. In playing the game, you need to locate point P.
Step 1 Find point B so that � and � . B is called the reflected
image of C in .
Step 2 Draw .
Step 3 intersects at the desired point P.
For each billiards problem, the cue ball at point C must strike the indicatedcushion(s) and then strike the ball at point A. Draw and label the correct path for the cue ball using the process described above.
1. cushion 2. cushion
3. cushion , then cushion 4. cushion , then cushion
C
A
RK
T S
C
ARK
T S
RSKTRSTS
C
A
RK
T S
C
A
RK
T S
RSKR
STAB
AB
STCHBH
STBC
K R
C
A
B
P
HT S
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
3-53-5
Chapter 3 Test, Form 1
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 149 Glencoe Algebra 2
Ass
essm
entWrite the letter for the correct answer in the blank at the right of each question.
1. A system of equations may not have A. exactly one solution. B. no solution.C. infinitely many solutions. D. exactly two solutions. 1.
Choose the correct description of each system of equations.A. consistent and independent B. consistent and dependentC. inconsistent D. inconsistent and dependent 2.
2. 4x � 2y � �6 3. 3x � y � 32x � y � 8 x � 2y � 4 3.
To solve each system of equations, which expression could be substituted for x into the first equation?
4. 5x � 2y � 8 A. y � 1 B. y � 1x � y � 1
C. ��25�x � �
58� D. x � 1 4.
5. 4x � 3y � 12 A. 3y � 5 B. y � 35x � 3y � �5
C. �3y � 5 D. ��13�x � �
53� 5.
The first equation of each system is multiplied by 4. By what number would you multiply the second equation in order to eliminate the x variable by adding?
6. 3x � 2y � 4 A. 3 B. �34x � 5y � 28 C. 4 D. �4 6.
7. �6x � 3y � 12 A. 3 B. �38x � 2y � 16 C. 6 D. �6 7.
Solve each system of equations.8. 3x � 2y � 5 A. (1, 1) B. (2, 0)
x � y � 2 C. (0, �2) D. (1, �1) 8.
9. 2x � 3y � 5 A. (3, 4) B. (�2, 3)3x � 2y � 1 C. (1, 1) D. (4, �1) 9.
10. Which system of equations is graphed?
A. y � �13�x � 0 B. y � 3x � 0
x � y � �2 x � y � �2
C. y � 3x � 0 D. y � �13�x � 0 10.
x � y � 2 x � y � 2
11. Which system of inequalities is graphed?A. y � �1 B. y � �1
y � �2x�1 y � �2x � 1C. y � �1 D. y � �1 11.
y � �2x � 1 y � �2x � 1
33
y
xO
(1, 3)
y
xO
© Glencoe/McGraw-Hill 150 Glencoe Algebra 2
Chapter 3 Test, Form 1 (continued)
12. Find the coordinates of the vertices A. (0, 0), (3, 0), (3, 2), (0, 2)of the figure formed by the system y � 0, x � 0, y � 2, and x � 3.
12.Use the system of inequalities y � 0, x � 0, and y � �2x � 4.13. Find the coordinates of the vertices
of the feasible region.
13.
14. Find the maximum value of f(x, y) � 3x � y for the feasible region.A. 2 B. 4 C. 6 D. 12 14.
15. Find the minimum value of f(x, y) � 3x � y for the feasible region.A. 6 B. 4 C. 2 D. 0 15.
For Questions 16–18, use the following information. A college arena sells tickets to students and to the public. Student tickets are $8 each and general public tickets are $32 each. The college reserves at least 5000 tickets for students. The arena seats 18,000.16. Let s represent the number of student tickets and p represent the number
of general public tickets. Which system of inequalities represents the number of tickets sold? A. s � 0, p � 0, s � p � 18,000 B. s � 5000, p � 0, s � p � 18,000C. s � 8, p � 32, s � p � 40 D. s � 0, p � 0, s � p � 18,000 16.
17. How many general public tickets should the college sell to maximize revenue (amount collected)?
A. 18,000 B. 0 C. 13,000 D. 5000 17.
18. What is the maximum revenue?A. $456,000 B. $416,000 C. $40,000 D. $576,000 18.
19. What is the value of y in the solution of the system of equations?
A. 1 B. 2 C. 3 D. 4 19.
20. The 300 students at Holmes School work a total of 5000 hours each month.Each student in group A works 10 hours, each in group B works 15 hours,and each in group C works 20 hours each month. There are twice as many students in group B as in group A. Which equation would not be included in the system used to solve this problem?A. A � 2B B. 10A � 15B � 20C � 5000C. A � B � C � 300 D. B � 2A 20.
Bonus Find the area of the region defined by the system of inequalities x � 0, y � 0, and x � 2y � 4. B:
2x � y � z � 132x � y � 3z � �3x � 2y � 4z � 20
NAME DATE PERIOD
33
A. (0, 0), (3, 0), (3, 2), (0, 2)B. (0, 0), (2, 0), (2, 3), (0, 3)C. (0, 0), (�3, 0), (�3, �2), (0, �2)D. (0, 0), (�2, 0), (�2, �3), (0, �3)
A. (0, 0), (�2, 0), (0, �4)B. (0, 0), (2, 0), (0, 4)C. (0, 0), (4, 0), (0, 2)D. (0, 0), (�4, 0), (0, 2)
Chapter 3 Test, Form 2A
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 151 Glencoe Algebra 2
Ass
essm
entWrite the letter for the correct answer in the blank at the right of each question.
1. The system of equations y � �3x � 5 and y � �3x � 7 has A. exactly one solution. B. no solution.C. infinitely many solutions. D. exactly two solutions. 1.
Choose the correct description of each system of equations.A. consistent and independent B. consistent and dependentC. inconsistent D. inconsistent and dependent 2.
2. 2x � y � 4 3. 9x � 3y � 15 3.4x � 2y � 6 6x � 2y � 10
To solve each system of equations, which expression could be substituted for y into the first equation?
4. 5x � 3y � 9 A. 12x � 3y B. ��35�x � 3
4x � y � 8 C. 4x � 8 D. 8 � 4x 4.
5. 3x � 6y � 12 A. �12�y � �
52� B. 2x � 5
2x � y � 5 C. 2x � 5 D. 12y � 5 5.
6. The first equation of the system is multiplied by 3. 4x � 3y � 6By what number would you multiply the second 6x � 1y � 10equation to eliminate the x variable by adding?A. �2 B. 2 C. 9 D. �9 6.
7. The first equation of the system is multiplied by 2. 4x � 3y � 6By what number would you multiply the second 6x � 1y � 10equation to eliminate the y variable by adding?A. �2 B. �1 C. 6 D. �3 7.
For Questions 8 and 9, solve each system of equations.8. 5x � 2y � 1 A. (1, �2) B. (1, 2)
y � 1 � 3x C. �0, �12�� D. (�2, 1) 8.
9. 3x � 4y � 12 A. (3, 0) B. (4, 0)2x � 3y � �9 C. (�1, 4) D. (0, 3) 9.
10. Which system of equations is graphed?A. x � y � 5 B. x � y � 5
x � 2y � 2 x � 2y � 2
C. x � y � 5 D. x � y � 5 10.x � 2y � 2 x � 2y � 2
11. Which system of inequalities is graphed?A. 2x � y � 2 B. 2x � y � 2
x � 3y � 6 x � 3y � 6C. 2x � y � 2 D. 2x � y � 2 11.
x � 3y � 6 x � 3y � 6
33
y
xO(4, �1)
y
xO
© Glencoe/McGraw-Hill 152 Glencoe Algebra 2
Chapter 3 Test, Form 2A (continued)
12. Find the coordinates of the vertices A. (0, 0), (3, 0), (3, 2), (0, 2)of the figure formed by the system x � �1, y � �2, and 2x � y � 6.
12.For Questions 13–15, use the system of inequalities x � 2, y � x � �3,and x � y � 5.13. Find the coordinates of the vertices of the feasible region.
A. (2, �1), (2, 3), (4, 1) B. (0, �3), (0, 5), (4, 1)C. (2, 0), (3, 0), (4, 1), (2, 3) D. (0, 0), (0, 5), (3, 0), (4, 1) 13.
14. Find the maximum value of f(x, y) � x � 4y for the feasible region.A. 14 B. 0 C. 8 D. 6 14.
15. Find the minimum value of f(x, y) � x � 4y for the feasible region.A. �2 B. 0 C. �10 D. �4 15.
16. What is the value of z in the solution of the system of equations?
A. 4 B. 1 C. �2 D. �34� 16.
An office building containing 96,000 square feet of space is to be made into apartments. There will be at most 15 one-bedroom units, each with 800 square feet of space. The remaining units, each with 1200 square feet of space, will have two bedrooms. Rent for each one-bedroom unit will be $650 and for each two-bedroom unit will be $900.17. Let x represent the number of one-bedroom apartments and y represent
the number of two-bedroom apartments. Which system of inequalities represents the number of apartments to be built? A. x � 15, y � 0, 650x � 900y � 96,000B. x � 15, y � 0, 800x � 1200y � 96,000C. x � 650, y � 900, 800x � 1200y � 96,000D. x � 15, y � 0, 800x � 1200y � 96,000 17.
18. How many two-bedroom apartments should be built to maximize revenue? A. 70 B. 15 C. 80 D. 120 18.
At a university, 1200 students are enrolled in engineering. There are twice as many in electrical engineering as in mechanical engineering,and three times as many in chemical engineering as in mechanical engineering.19. Which system of equations represents the number of students in each program?
A. c � m � e � 1200, 2m � e, 3m � c B. c � m � e � 1200, 3m � e, 2m � cC. c � m � e � 1200, 2e � m, 3c � m D. c � m � e � 1200, 2m � e, 3m � 2e 19.
20. How many students are enrolled in the mechanical engineering program?A. 200 B. 400 C. 600 D. 1200 20.
Bonus Find the value of x in the solution of the system of
equations x � y � �98� and x � 2y � �
98�. B:
2x � 3y � z � 9x � 2y � z � 4x � 3y � 2z � �3
NAME DATE PERIOD
33
A. (0, 0), (3, 0), (0, 6)B. (�1, 8), (�1, �2), (4, �2)C. (0, 0), (0, 3), (6, 0)D. (�1, �2), (�1, 6), (4, 0)
Chapter 3 Test, Form 2B
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 153 Glencoe Algebra 2
Ass
essm
entWrite the letter for the correct answer in the blank at the right of each question.
1. The system of equations y � 2x � 3 and y � 4x � 3 has A. exactly one solution. B. no solution.C. infinitely many solutions. D. exactly two solutions. 1.
Choose the correct description of each system of equations.A. consistent and independent B. consistent and dependentC. inconsistent D. inconsistent and dependent 2.
2. x � 2y � 7 3. 2x � 3y � 10 3.3x � 2y � 5 4x � 6y � 20
To solve each system of equations, which expression could be substituted for x into the first equation?
4. 3x � 5y � 14 A. 10 � 4y B. 4y � 10x � 4y � 10 C. �
14�x � �
52� D. ��
14�x � �
52� 4.
5. 2x � 7y � 10 A. �12�x � 15 B. �
12�x � 15
x � 2y � 15 C. 2y � 15 D. 2y � 15 5.
6. The first equation of the system is multiplied by 2.By what number would you multiply the second equation to eliminate the x variable by adding? A. 3 B. �3 C. 2 D. �2 6.
7. The first equation of the system is multiplied by 4.By what number would you multiply the second equation to eliminate the y variable by adding?A. 5 B. �5 C. 2 D. �2 7.
For Questions 8 and 9, solve each system of equations.8. 4x � 3y � 14 A. (1, 1) B. (5, 2)
y � �3x � 4 C. (�4, �10) D. (2, �2) 8.
9. 4x � 3y � 8 A. (�2, 1) B. (2, 0)2x � 5y � �9 C. (0, �83) D. ��
12�, �2� 9.
10. Which system of equations is graphed?A. 2x � y � 1 B. 2x � y � �1
�3x � y � 3 3x � y � 3C. 2x � y � 1 D. 2x � y � �1 10.
3x � y � 3 �3x � y � 3
11. Which system of inequalities is graphed?A. 2x � y � 5 B. 2x � y � 5
3x � 2y � 9 3x � 2y � 9C. 2x � y � �5 D. �2x � y � 5 11.
3x � 2y � 9 3x � 2y � 9
2x � 5y � 168x � 4y � 10
6x � 5y � 214x � 7y � 15
33
y
xO
(�2, 3)
yxO
© Glencoe/McGraw-Hill 154 Glencoe Algebra 2
Chapter 3 Test, Form 2B (continued)
12. Find the coordinates of the vertices of the figure formed by the system x � 0, y � �2, and 2x � y � 4.A. (3, �2), (0, 4), (0, �2) B. (�2, 0), (4, 0), (�2, 3)C. (0, 0), (0, 4), (2, 0) D. (�2, 3), (0, 4), (0, �2) 12.
For Questions 13–15, use the system of inequalities y � 1, y � x � 6,and x � 2y � 6.13. Find the coordinates of the vertices of the feasible region.
A. (�6, 0), (�2, 4), (6, 0) B. (�5, 1), (�2, 4), (4, 1)C. (0, 1), (0, 3), (4, 1) D. (�5, 1), (�2, 4), (0, 3), (0, 1) 13.
14. Find the maximum value of f(x, y) � 2x � y for the feasible region.A. 0 B. 11 C. 9 D. 8 14.
15. Find the minimum value of f(x, y) � 2x � y for the feasible region.A. �10 B. 0 C. �9 D. �4 15.
16. What is the value of z in the solution of the system of equations?
A. �1 B. 12 C. 3 D. �2 16.
Tickets to a golf tournament are sold in advance for $40 each, and on the day of the event for $50 each. For the tournament to occur, at least 2000 of the 8000 tickets must be sold in advance.17. Let a represent the number of advance tickets sold and d represent the
number sold on the day of the tournament. Which system of inequalities represents the number of tickets sold? A. a � 2000, d � 0, a � d � 8000 B. a � 0, d � 0, a � d � 8000C. a � 0, d � 0, a � d � 2000 D. a � 40, d � 50, a � d � 2000 17.
18. How many advance tickets should be sold to maximize revenue?A. 6000 B. 2000 C. 4000 D. 8000 18.
A local gas station sells low-grade (�), mid-grade (m), and premium (p)gasoline. Mid-grade gasoline costs $0.10 per gallon more than low-grade,and premium gasoline costs $0.10 per gallon more than mid-grade gasoline. Five gallons of low-grade gas cost $9.19. Which system of equations represents the cost of each type of gasoline?
A. 5� � m � 9, m � � � 0.10, p � m � 0.10B. 5� � 9, m � � � 0.10, p � m � 0.10C. 5� � 9, m � ��0.10, p � m � 0.10D. 0.10� � 0.10m � 5p � 9, 0.10� � m � 0, 0.10m � p � 0 19.
20. What is the cost of one gallon of premium gasoline?A. $1.80 B. $1.90 C. $2.00 D. $2.10 20.
Bonus Solve the system of equations. B:a � b � 6 c � d � 4 f � a � 2b � c � 5 d � f � 3
2x � 3y � z � 124x � y � z � �3�2x � 2y � z � 3
NAME DATE PERIOD
33
Chapter 3 Test, Form 2C
© Glencoe/McGraw-Hill 155 Glencoe Algebra 2
Solve each system of equations by graphing.
1. 3x � 2y � 62x � y � 4
2. 3x � y � 13y � 9x � 6
Describe each system of equations as consistent andindependent, consistent and dependent, or inconsistent.
3. y � 2x � 5 4. 2x � y � 5y � �3x � 4 6x � 3y � 15
Solve each system of equations by using substitution.
5. 3x � 7y � 19 6. x � 3y � 12x � y � 5 5x � y � 4
Solve each system of equations by using elimination.
7. 3x � 2y � 4 8. 4x � y � 102x � 3y � 7 5x � 2y � 6
Solve each system of inequalities by graphing.
9. 4x � 3y � 92x � y � 5
10. y � �32�x � 2
2y � x � 4
Find the coordinates of the vertices of the figure formed by each system of inequalities.
11. y � �3 12. x � 3y � 2x � 1 y � 2x � 4x � 2 x � y � �2
3y � �2x � 12
NAME DATE PERIOD
SCORE 33
Ass
essm
ent1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
y
xO
y
xO
y
xO
y
xO
© Glencoe/McGraw-Hill 156 Glencoe Algebra 2
Chapter 3 Test, Form 2C (continued)
Use the system of inequalities x � �2, x � y � 7, and y � 2x � 1.
13. Find the coordinates of the vertices of the feasible region.
14. Find the maximum and minimum values of the function f(x, y) � 3x � y for the feasible region.
The area of a parking lot is 600 square meters. A car requires 6 square meters and a bus requires 30 square meters of space. The lot can handle a maximum of 60 vehicles.
15. Let c represent the number of cars and b represent the number of buses. Write a system of inequalities to represent the number of vehicles that can be parked.
16. If a car costs $3 and a bus costs $8 to park in the lot, determine the number of each vehicle to maximize the amount collected.
Solve each system of equations.
17. x � 2y � 3z � 5x � y � 2z � �3x � y � z � 2
18. 3x � y � 2z � 12x � y � z � �3x � y � 4z � �3
A printing company sells small packages of personalized stationery for $7 each, medium packages for $12 each,and large packages for $15 each. Yesterday, the company sold 9 packages of stationery, collecting a total of $86.Three times as many medium packages were sold as large packages.
19. Let s represent the number of small packages, m the number of medium packages, and � the number of large packages. Write a system of three equations that represents the number of packages sold.
20. Find the number of each size package sold.
Bonus Find the perimeter of the region defined by the system of inequalities:�2 � x � 5�4 � y � �1 B:
NAME DATE PERIOD
33
13.
14.
15.
16.
17.
18.
19.
20.
Chapter 3 Test, Form 2D
© Glencoe/McGraw-Hill 157 Glencoe Algebra 2
Solve each system of equations by graphing.
1. x � y � 52y � x � 2
2. y � �23�x � 1
2x � y � �1
Describe each system of equations as consistent andindependent, consistent and dependent, or inconsistent.
3. 3x � 4y � 5 4. 2x � 7y � 146x � 8y � �5 x � 3y � 6
Solve each system of equations by using substitution.
5. 4x � y � 10 6. x � y � 6y � 3x � 6 3x � 2y � �22
Solve each system of equations by using elimination.
7. 5x � 2y � 1 8. 5x � 3y � 162x � 3y � 7 2x � 7y � �10
Solve each system of inequalities by graphing.
9. 2x � 3y � �33y � �2x � 6
10. x � 3y � 6
y � �32�x � 2
Find the coordinates of the vertices of the figure formed by the solution of each system of inequalities.
11. x � �3 12. y � 3y � �2 4y � 3x � 122x � y � �2 x � y � �4
y � 2x � 1
NAME DATE PERIOD
SCORE 33
Ass
essm
ent1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
y
xO
y
xO
y
xO
y
xO
© Glencoe/McGraw-Hill 158 Glencoe Algebra 2
Chapter 3 Test, Form 2D (continued)
For Questions 13 and 14, use the system of inequalities y � 7, x � y � 2, y � 2x � 5.
13. Find the coordinates of the vertices of the feasible region. 13.
14. Find the maximum and minimum values of the function 14.f(x, y) � 3x � y for the feasible region.
Kristin earns $7 per hour at a video store and $10 per hour at a landscaping company. She must work at least 4 hours per week at the video store, but the total number of hours she works at both jobs cannot be greater than 15.
15. Let v represent the number of hours working at the video 15.store and � represent the number of hours working at the landscaping company. Write a system of inequalities to represent the number of hours worked in one week.
16. Determine Kristin’s maximum weekly earnings (before 16.deductions).
Solve each system of equations.
17. 2x � y � z � 4 18. 3x � y � z � 12 17.3x � y � 4z � 11 2x � 3y � z � 5x � y � 5z � 20 x � 2y � z � 9 18.
The price of a sweatshirt at a local shop is twice the price of a pair of shorts. The price of a T-shirt at the shop is $4 less than the price of a pair of shorts. Brad purchased 3 sweatshirts, 2 pairs of shorts, and 5 T-shirts for a total cost of $136.
19. Let w represent the price of one sweatshirt, t represent the 19.price of one T-shirt, and h represent the price of one pair of shorts. Write a system of three equations that represents the prices of the clothing.
20. Find the cost of one sweatshirt. 20.
Bonus Solve the system of equations. B:
�23�x � �
12�y � 4
��16�x � �
18�y � �3
NAME DATE PERIOD
33
Chapter 3 Test, Form 3
© Glencoe/McGraw-Hill 159 Glencoe Algebra 2
Solve each system of equations by graphing.
1. �13�x � �
12�y � 1
�12�x � �
14�y � ��
12�
2. x � 0.5y � 0.52.5 � 1.5x � y
Describe each system of equations as consistent andindependent, consistent and dependent, or inconsistent.
3. 2x � �23�y � �
130� 4. �1
34�
x � �114�
y � �12�
9x � 3y � 15 6x � 2( y � 5)
Solve each system of equations by using substitution.
5. �12�x � �
13�y � 1 6. �
45�a � b � �
34� � 0
�23�y � x � 6 2a � �
54� � �
52�b
Solve each system of equations by using elimination.
7. �23�x � 4y � �1 8. 0.2c � 1.5d � �2.7
�12�x � 3y � ��
94�
1.2c � 0.5d � 2.8
Solve each system of inequalities by graphing.
9. x � y � 5x � y � �3x � 2y � 6x � 2y � �2
10. � x � � 1� y � 1 � � 3x � 3y � �6
Find the coordinates of the vertices of the figure formed by the solution of each system of inequalities.
11. x � 2 12. �2 � x � 5�4 � y � 3
�12�x � �
12�y � 1
x � y � �32x � y � � 2y � �4
NAME DATE PERIOD
SCORE 33
Ass
essm
ent1.
2.
3.
4.
5.
6.
7.
8.9.
10.
11.
12.
y
xO
y
xO
y
xO
y
xO
© Glencoe/McGraw-Hill 160 Glencoe Algebra 2
Chapter 3 Test, Form 3 (continued)
For Questions 13 and 14, use the system of inequalities
x � �3, x � y � �4, x � y � 1, and �13�x � �
12�y � 2.
13. Find the coordinates of the vertices of the feasible region. 13.
14. Find the maximum and minimum values of the function
f(x, y) � 3x � �12�y for the feasible region. 14.
A dog food manufacturer wants to advertise its products.A magazine charges $60 per ad and requires the purchase of at least three ads. A radio station charges $150 per commercial minute and requires the purchase of at least four minutes. Each magazine ad reaches 12,000 people while each commercial minute reaches 16,000 people. At most $900 can be spent on advertising.
15. Let a represent the number of magazine ads and m represent 15.the number of commercial minutes. Write a system of inequalities that represents the advertising plan for the company.
16. How many ads and commercial minutes should be purchased 16.to reach the most people? How many people would this be?
Solve each system of equations.
17. 4x � 6y � z � 1 18. 10a � b � �13�c � �6
17.3x � �
12�y � �
23�z � �
92� 2a � 3b � �
17�c � �
352�
5x � 3y � 2z � ��121� ��
52�a � �
14�c � �
149� 18.
An electronic repair shop offers three types of service:on-site, at-store, and by-mail. On-site service costs 3 times as much as at-store service. By-mail service costs $10 less than at-store service. Last week, the shop completed 15 services on-site, 40 services at-store, and 5 services by mail for total sales of $2650.
19. Let s represent the cost of one on-site repair, a represent 19.the cost of one at-store repair, and m represent the cost of one by-mail repair service. Write a system of three equations that represents the cost of each repair service.
20. Determine the cost of one on-site repair service. 20.
Bonus Solve the system of equations. B:
�3x� � �
4y� � �
52�
�6x� � �
1y� � �
12�
NAME DATE PERIOD
33
Chapter 3 Open-Ended Assessment
© Glencoe/McGraw-Hill 161 Glencoe Algebra 2
Demonstrate your knowledge by giving a clear, concise solutionto each problem. Be sure to include all relevant drawings andjustify your answers. You may show your solution in more thanone way or investigate beyond the requirements of the problem.
1. The square of Janet’s age is 400 more than the square of the sumof Kim’s and Sue’s ages. Kim’s and Sue’s ages total 10 less thanJanet’s age. Find the square of the sum of the ages of Janet, Kim,and Sue. Explain your reasoning.
2. A feasible region has vertices at (0, 8), (6, 0), and (0, 0).a. Write a system of inequalities whose graph forms this feasible
region.b. Explain how to find the maximum and minimum values of
f(x, y) � x � y for the region.
3. Explain what the algebraic solution of a system of two linearequations is and what that means in terms of the graphs of theequations of the system.
4. When describing a system of two linear equations, a studentindicated that the system was inconsistent and dependent.Discuss the meaning of the student’s description. Then assess thestudent’s understanding of these terms.
5. A business owner asks the company finance manager to developa formula, or function, for the cost incurred in the production oftheir products, and another function for the revenue (moneycollected) that the company earns when the products are sold. Inpreparing the report for the owner, the finance manager preparesa graph which shows both the cost function and the revenuefunction. The business owner, on seeing the graph of two parallellines, makes a major business decision. What might that decisionbe and why would it be made?
6. Explain what it means for a system of two linear inequalities tohave no solution. Sketch a graph of such a system and write thesystem of inequalities represented by your graph.
NAME DATE PERIOD
SCORE 33
Ass
essm
ent
© Glencoe/McGraw-Hill 162 Glencoe Algebra 2
Chapter 3 Vocabulary Test/Review
Choose from the terms above to complete each sentence.
1. A system of equations with no solutions is called a(n)
.
2. A(n) is a system of equations withan infinite number of solutions.
3. (3, �2, 7) is an example of a(n) .
4. If your first step in solving a system of equations is to solve one of the equations for one variable in terms of the others, you are using the
.
5. In a linear programming problem, the inequalities are called
.
6. A set of two or more inequalities that are considered together is called
a(n) .
7. If you are solving a system of equations and one of your steps is to add
the equations, you are using the .
8. If you are solving a system of equations by graphing and your graph shows two intersecting lines, the system can be described both as a(n)
and as a(n) .
9. Graphing, the substitution method, and the elimination method are all
methods for solving a(n) .
10. The process of finding the maximum or minimum values of a function
defined by a feasible region is called .
In your own words—Define each term.
11. feasible region
12. unbounded region
bounded regionconsistent systemconstraintsdependent system
elimination methodfeasible regioninconsistent systemindependent system
linear programmingordered triplesubstitution methodsystem of equations
system of inequalitiesunbounded regionvertices
NAME DATE PERIOD
SCORE 33
Chapter 3 Quiz (Lessons 3–1 and 3–2)
33
© Glencoe/McGraw-Hill 163 Glencoe Algebra 2
1. Solve the system x � y � 4 and x � 2y � 1 by graphing.
2. Graph the system y � x � 2 and 2y � 2x � 4. Describe it as consistent and independent, consistent and dependent,or inconsistent.
3. Solve the system 3y � 2x, y � �23�x � 2, by using substitution.
4. Solve the system of equations by 4x � 5y � �2, and 3x � 2y � �13 using elimination.
5. Standardized Test Practice Compare the quantity in Column A and the quantity in Column B. Then determine whether:A. the quantity in Column A is greater,B. the quantity in Column B is greater,C. the two quantities are equal, or D. the relationship cannot be determined from the
information given.2x � 3y � 25x � y � 22
Column A Column B
NAME DATE PERIOD
SCORE
Chapter 3 Quiz (Lesson 3–3)
Solve each system of inequalities by graphing.
1. x � y � �3 2. y � �13�x � 1
2x � y � 6y � �
13�x � 2
Find the coordinates of the vertices of the figure formed by each system of inequalities.
3. x � 0 4. y � �2y � 0 x � 32x � y � 4 x � y � 2
y � 2x � 4
NAME DATE PERIOD
SCORE 33
Ass
essm
ent1.
2.
3.
4.
5.
y
xO
y
xO
1.
2.
3.
4.
y
xO
y
xO
yx
© Glencoe/McGraw-Hill 164 Glencoe Algebra 2
1. A feasible region has vertices at (�2, 3), (1, 6), (1, �1), and (�3, �2). Find the maximum and minimum of the function f(x, y) � �x � 2y over this region.
2. Graph the system of inequalities. Name the coordinates of the vertices of the feasible region. Find the maximum and minimum values of the given function for this region.4y � x � 12�4y � 3x � 45x � 4y � 4f(x, y) � x � 5y
A clothing company makes jackets and pants. Each jacket requires 1 hour of cutting and 4 hours of sewing. Each pair of pants requires 2 hours of cutting and 2 hours of sewing.The total time per day available for cutting is 20 hours and for sewing is 32 hours.
3. Let j represent the number of jackets and let p represent the number of pairs of pants. Write a system of inequalities to represent the number of items that can be produced.
4. If the profit on a jacket is $14 and the profit on a pair of pants is $8, determine the number of each that should be made each day to maximize profit. What is the maximum profit?
Chapter 3 Quiz (Lesson 3–5)
Solve each system of equations.1. 4x � 6y � 3z � 20 2. 3x � y � 2z � 1 1.
x � 5y � z � �15 �2x � �4�7x � y � 2z � 1 x � 3y � 11 2.
3. x � 2y � z � �73.3x � 3y � z � 13
2x � 5y � 2z � 0
During one month, a rental car agency rented a total of 155 cars, vans, and trucks. Nine times as many cars were rented as vans, and three times as many vans were rented as trucks.
4. Let x represent the number of cars, let y represent the 4.number of vans, and let z represent the number of trucks.Write a system of three equations that represents the number of vehicles rented. 5.
5. Find the number of each type of vehicle rented.
NAME DATE PERIOD
SCORE
Chapter 3 Quiz (Lesson 3–4)
33
NAME DATE PERIOD
SCORE
33
1.
2.
3.
4.
y
xO
Chapter 3 Mid-Chapter Test (Lessons 3–1 through 3–3)
© Glencoe/McGraw-Hill 165 Glencoe Algebra 2
Write the letter for the correct answer in the blank at the right of each question.
1. Choose the correct description 3x � y � 5of the system of equations. 6x � 2y � 5 A. consistent and independentB. consistent and dependentC. inconsistentD. inconsistent and dependent 1.
2. Solve 3x � 2y � 7 and x � 4y � �21 by using substitution.
A. (3, �1) B. ��73�, �
72�� C. (�1, 5) D. (1, 5) 2.
Solve each system of equations by using elimination.
3. 2x � 5y � 18 A. (�1, 4) B. �9, �158��
3x � 2y � �11 C. (1, 4) D. (�3, 1) 3.
4. 3x � 5y � 14 A. (3, �1) B. (8, 2)2x � 3y � 3 C. (0, 1) D. (6, �3) 4.
5. Solve 3x � y � �4 and x � 2y � �6 by graphing.
6. Solve 2x � y � �1 and x � y � 4 by graphing.
7. Classify the system as consistent 3x � y � 5and independent, consistent and 6x � 2( y � 5)dependent, or inconsistent.
8. Solve the system of inequalities 4x � y � 4by graphing. 3y � �x � 6
9. Find the coordinates of the x � �2vertices of the figure formed y � 6by the system of inequalities. y � ��
32�x � 6
x � 2y � 4
Part II
Part I
NAME DATE PERIOD
SCORE 33
Ass
essm
ent
5.
6.
7.
8.
9.
y
xO
y
xO
y
xO
© Glencoe/McGraw-Hill 166 Glencoe Algebra 2
Chapter 3 Cumulative Review (Chapters 1–3)
1. Simplify �13�(6x � 21) � 4(x � 5). (Lesson 1-2) 1.
2. Solve 7 � 2(m � 3) � 4 � m. (Lesson 1-3) 2.
3. Solve the inequality � 3 � 2x � � 7. Then graph the solution 3.set. (Lesson 1-6)
4. Find f(2a) if f(x) � � x3 � 2x � 5. (Lesson 2-1) 4.
For Questions 5 and 6, state whether each equation or function is linear. If not, explain. (Lesson 2-2)
5. f(x) � �1x� � 5 6. y � x2 � 2
7. Write an equation for the line that passes through (2, �3) 7.and is parallel to the line whose equation is y � �4x � 3.(Lesson 2-4)
8. Evaluate f��18�� if f(x) � 5 � 3x. (Lesson 2-6) 8.
9. Describe the system of equations as consistent and 9.independent, consistent and dependent, or inconsistent.2x � 3y � 114x � 6y � 22 (Lesson 3-1)
10. Solve the system of equations y � 2x � 5 10.by using substitution. 4x � 5y � �1(Lesson 3-2)
11. Solve the system of equations y � 3x � 5 11.by using elimination. 4x � 9y � �22(Lesson 3-2)
For Questions 12 and 13, use the system of inequalities x � 1, y � �2, and x � y � 4.
12. Find the coordinates of the vertices of the figure formed by 12.the system of inequalities. (Lesson 3-3)
13. Find the maximum and minimum values of the function 13.f(x, y) � y � 3x for the feasible region. (Lesson 3-4)
14. Solve the system of equations. 2x � y � 3z � 9 14.(Lesson 3-5) x � 2y � z � �8
x � 3y � 2z � 11
�1�2�3�4�5�6 0 1 2
NAME DATE PERIOD
33
5.
6.
Standardized Test Practice (Chapters 1–3)
© Glencoe/McGraw-Hill 167 Glencoe Algebra 2
1. A number is 8 more than the ratio of q and 2. What is three-fourths of the number in terms of q?
A. 6 B. 5 � �38q� C. �
38q� D. 6 � �
38q� 1.
2. Carlisle’s salary was raised from $19,000 to $23,750. Find the percent of increase.E. 20% F. 25% G. 30% H. 47.5% 2.
3. Which of the following fractions is more than one-fourth but less than three-eights?
A. �27� B. �
28� C. �
34� D. �
12� 3.
4. If the area of a square is 49, the difference between the length of a diagonal and the length of a side is _____.E. 7�2� � 7 F. 0 G. �2� H. 1 � �2� 4.
5. Which is not equal to �2�?
A. 2�12�
B. �4
4� C. ��2��2 D. �6
8� 5.
6. Find the area of the triangle at the right.E. 32 F. 48G. 60 H. 120 6.
7. If m is an integer greater than 5, then which of the following must represent an odd integer?A. m2 B. m � 1 C. 2m � 3 D. m � 2 7.
8. Desiree and Mario together have $30. Mario and Scott together have $26. Desiree and Scott together have $34. What is the least amount of money any of them has?E. $11 F. $15 G. $19 H. $26 8.
9. If x, y, and z are negative integers, which of the following must be true?A. x(y � z) � 0 B. xyz � 0C. x � (y � z) � 0 D. x � y � z 9.
10. If �r �r
2� � �
43�, then �2
r� � _____.
E. �14� F. �
23� G. 1 H. 3 10. HGFE
DCBA
HGFE
DCBA
HGFE
DCBA
HGFE
DCBA
HGFE
DCBA
NAME DATE PERIOD
33
Ass
essm
ent
10
12
10
Part 1: Multiple Choice
Instructions: Fill in the appropriate oval for the best answer.
© Glencoe/McGraw-Hill 168 Glencoe Algebra 2
Standardized Test Practice (continued)
11. If the operation � is defined by the equation 11. 12.a � b � ab � b2, what is the value of n in theequation 4 � n � 4?
12. In a survey of high school sophomores,60 students named Science as their favorite subject. If 70% of the students surveyed had a favorite subject other than Science, how many sophomores were surveyed?
13. If figure RSTV is 13. 14.a parallelogram,what is the value of x?
14. What number is in the hundreds place in the product of 540 and 9?
Column A Column B
15. p � r3 � 4, q � r3 � 4 15.
16. 16.
17. 17.
18. x � 0 18.
19. 2x � 1 � 13 and 5y � 14 19.
5y2x
DCBA
�xx
1
6
0��
xx
6
2�
DCBA
DCBA0.09�0.09�
DCBA0.4 2c�25
� of 6c
qp
DCBA
Part 3: Quantitative Comparison
Instructions: Compare the quantities in columns A and B. Shade in if the quantity in column A is greater; if the quantity in column B is greater; if the quantities are equal; or if the relationship cannot be determined from the information given.
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
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87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
NAME DATE PERIOD
33
NAME DATE PERIOD
Part 2: Grid In
Instructions: Enter your answer by writing each digit of the answer in a column boxand then shading in the appropriate oval that corresponds to that entry.
R U
S T3y˚
5x˚
y˚
A
D
C
B
Standardized Test PracticeStudent Record Sheet (Use with pages 150–151 of the Student Edition.)
© Glencoe/McGraw-Hill A1 Glencoe Algebra 2
NAME DATE PERIOD
33
An
swer
s
Select the best answer from the choices given and fill in the corresponding oval.
1 4 7 9
2 5 8 10
3 6
Solve the problem and write your answer in the blank.
For Questions 14–18, also enter your answer by writing each number or symbol ina box. Then fill in the corresponding oval for that number or symbol.
11 14 16 18
12
13
15 17
Select the best answer from the choices given and fill in the corresponding oval.
19 21
20 22 DCBADCBA
DCBADCBA
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
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0 0 0
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.
99 9 987654321
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0 0 0
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.
99 9 987654321
87654321
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DCBADCBA
DCBADCBADCBADCBA
DCBADCBADCBADCBA
Part 2 Short Response/Grid InPart 2 Short Response/Grid In
Part 1 Multiple ChoicePart 1 Multiple Choice
Part 3 Quantitative ComparisonPart 3 Quantitative Comparison
© Glencoe/McGraw-Hill A2 Glencoe Algebra 2
Answers (Lesson 3-1)
Stu
dy G
uid
e a
nd I
nte
rven
tion
So
lvin
g S
yste
ms
of
Eq
uat
ion
s by
Gra
ph
ing
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-1
3-1
©G
lenc
oe/M
cGra
w-H
ill11
9G
lenc
oe A
lgeb
ra 2
Lesson 3-1
Gra
ph
Sys
tem
s o
f Eq
uat
ion
sA
sys
tem
of
equ
atio
ns
is a
set
of
two
or m
ore
equ
atio
ns
con
tain
ing
the
sam
e va
riab
les.
You
can
sol
ve a
sys
tem
of
lin
ear
equ
atio
ns
bygr
aph
ing
the
equ
atio
ns
on t
he
sam
e co
ordi
nat
e pl
ane.
If t
he
lin
es i
nte
rsec
t,th
e so
luti
on i
sth
at i
nte
rsec
tion
poi
nt.
Sol
ve t
he
syst
em o
f eq
uat
ion
s b
y gr
aph
ing.
x �
2y�
4x
�y
��
2W
rite
eac
h e
quat
ion
in
slo
pe-i
nte
rcep
t fo
rm.
x �
2y�
4→
y�
�2
x�
y�
�2
→y
��
x�
2
Th
e gr
aph
s ap
pear
to
inte
rsec
t at
(0,
�2)
.
CH
ECK
Su
bsti
tute
th
e co
ordi
nat
es i
nto
eac
h e
quat
ion
.x
�2y
�4
x�
y�
�2
0 �
2(�
2) �
40
�(�
2)�
�2
4 �
4✓
�2
��
2✓
Th
e so
luti
on o
f th
e sy
stem
is
(0,�
2).
Sol
ve e
ach
sys
tem
of
equ
atio
ns
by
grap
hin
g.
1.y
��
�1
2.y
�2x
�2
3.y
��
�3
y�
�4
(6,�
1)y
��
x�
4(2
,2)
y�
(4,1
)
4.3x
�y
�0
5.2x
��
�7
6.�
y�
2
x�
y�
�2
(1,3
)�
y�
1(�
4,3)
2x�
y�
�1
(�2,
�3)
( –2,
–3)
x
y
O
( –4,
3)
x
y
O
( 1, 3
)
x
y
O
x � 2
x � 2y � 3
( 4, 1
)
x
y
O
( 2, 2
)
x
y
O
( 6, –
1)
x
y
O
x � 4x � 2
x � 2x � 3
x
y
O
( 0, –
2)
x � 2
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill12
0G
lenc
oe A
lgeb
ra 2
Cla
ssif
y Sy
stem
s o
f Eq
uat
ion
sT
he
foll
owin
g ch
art
sum
mar
izes
th
e po
ssib
ilit
ies
for
grap
hs
of t
wo
lin
ear
equ
atio
ns
in t
wo
vari
able
s.
Gra
ph
s o
f E
qu
atio
ns
Slo
pes
of
Lin
esC
lass
ific
atio
n o
f S
yste
mN
um
ber
of
So
luti
on
s
Line
s in
ters
ect
Diff
eren
t sl
opes
Con
sist
ent
and
inde
pend
ent
One
Line
s co
inci
de (
sam
e lin
e)S
ame
slop
e, s
ame
Con
sist
ent
and
depe
nden
tIn
finite
ly m
any
y-in
terc
ept
Line
s ar
e pa
ralle
lS
ame
slop
e, d
iffer
ent
Inco
nsis
tent
Non
e y-
inte
rcep
ts
Gra
ph
th
e sy
stem
of
equ
atio
ns
x�
3y�
6an
d d
escr
ibe
it a
s co
nsi
sten
t a
nd
in
dep
end
ent,
2x�
y�
�3
con
sist
ent
an
d d
epen
den
t,or
in
con
sist
ent.
Wri
te e
ach
equ
atio
n i
n s
lope
-in
terc
ept
form
.
x�
3y�
6→
y�
x�
2
2x�
y�
�3
→y
�2x
�3
Th
e gr
aph
s in
ters
ect
at (
�3,
�3)
.Sin
ce t
her
e is
on
e so
luti
on,t
he
syst
em i
s co
nsi
sten
t an
d in
depe
nde
nt.
Gra
ph
th
e sy
stem
of
equ
atio
ns
and
des
crib
e it
as
con
sist
ent
an
d i
nd
epen
den
t,co
nsi
sten
t a
nd
dep
end
ent,
or i
nco
nsi
sten
t.
1.3x
�y
��
2 2.
x�
2y�
5 3.
2x�
3y�
06x
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�10
3x�
15 �
�6y
co
nsi
sten
t4x
�6y
�3
inco
nsi
sten
tan
d d
epen
den
tin
con
sist
ent
4.2x
�y
�3
5.4x
�y
��
26.
3x�
y�
2
x�
2y�
4co
nsi
sten
t 2x
��
�1
con
sist
ent
x�
y�
6 co
nsi
sten
t
and
ind
epen
den
t an
d d
epen
den
t an
d in
dep
end
ent
x
y
O
x
y
Ox
y
O
y � 2
x
y
Ox
y
Ox
y
O
1 � 3
x
y O
( –3,
–3)
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
So
lvin
g S
yste
ms
of
Eq
uat
ion
s by
Gra
ph
ing
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-1
3-1
Exer
cises
Exer
cises
Exam
ple
Exam
ple
© Glencoe/McGraw-Hill A3 Glencoe Algebra 2
An
swer
s
Answers (Lesson 3-1)
Skil
ls P
ract
ice
So
lvin
g S
yste
ms
of
Eq
uat
ion
s B
y G
rap
hin
g
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-1
3-1
©G
lenc
oe/M
cGra
w-H
ill12
1G
lenc
oe A
lgeb
ra 2
Lesson 3-1
Sol
ve e
ach
sys
tem
of
equ
atio
ns
by
grap
hin
g.
1.x
�2
2.y
��
3x�
63.
y�
4 �
3x
y�
0(2
,0)
y�
2x�
4(2
,0)
y�
�x
�1
(2,�
2)
4.y
�4
�x
5.y
��
2x�
26.
y�
x
y�
x�
2(3
,1)
y�
x�
5(3
,�4)
y�
�3x
�4
(1,1
)
7.x
�y
�3
8.x
�y
�4
9.3x
�2y
�4
x�
y�
1(2
,1)
2x�
5y�
8(4
,0)
2x�
y�
1(�
2,�
5)
Gra
ph
eac
h s
yste
m o
f eq
uat
ion
s an
d d
escr
ibe
it a
s co
nsi
sten
t a
nd
ind
epen
den
t,co
nsi
sten
t a
nd
dep
end
ent,
or i
nco
nsi
sten
t.
10.y
��
3x11
.y�
x�
512
.2x
�5y
�10
y�
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103x
�y
�15
inco
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sten
tco
nsi
sten
t an
d
con
sist
ent
and
d
epen
den
tin
dep
end
ent
( 5, 0
)
2
2x
y
O
x
y
O
x
y
O
( –2,
–5)
x
y
O( 4
, 0)
x
y
O( 2
, 1)
x
y
O
( 1, 1
)
x
y
O( 3
, –4)
x
y
O
( 3, 1
)
x
y
O
1 � 3
( 2, –
2)x
y
O( 2
, 0)
x
y
O( 2
, 0)
x
y
O
1 � 2
©G
lenc
oe/M
cGra
w-H
ill12
2G
lenc
oe A
lgeb
ra 2
Sol
ve e
ach
sys
tem
of
equ
atio
ns
by
grap
hin
g.
1.x
�2y
�0
2.x
�2y
�4
3.2x
�y
�3
y�
2x�
3(2
,1)
2x�
3y�
1(2
,1)
y�
x�
(3,�
3)
4.y
�x
�3
5.2x
�y
�6
6.5x
�y
�4
y�
1(�
2,1)
x�
2y�
�2
(2,�
2)�
2x�
6y�
4(1
,1)
Gra
ph
eac
h s
yste
m o
f eq
uat
ion
s an
d d
escr
ibe
it a
s co
nsi
sten
t a
nd
ind
epen
den
t,co
nsi
sten
ta
nd
dep
end
ent,
or i
nco
nsi
sten
t.
7.2x
�y
�4
8.y
��
x�
29.
2y�
8 �
x
x�
y�
2x
�y
��
4y
�x
�4
con
sist
ent
and
ind
ep.
inco
nsi
sten
tco
nsi
sten
t an
d d
ep.
SOFT
WA
RE
For
Exe
rcis
es 1
0–12
,use
th
e fo
llow
ing
info
rmat
ion
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rite
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20,0
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8,00
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4,00
0
x
y
O
x
y
O( 2
, 0)
x
y
O
1 � 2
y
( 1, 1
)x
O( 2
, –2)
x
y O( –
2, 1
)
x
y
O
( 3, –
3)
x
y
O( 2
, 1) x
y
O
( 2, 1
)
x
y
O
9 � 21 � 2
Pra
ctic
e (
Ave
rag
e)
So
lvin
g S
yste
ms
of
Eq
uat
ion
s B
y G
rap
hin
g
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-1
3-1
© Glencoe/McGraw-Hill A4 Glencoe Algebra 2
Answers (Lesson 3-1)
Readin
g t
o L
earn
Math
em
ati
csS
olv
ing
Sys
tem
s o
f E
qu
atio
ns
by G
rap
hin
g
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-1
3-1
©G
lenc
oe/M
cGra
w-H
ill12
3G
lenc
oe A
lgeb
ra 2
Lesson 3-1
Pre-
Act
ivit
yH
ow c
an a
sys
tem
of
equ
atio
ns
be
use
d t
o p
red
ict
sale
s?
Rea
d th
e in
trod
uct
ion
to
Les
son
3-1
at
the
top
of p
age
110
in y
our
text
book
.
•W
hic
h a
re g
row
ing
fast
er,i
n-s
tore
sal
es o
r on
lin
e sa
les?
on
line
sale
s
•In
wh
at y
ear
wil
l th
e in
-sto
re a
nd
onli
ne
sale
s be
th
e sa
me?
2005
Rea
din
g t
he
Less
on
1.T
he
Stu
dy T
ip o
n p
age
110
of y
our
text
book
say
s th
at w
hen
you
sol
ve a
sys
tem
of
equ
atio
ns
by g
raph
ing
and
fin
d a
poin
t of
in
ters
ecti
on o
f th
e tw
o li
nes
,you
mu
st a
lway
sch
eck
the
orde
red
pair
in
bot
hof
th
e or
igin
al e
quat
ion
s.W
hy
is i
t n
ot g
ood
enou
gh t
och
eck
the
orde
red
pair
in
just
on
e of
th
e eq
uat
ion
s?S
amp
le a
nsw
er:T
o b
e a
solu
tio
n o
f th
e sy
stem
,th
e o
rder
ed p
air
mu
stm
ake
bo
th o
f th
e eq
uat
ion
s tr
ue.
2.U
nde
r ea
ch s
yste
m g
raph
ed b
elow
,wri
te a
ll o
f th
e fo
llow
ing
wor
ds t
hat
app
ly:c
onsi
sten
t,in
con
sist
ent,
dep
end
ent,
and
ind
epen
den
t.
inco
nsi
sten
tco
nsi
sten
t;co
nsi
sten
t;d
epen
den
tin
dep
end
ent
Hel
pin
g Y
ou
Rem
emb
er
3.L
ook
up
the
wor
ds c
onsi
sten
tan
d in
con
sist
ent
in a
dic
tion
ary.
How
can
th
e m
ean
ing
ofth
ese
wor
ds h
elp
you
dis
tin
guis
h b
etw
een
con
sist
ent
and
inco
nsi
sten
t sy
stem
s of
equ
atio
ns?
Sam
ple
an
swer
:O
ne
mea
nin
g o
f co
nsi
sten
tis
“b
ein
g in
ag
reem
ent,”
or
“co
mp
atib
le,”
wh
ile o
ne
mea
nin
g o
f in
con
sist
ent
is “
no
t b
ein
g in
agre
emen
t”o
r “i
nco
mp
atib
le.”
Wh
en a
sys
tem
is c
on
sist
ent,
the
equ
atio
ns
are
com
pat
ible
bec
ause
bo
th c
an b
e tr
ue
at t
he
sam
e ti
me
(fo
rth
e sa
me
valu
es o
f x
and
y).
Wh
en a
sys
tem
is in
con
sist
ent,
the
equ
atio
ns
are
inco
mp
atib
le b
ecau
se t
hey
can
nev
er b
e tr
ue
at t
he
sam
e ti
me.
x
y
Ox
y
Ox
y
O
©G
lenc
oe/M
cGra
w-H
ill12
4G
lenc
oe A
lgeb
ra 2
Inve
stm
ents
The
fol
low
ing
grap
h sh
ows
the
valu
e of
tw
o di
ffer
ent
inve
stm
ents
ove
r ti
me.
Lin
e A
repr
esen
ts a
n i
nit
ial
inve
stm
ent
of $
30,0
00 w
ith
a b
ank
payi
ng
pass
book
sav
ings
inte
rest
.Lin
e B
rep
rese
nts
an in
itia
l inv
estm
ent
of $
5,00
0 in
a
prof
itab
le m
utua
l fun
d w
ith
divi
dend
s re
inve
sted
and
capi
tal g
ains
acc
epte
d in
sh
ares
.By
deri
ving
the
line
ar e
quat
ion
y�
mx
�b
for
A a
nd
B,y
ou c
an p
redi
ct
the
valu
e of
th
ese
inve
stm
ents
for
yea
rs t
o co
me.
1.T
he
y-in
terc
ept,
b,is
th
e in
itia
l in
vest
men
t.F
ind
bfo
r ea
ch o
f th
e fo
llow
ing.
a.li
ne
A30
,000
b.
lin
e B
5000
2.T
he
slop
e of
th
e li
ne,
m,i
s th
e ra
te o
f re
turn
.Fin
d m
for
each
of
the
foll
owin
g.a.
lin
e A
0.58
8b
.li
ne
B2.
73
3.W
hat
are
th
e eq
uat
ion
s of
eac
h o
f th
e fo
llow
ing
lin
es?
a.li
ne
Ay
�0.
588x
�30
b.
lin
e B
y�
2.73
x�
5
4.W
hat
wil
l be
th
e va
lue
of t
he
mu
tual
fu
nd
afte
r 11
yea
rs o
f in
vest
men
t?$3
5,03
0
5.W
hat
wil
l be
th
e va
lue
of t
he
ban
k ac
cou
nt
afte
r 11
yea
rs o
f in
vest
men
t?$3
6,46
8
6.W
hen
wil
l th
e m
utu
al f
un
d an
d th
e ba
nk
acco
un
t h
ave
equ
al v
alu
e?af
ter
11.6
7 ye
ars
of
inve
stm
ent
7.W
hich
inve
stm
ent
has
the
grea
test
pay
off a
fter
11
year
s of
inve
stm
ent?
the
mu
tual
fu
nd
A B25 203035 10 5 0
15
12
34
56
78
9
Amount Invested (thousands)
Year
s In
vest
ed
x
y
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-1
3-1
© Glencoe/McGraw-Hill A5 Glencoe Algebra 2
An
swer
s
Answers (Lesson 3-2)
Stu
dy G
uid
e a
nd I
nte
rven
tion
So
lvin
g S
yste
ms
of
Eq
uat
ion
s A
lgeb
raic
ally
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-2
3-2
©G
lenc
oe/M
cGra
w-H
ill12
5G
lenc
oe A
lgeb
ra 2
Lesson 3-2
Sub
stit
uti
on
To
solv
e a
syst
em o
f li
nea
r eq
uat
ion
s by
su
bst
itu
tion
,fir
st s
olve
for
on
eva
riab
le i
n t
erm
s of
th
e ot
her
in
on
e of
th
e eq
uat
ion
s.T
hen
su
bsti
tute
th
is e
xpre
ssio
n i
nto
the
oth
er e
quat
ion
an
d si
mpl
ify.
Use
su
bst
itu
tion
to
solv
e th
e sy
stem
of
equ
atio
ns.
2x�
y�
9x
�3y
��
6S
olve
th
e fi
rst
equ
atio
n f
or y
in t
erm
s of
x.
2x�
y�
9F
irst
equa
tion
�y
��
2x�
9S
ubtr
act
2xfr
om b
oth
side
s.
y�
2x�
9M
ultip
ly b
oth
side
s by
�1.
Su
bsti
tute
th
e ex
pres
sion
2x
�9
for
yin
to t
he
seco
nd
equ
atio
n a
nd
solv
e fo
r x.
x�
3y�
�6
Sec
ond
equa
tion
x�
3(2x
�9)
��
6S
ubst
itute
2x
�9
for
y.
x�
6x�
27 �
�6
Dis
trib
utiv
e P
rope
rty
7x�
27 �
�6
Sim
plify
.
7x�
21A
dd 2
7 to
eac
h si
de.
x�
3D
ivid
e ea
ch s
ide
by 7
.
Now
,su
bsti
tute
th
e va
lue
3 fo
r x
in e
ith
er o
rigi
nal
equ
atio
n a
nd
solv
e fo
r y.
2x�
y�
9F
irst
equa
tion
2(3)
�y
�9
Rep
lace
xw
ith 3
.
6 �
y�
9S
impl
ify.
�y
�3
Sub
trac
t 6
from
eac
h si
de.
y�
�3
Mul
tiply
eac
h si
de b
y �
1.
Th
e so
luti
on o
f th
e sy
stem
is
(3,�
3).
Sol
ve e
ach
sys
tem
of
lin
ear
equ
atio
ns
by
usi
ng
sub
stit
uti
on.
1.3x
�y
�7
2.2x
�y
�5
3.2x
�3y
��
34x
�2y
�16
3x�
3y�
3x
�2y
�2
(�1,
10)
(2,1
)(�
12,7
)
4.2x
�y
�7
5.4x
�3y
�4
6.5x
�y
�6
6x�
3y�
142x
�y
��
83
�x
�0
no
so
luti
on
(�2,
�4)
(3,�
9)
7.x
�8y
��
28.
2x�
y�
�4
9.x
�y
��
2x
�3y
�20
4x�
y�
12x
�3y
�2
(14,
�2)
��,3
�(�
8,�
6)
10.x
�4y
�4
11.x
�3y
�2
12.2
x�
2y�
42x
�12
y�
134x
�12
y�
8x
�2y
�0
�5,�
infi
nit
ely
man
y�
,�
2 � 34 � 3
1 � 4
1 � 2
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill12
6G
lenc
oe A
lgeb
ra 2
Elim
inat
ion
To
solv
e a
syst
em o
f li
nea
r eq
uat
ion
s by
eli
min
atio
n,a
dd o
r su
btra
ct t
he
equ
atio
ns
to e
lim
inat
e on
e of
th
e va
riab
les.
You
may
fir
st n
eed
to m
ult
iply
on
e or
bot
h o
f th
eeq
uat
ion
s by
a c
onst
ant
so t
hat
on
e of
th
e va
riab
les
has
th
e sa
me
(or
oppo
site
) co
effi
cien
t in
one
equ
atio
n a
s it
has
in
th
e ot
her
.
Use
th
e el
imin
atio
n m
eth
od t
o so
lve
the
syst
em o
f eq
uat
ion
s.2x
�4y
��
263x
�y
��
24
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
So
lvin
g S
yste
ms
of
Eq
uat
ion
s A
lgeb
raic
ally
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-2
3-2
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Mu
ltip
ly t
he
seco
nd
equ
atio
n b
y 4.
Th
en s
ubt
ract
the
equ
atio
ns
to e
lim
inat
e th
e y
vari
able
.2x
�4y
��
262x
�4y
��
263x
�y
��
24M
ultip
ly b
y 4.
12x
�4y
��
96�
10x
�70
x�
�7
Rep
lace
xw
ith
�7
and
solv
e fo
r y.
2x�
4y�
�26
2(�
7) �
4y�
�26
�14
�4y
��
26�
4y�
�12
y�
3T
he
solu
tion
is
(�7,
3).
Mu
ltip
ly t
he
firs
t eq
uat
ion
by
3 an
d th
e se
con
deq
uat
ion
by
2.T
hen
add
th
e eq
uat
ion
s to
elim
inat
e th
e y
vari
able
.3x
�2y
�4
Mul
tiply
by
3.9x
�6y
�12
5x�
3y�
�25
Mul
tiply
by
2.10
x�
6y�
�50
19x
��
38x
��
2
Rep
lace
xw
ith
�2
and
solv
e fo
r y.
3x�
2y�
43(
�2)
�2y
�4
�6
�2y
�4
�2y
�10
y�
�5
Th
e so
luti
on i
s (�
2,�
5).
Use
th
e el
imin
atio
n m
eth
od t
o so
lve
the
syst
em o
f eq
uat
ion
s.3x
�2y
�4
5x�
3y�
�25
Exer
cises
Exer
cises
Sol
ve e
ach
sys
tem
of
equ
atio
ns
by
usi
ng
elim
inat
ion
.
1.2x
�y
�7
2.x
�2y
�4
3.3x
�4y
��
104.
3x�
y�
123x
�y
�8
�x
�6y
�12
x�
4y�
25x
�2y
�20
(3,�
1)(1
2,4)
(�2,
�1)
(4,0
)
5.4x
�y
�6
6.5x
�2y
�12
7.2x
�y
�8
8.7x
�2y
��
1
2x�
�4
�6x
�2y
��
143x
�y
�12
4x�
3y�
�13
no
so
luti
on
(2,1
)in
fin
itel
y m
any
(�1,
3)
9.3x
�8y
��
610
.5x
�4y
�12
11.�
4x�
y�
�12
12.5
m�
2n�
�8
x�
y�
97x
�6y
�40
4x�
2y�
64m
�3n
�2
(6,�
3)(4
,�2)
�,�
2 �(�
4,6)
5 � 2
3 � 2y � 2
© Glencoe/McGraw-Hill A6 Glencoe Algebra 2
Answers (Lesson 3-2)
Skil
ls P
ract
ice
So
lvin
g S
yste
ms
of
Eq
uat
ion
s A
lgeb
raic
ally
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-2
3-2
©G
lenc
oe/M
cGra
w-H
ill12
7G
lenc
oe A
lgeb
ra 2
Lesson 3-2
Sol
ve e
ach
sys
tem
of
equ
atio
ns
by
usi
ng
sub
stit
uti
on.
1.m
�n
�20
2.x
�3y
��
33.
w�
z�
1m
�n
��
4(8
,12)
4x�
3y�
6(3
,�2)
2w�
3z�
12(3
,2)
4.3r
�s
�5
5.2b
�3c
��
46.
x�
y�
12r
�s
�5
(2,�
1)b
�c
�3
(13,
�10
)2x
�3y
�12
(3,2
)
Sol
ve e
ach
sys
tem
of
equ
atio
ns
by
usi
ng
elim
inat
ion
.
7.2p
�q
�5
8.2j
�k
�3
9.3c
�2d
�2
3p�
q�
5(2
,�1)
3j�
k�
2(1
,�1)
3c�
4d�
50(6
,8)
10.2
f�
3g�
911
.�2x
�y
��
112
.2x
�y
�12
f�
g�
2(3
,1)
x�
2y�
3(1
,1)
2x�
y�
6n
o s
olu
tio
n
Sol
ve e
ach
sys
tem
of
equ
atio
ns
by
usi
ng
eith
er s
ub
stit
uti
on o
r el
imin
atio
n.
13.�
r�
t�
514
.2x
�y
��
515
.x�
3y�
�12
�2r
�t
�4
(1,6
)4x
�y
�2
��,4
�2x
�y
�11
(3,5
)
16.2
p�
3q�
617
.6w
�8z
�16
18.c
�d
�6
�2p
�3q
��
6(3
,0)
3w�
4z�
8c
�d
�0
(3,3
)in
fin
itel
y m
any
19.2
u�
4v�
�6
20.3
a�
b�
�1
21.2
x�
y�
6u
�2v
�3
no
so
luti
on
�3a
�b
�5
(�1,
2)3x
�2y
�16
(4,�
2)
22.3
y�
z�
�6
23.c
�2d
��
224
.3r
�2s
�1
�3y
�z
�6
(�2,
0)�
2c�
5d�
3(�
4,1)
2r�
3s�
9(�
3,�
5)
25.T
he
sum
of
two
nu
mbe
rs i
s 12
.Th
e di
ffer
ence
of
the
sam
e tw
o n
um
bers
is
�4.
Fin
d th
e n
um
bers
.4,
8
26.T
wic
e a
nu
mbe
r m
inu
s a
seco
nd
nu
mbe
r is
�1.
Tw
ice
the
seco
nd
nu
mbe
r ad
ded
to t
hre
e ti
mes
th
e fi
rst
nu
mbe
r is
9.F
ind
the
two
nu
mbe
rs.
1,3
1 � 2
©G
lenc
oe/M
cGra
w-H
ill12
8G
lenc
oe A
lgeb
ra 2
Sol
ve e
ach
sys
tem
of
equ
atio
ns
by
usi
ng
sub
stit
uti
on.
1.2x
�y
�4
2.x
�3y
�9
3.g
�3h
�8
no
3x�
2y�
1(7
,�10
)x
�2y
��
1(3
,�2)
g�
h�
9so
luti
on
4.2a
�4b
�6
infi
nit
ely
5.2m
�n
�6
6.4x
�3y
��
6�
a�
2b�
�3
man
y5m
�6n
�1
(5,�
4)�
x�
2y�
7(�
3,�
2)
7.u
�2v
�8.
x�
3y�
169.
w�
3z�
1
�u
�2v
�5
no
so
luti
on
4x�
y�
9(1
,�5)
3w�
5z�
�4
��,
�
Sol
ve e
ach
sys
tem
of
equ
atio
ns
by
usi
ng
elim
inat
ion
.
10.2
r�
s�
511
.2m
�n
��
112
.6x
�3y
�6
3r�
s�
20(5
,�5)
3m�
2n�
30(4
,9)
8x�
5y�
12(�
1,4)
13.3
j�
k�
1014
.2x
�y
��
4n
o
15.2
g�
h�
64j
�k
�16
(6,8
)�
4x�
2y�
6so
luti
on
3g�
2h�
16(4
,�2)
16.2
t�
4v�
6in
fin
itel
y17
.3x
�2y
�12
18.
x�
3y�
11�
t�
2v�
�3
man
y2x
�y
�14
(6,3
)8x
�5y
�17
(4,3
)
Sol
ve e
ach
sys
tem
of
equ
atio
ns
by
usi
ng
eith
er s
ub
stit
uti
on o
r el
imin
atio
n.
19.8
x�
3y�
�5
20.8
q�
15r
��
4021
.3x
�4y
�12
infi
nit
ely
10x
�6y
��
13�
,�3 �
4q�
2r�
56(1
0,8)
x�
y�
man
y
22.4
b�
2d�
5n
o23
.s�
3y�
424
.4m
�2p
�0
�2b
�d
�1
solu
tio
ns
�1
(1,1
)�
3m�
9p�
5�
,�
25.5
g�
4k�
1026
.0.5
x�
2y�
527
.h�
z�
3n
o�
3g�
5k�
7(6
,�5)
x�
2y�
�8
(�2,
3)�
3h�
3z�
6so
luti
on
SPO
RTS
For
Exe
rcis
es 2
8 an
d 2
9,u
se t
he
foll
owin
g in
form
atio
n.
Las
t ye
ar t
he
voll
eyba
ll t
eam
pai
d $5
per
pai
r fo
r so
cks
and
$17
per
pair
for
sh
orts
on
ato
tal
purc
has
e of
$31
5.T
his
yea
r th
ey s
pen
t $3
42 t
o bu
y th
e sa
me
nu
mbe
r of
pai
rs o
f so
cks
and
shor
ts b
ecau
se t
he
sock
s n
ow c
ost
$6 a
pai
r an
d th
e sh
orts
cos
t $1
8.
28.W
rite
a s
yste
m o
f tw
o eq
uat
ion
s th
at r
epre
sen
ts t
he
nu
mbe
r of
pai
rs o
f so
cks
and
shor
tsbo
ugh
t ea
ch y
ear.
5x�
17y
�31
5,6x
�18
y�
342
29.H
ow m
any
pair
s of
soc
ks a
nd
shor
ts d
id t
he
team
bu
y ea
ch y
ear?
sock
s:12
,sh
ort
s:15
2 � 31 � 3
4 � 34 � 9
1 � 31 � 2
2 � 3
1 � 2
1 � 21 � 2
1 � 2
1 � 3
Pra
ctic
e (
Ave
rag
e)
So
lvin
g S
yste
ms
of
Eq
uat
ion
s A
lgeb
raic
ally
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-2
3-2
© Glencoe/McGraw-Hill A7 Glencoe Algebra 2
An
swer
s
Answers (Lesson 3-2)
Readin
g t
o L
earn
Math
em
ati
csS
olv
ing
Sys
tem
s o
f E
qu
atio
ns
Alg
ebra
ical
ly
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-2
3-2
©G
lenc
oe/M
cGra
w-H
ill12
9G
lenc
oe A
lgeb
ra 2
Lesson 3-2
Pre-
Act
ivit
yH
ow c
an s
yste
ms
of e
qu
atio
ns
be
use
d t
o m
ake
con
sum
erd
ecis
ion
s?
Rea
d th
e in
trod
uct
ion
to
Les
son
3-2
at
the
top
of p
age
116
in y
our
text
book
.
•H
ow m
any
mor
e m
inu
tes
of l
ong
dist
ance
tim
e di
d Yo
lan
da u
se i
nF
ebru
ary
than
in
Jan
uar
y?13
min
ute
s
•H
ow m
uch
mor
e w
ere
the
Feb
ruar
y ch
arge
s th
an t
he
Jan
uar
y ch
arge
s?$1
.04
•U
sin
g yo
ur
answ
ers
for
the
ques
tion
s ab
ove,
how
can
you
fin
d th
e ra
tepe
r m
inu
te?
Fin
d $
1.04
�13
.
Rea
din
g t
he
Less
on
1.S
upp
ose
that
you
are
ask
ed t
o so
lve
the
syst
em o
f eq
uat
ion
s 4x
�5y
�7
at t
he
righ
t by
th
e su
bsti
tuti
on m
eth
od.
3x�
y�
�9
Th
e fi
rst
step
is
to s
olve
on
e of
th
e eq
uat
ion
s fo
r on
e va
riab
le
in t
erm
s of
th
e ot
her
.To
mak
e yo
ur
wor
k as
eas
y as
pos
sibl
e,w
hic
h e
quat
ion
wou
ld y
ou s
olve
for
wh
ich
var
iabl
e? E
xpla
in.
Sam
ple
an
swer
:S
olv
e th
e se
con
d e
qu
atio
n f
or
yb
ecau
se in
th
ateq
uat
ion
th
e va
riab
le y
has
a c
oef
fici
ent
of
1.
2.S
upp
ose
that
you
are
ask
ed t
o so
lve
the
syst
em o
f eq
uat
ion
s 2x
�3y
��
2at
th
e ri
ght
by t
he
elim
inat
ion
met
hod
.7x
�y
�39
To
mak
e yo
ur
wor
k as
eas
y as
pos
sibl
e,w
hic
h v
aria
ble
wou
ld
you
eli
min
ate?
Des
crib
e h
ow y
ou w
ould
do
this
.S
amp
le a
nsw
er:
Elim
inat
e th
e va
riab
le y
;m
ult
iply
th
e se
con
d e
qu
atio
n b
y3
and
th
en a
dd
th
e re
sult
to
th
e fi
rst
equ
atio
n.
Hel
pin
g Y
ou R
emem
ber
3.T
he
subs
titu
tion
met
hod
an
d el
imin
atio
n m
eth
od f
or s
olvi
ng
syst
ems
both
hav
e se
vera
lst
eps,
and
it m
ay b
e di
ffic
ult
to
rem
embe
r th
em.Y
ou m
ay b
e ab
le t
o re
mem
ber
them
mor
e ea
sily
if
you
not
ice
wh
at t
he
met
hod
s h
ave
in c
omm
on.W
hat
ste
p is
th
e sa
me
inbo
th m
eth
ods?
Sam
ple
an
swer
:A
fter
fin
din
g t
he
valu
e o
f o
ne
of
the
vari
able
s,yo
u f
ind
the
valu
e o
f th
e o
ther
var
iab
le b
y su
bst
itu
tin
g t
he
valu
e yo
u h
ave
fou
nd
in o
ne
of
the
ori
gin
al e
qu
atio
ns.
©G
lenc
oe/M
cGra
w-H
ill13
0G
lenc
oe A
lgeb
ra 2
Usi
ng
Co
ord
inat
esF
rom
on
e ob
serv
atio
n p
oin
t,th
e li
ne
of s
igh
t to
a d
own
ed p
lan
e is
giv
en b
y y
�x
�1.
Th
is e
quat
ion
des
crib
es t
he
dist
ance
fro
m t
he
obse
rvat
ion
poi
nt
to t
he
plan
e in
a s
trai
ght
lin
e.F
rom
an
oth
er o
bser
vati
on p
oin
t,th
e li
ne
of
sigh
t is
giv
en b
y x
�3y
�21
.Wh
at a
re t
he
coor
din
ates
of
the
poin
t at
wh
ich
th
e cr
ash
occ
urr
ed?
Sol
ve t
he
syst
em o
f eq
uat
ion
s�y
�x
�1
.x
�3y
�21
x�
3y�
21x
�3(
x�
1) �
21S
ubst
itute
x�
1 fo
r y.
x�
3x�
3 �
214x
�24
x�
6
x�
3y�
216
�3y
�21
Sub
stitu
te 6
for
x.
3y�
15y
�5
Th
e co
ordi
nat
es o
f th
e cr
ash
are
(6,
5).
Sol
ve t
he
foll
owin
g.
1.T
he
lin
es o
f si
ght
to a
for
est
fire
are
as
foll
ows.
Fro
m R
ange
r S
tati
on A
:3x
�y
�9
Fro
m R
ange
r S
tati
on B
:2x
�3y
�13
Fin
d th
e co
ordi
nat
es o
f th
e fi
re.
(2,3
)
2.A
n a
irpl
ane
is t
rave
lin
g al
ong
the
lin
e x
�y
��
1 w
hen
it
sees
an
oth
er
airp
lan
e tr
avel
ing
alon
g th
e li
ne
5x�
3y�
19.I
f th
ey c
onti
nu
e al
ong
the
sam
e li
nes
,at
wh
at p
oin
t w
ill
thei
r fl
igh
t pa
ths
cros
s?(2
,3)
3.T
wo
min
e sh
afts
are
du
g al
ong
the
path
s of
th
e fo
llow
ing
equ
atio
ns.
x�
y�
1400
2x�
y�
1300
If t
he
shaf
ts m
eet
at a
dep
th o
f 20
0 fe
et,w
hat
are
th
e co
ordi
nat
es o
f th
e po
int
at w
hic
h t
hey
mee
t?(9
00,�
500)
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-2
3-2
© Glencoe/McGraw-Hill A8 Glencoe Algebra 2
Answers (Lesson 3-3)
Stu
dy G
uid
e a
nd I
nte
rven
tion
So
lvin
g S
yste
ms
of
Ineq
ual
itie
s by
Gra
ph
ing
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-3
3-3
©G
lenc
oe/M
cGra
w-H
ill13
1G
lenc
oe A
lgeb
ra 2
Lesson 3-3
Gra
ph
Sys
tem
s o
f In
equ
alit
ies
To s
olve
a s
yste
m o
f in
equa
litie
s,gr
aph
the
ineq
ualit
ies
in t
he s
ame
coor
dina
te p
lane
.The
sol
utio
n se
t is
rep
rese
nted
by
the
inte
rsec
tion
of
the
grap
hs.
Sol
ve t
he
syst
em o
f in
equ
alit
ies
by
grap
hin
g.y
�2x
�1
and
y�
�2
Th
e so
luti
on o
f y
�2x
�1
is R
egio
ns
1 an
d 2.
Th
e so
luti
on o
f y
��
2 is
Reg
ion
s 1
and
3.
Th
e in
ters
ecti
on o
f th
ese
regi
ons
is R
egio
n 1
,wh
ich
is
the
solu
tion
set
of
the
syst
em o
f in
equ
alit
ies.
Sol
ve e
ach
sys
tem
of
ineq
ual
itie
s b
y gr
aph
ing.
1.x
�y
�2
2.3x
�2y
��
13.
y
�1
x�
2y�
1x
�4y
��
12x
�2
4.y
��
35.
y�
�2
6.y
��
�1
y�
2xy
��
2x�
1y
�3x
�1
7.x
�y
�4
8.x
�3y
�3
9.x
�2y
�6
2x�
y�
2x
�2y
�4
x�
4y�
�4
x
y
Ox
y
O
x
y
O
x
y
Ox
y
O
x
y
O
x � 4x � 3
x � 2
x
y
Ox
y
Ox
y
O
x � 3
x � 3
x
y
O
Regi
on 3
Regi
on 1
Regi
on 2
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill13
2G
lenc
oe A
lgeb
ra 2
Fin
d V
erti
ces
of
a Po
lyg
on
al R
egio
nS
omet
imes
th
e gr
aph
of
a sy
stem
of
ineq
ual
itie
s fo
rms
a bo
un
ded
regi
on.Y
ou c
an f
ind
the
vert
ices
of
the
regi
on b
y a
com
bin
atio
nof
th
e m
eth
ods
use
d ea
rlie
r in
th
is c
hap
ter:
grap
hin
g,su
bsti
tuti
on,a
nd/
or e
lim
inat
ion
.
Fin
d t
he
coor
din
ates
of
the
vert
ices
of
the
figu
re f
orm
ed b
y 5x
�4y
�20
,y�
2x�
3,an
d x
�3y
�4.
Gra
ph t
he
bou
nda
ry o
f ea
ch i
neq
ual
ity.
Th
e in
ters
ecti
ons
of t
he
bou
nda
ry l
ines
are
th
e ve
rtic
es o
f a
tria
ngl
e.T
he
vert
ex (
4,0)
can
be
dete
rmin
ed f
rom
th
e gr
aph
.To
fin
d th
eco
ordi
nat
es o
f th
e se
con
d an
d th
ird
vert
ices
,sol
ve t
he
two
syst
ems
of e
quat
ion
s
y�
2x�
3an
dy
�2x
�3
5x�
4y�
20x
�3y
�4
x
y
O
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
So
lvin
g S
yste
ms
of
Ineq
ual
itie
s by
Gra
ph
ing
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-3
3-3
For
th
e fi
rst
syst
em o
f eq
uat
ion
s,re
wri
te t
he
firs
teq
uat
ion
in
sta
nda
rd f
orm
as
2x�
y�
�3.
Th
enm
ult
iply
th
at e
quat
ion
by
4 an
d ad
d to
th
e se
con
deq
uat
ion
.2x
�y
��
3M
ultip
ly b
y 4.
8x�
4y�
�12
5x�
4y�
20(�
)5x
�4y
�20
13x
�8
x�
Th
en s
ubs
titu
te x
�in
on
e of
th
e or
igin
al e
quat
ion
s an
d so
lve
for
y.
2 ���
y�
�3
�y
��
3
y�
Th
e co
ordi
nat
es o
f th
e se
con
d ve
rtex
are
�,4
�.3 � 13
8 � 13
55 � 13
16 � 138 � 13
8 � 13
8 � 13
For
th
e se
con
d sy
stem
of
equ
atio
ns,
use
su
bsti
tuti
on.
Su
bsti
tute
2x
�3
for
yin
th
ese
con
d eq
uat
ion
to
get
x�
3(2x
�3)
�4
x�
6x�
9 �
4�
5x�
13x
��
Th
en s
ubs
titu
te x
��
in t
he
firs
t eq
uat
ion
to
solv
e fo
r y.
y�
2 ����
3
y�
��
3
y�
�
Th
e co
ordi
nat
es o
f th
e th
ird
vert
ex a
re ��
2,�
2�.
1 � 53 � 5
11 � 526 � 5
13 � 5
13 � 5
13 � 5
Exer
cises
Exer
cises
Th
us,
the
coor
din
ates
of
the
thre
e ve
rtic
es a
re (
4,0)
, �,4
�,an
d ��2
,�2
�.
Fin
d t
he
coor
din
ates
of
the
vert
ices
of
the
figu
re f
orm
ed b
y ea
ch s
yste
m o
fin
equ
alit
ies.
1.y
��
3x�
72.
x�
�3
3.y
��
x�
3
y�
x(2
,1),
(�4,
�2)
,y
��
x�
3(�
3,4)
,y
�x
�1
(�2,
4),(
2,2)
,
y�
�2
(3,�
2)y
�x
�1
(�3,
�4)
,(3,
2)y
�3x
�10
��3
,��
4 � 53 � 5
1 � 21 � 3
1 � 2
1 � 2
1 � 53 � 5
3 � 138 � 13
Exam
ple
Exam
ple
© Glencoe/McGraw-Hill A9 Glencoe Algebra 2
An
swer
s
Answers (Lesson 3-3)
Skil
ls P
ract
ice
So
lvin
g S
yste
ms
of
Ineq
ual
itie
s by
Gra
ph
ing
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-3
3-3
©G
lenc
oe/M
cGra
w-H
ill13
3G
lenc
oe A
lgeb
ra 2
Lesson 3-3
Sol
ve e
ach
sys
tem
of
ineq
ual
itie
s b
y gr
aph
ing.
1.x
�1
2.x
��
33.
x�
2y
��
1y
��
3x
�4
no
so
luti
on
4.y
�x
5.y
��
4x6.
x�
y�
�1
y�
�x
y�
3x�
23x
�y
�4
7.y
�3
8.y
��
2x�
39.
x�
y�
4x
�2y
�12
y�
x�
22x
�y
�4
Fin
d t
he
coor
din
ates
of
the
vert
ices
of
the
figu
re f
orm
ed b
y ea
ch s
yste
m o
fin
equ
alit
ies.
10.y
�0
11.y
�3
�x
12.x
��
2x
�0
y�
3y
�x
�2
y�
�x
�1
x�
�5
x�
y�
2 (�
2,4)
,(0
,0),
(0,�
1),(
�1,
0)(0
,3),
(�5,
3),(
�5,
8)(�
2,�
4),(
2,0)
x
y
Ox
y
Ox
y
O2
2
x
y
Ox
y
Ox
y
O
x
y
Ox
y
Ox
y
O
©G
lenc
oe/M
cGra
w-H
ill13
4G
lenc
oe A
lgeb
ra 2
Sol
ve e
ach
sys
tem
of
ineq
ual
itie
s b
y gr
aph
ing.
1.y
�1
��
x2.
x�
�2
3.y
�2x
�3
y�
12y
�3x
�6
y�
�x
�2
4.x
�y
��
25.
y
�1
6.3y
�4x
3x�
y�
�2
y�
x�
12x
�3y
��
6
Fin
d t
he
coor
din
ates
of
the
vert
ices
of
the
figu
re f
orm
ed b
y ea
ch s
yste
m o
fin
equ
alit
ies.
7.y
�1
�x
8.x
�y
�2
9.y
�2x
�2
y�
x�
1x
�y
�2
2x�
3y�
6x
�3
x�
�2
y�
4
(1,0
),(3
,2),
(3,�
2)(�
2,4)
,(�
2,�
4),(
2,0)
(�3,
4), �
,1�,(
3,4)
DR
AM
AF
or E
xerc
ises
10
and
11,
use
th
e fo
llow
ing
info
rmat
ion
.T
he
dram
a cl
ub
is s
elli
ng
tick
ets
to i
ts p
lay.
An
adu
lt t
icke
t co
sts
$15
and
a st
ude
nt
tick
et c
osts
$11
.Th
e au
dito
riu
m w
ill
seat
300
tic
ket-
hol
ders
.Th
e dr
ama
clu
b w
ants
to
coll
ect
atle
ast
$363
0 fr
om t
icke
t sa
les.
10.W
rite
an
d gr
aph
a s
yste
m o
f fo
ur
ineq
ual
itie
s th
at
desc
ribe
how
man
y of
eac
h t
ype
of t
icke
t th
e cl
ub
mu
st
sell
to
mee
ts i
ts g
oal.
x
0,y
0,
x�
y�
300,
15x
�11
y
3630
11.L
ist
thre
e di
ffer
ent
com
bin
atio
ns
of t
icke
ts s
old
that
sati
sfy
the
ineq
ual
itie
s.S
amp
le a
nsw
er:
250
adu
lt a
nd
50
stu
den
t,20
0 ad
ult
an
d
100
stu
den
t,14
5 ad
ult
an
d 1
48 s
tud
ent
Pla
y T
ickets
Ad
ult
Tic
kets
Student Tickets
100
020
030
040
0
400
350
300
250
200
150
100 50
3 � 2
x
y O
y
xO
y
xO
x
y
Ox
y
Ox
y
O
1 � 2
Pra
ctic
e (
Ave
rag
e)
So
lvin
g S
yste
ms
of
Ineq
ual
itie
s by
Gra
ph
ing
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-3
3-3
© Glencoe/McGraw-Hill A10 Glencoe Algebra 2
Answers (Lesson 3-3)
Readin
g t
o L
earn
Math
em
ati
csS
olv
ing
Sys
tem
s o
f In
equ
alit
ies
by G
rap
hin
g
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-3
3-3
©G
lenc
oe/M
cGra
w-H
ill13
5G
lenc
oe A
lgeb
ra 2
Lesson 3-3
Pre-
Act
ivit
yH
ow c
an y
ou d
eter
min
e w
het
her
you
r b
lood
pre
ssu
re i
s in
a
nor
mal
ran
ge?
Rea
d th
e in
trod
uct
ion
to
Les
son
3-3
at
the
top
of p
age
123
in y
our
text
book
.
Sat
ish
is
37 y
ears
old
.He
has
a b
lood
pre
ssu
re r
eadi
ng
of 1
35/9
9.Is
his
bloo
d pr
essu
re w
ith
in t
he
nor
mal
ran
ge?
Exp
lain
.S
amp
le a
nsw
er:
No
;h
is s
ysto
lic p
ress
ure
is n
orm
al,b
ut
his
dia
sto
lic p
ress
ure
is t
oo
hig
h.I
t sh
ou
ld b
e b
etw
een
60
and
90.
Rea
din
g t
he
Less
on
1.W
ith
out
actu
ally
dra
win
g th
e gr
aph
,des
crib
e th
e bo
un
dary
lin
es
x
�3
for
the
syst
em o
f in
equ
alit
ies
show
n a
t th
e ri
ght.
y
�5
Two
das
hed
ver
tica
l lin
es (
x�
3 an
d x
��
3) a
nd
tw
o s
olid
ho
rizo
nta
llin
es (
y�
�5
and
y�
5)
2.T
hin
k ab
out
how
th
e gr
aph
wou
ld l
ook
for
the
syst
em g
iven
abo
ve.W
hat
wil
l be
th
esh
ape
of t
he
shad
ed r
egio
n?
(It
is n
ot n
eces
sary
to
draw
th
e gr
aph
.See
if
you
can
imag
ine
it w
ith
out
draw
ing
anyt
hin
g.If
th
is i
s di
ffic
ult
to
do,m
ake
a ro
ugh
ske
tch
to
hel
p yo
u a
nsw
er t
he
ques
tion
.)a
rect
ang
le
3.W
hic
h s
yste
m o
f in
equ
alit
ies
mat
ches
th
e gr
aph
sh
own
at
th
e ri
ght?
BA
.x
�y
��
2B
.x
�y
��
2x
�y
�2
x�
y�
2
C.
x�
y�
�2
D.
x�
y�
�2
x�
y�
2x
�y
�2
Hel
pin
g Y
ou
Rem
emb
er
4.T
o gr
aph
a s
yste
m o
f in
equ
alit
ies,
you
mu
st g
raph
tw
o or
mor
e bo
un
dary
lin
es.W
hen
you
gra
ph e
ach
of
thes
e li
nes
,how
can
th
e in
equ
alit
y sy
mbo
ls h
elp
you
rem
embe
rw
het
her
to
use
a d
ash
ed o
r so
lid
lin
e?U
se a
das
hed
lin
e if
th
e in
equ
alit
y sy
mb
ol i
s �
or
�,b
ecau
se t
hes
esy
mb
ols
do
no
t in
clu
de
equ
alit
y an
d t
he
das
hed
lin
e re
min
ds
you
th
atth
e lin
e it
self
is n
ot
incl
ud
ed in
th
e g
rap
h.U
se a
so
lid li
ne
if t
he
sym
bo
lis
o
r �
,bec
ause
th
ese
sym
bo
ls in
clu
de
equ
alit
y an
d t
ell y
ou
th
at t
he
line
itse
lf is
incl
ud
ed in
th
e g
rap
h.
x
y
O
©G
lenc
oe/M
cGra
w-H
ill13
6G
lenc
oe A
lgeb
ra 2
Trac
ing
Str
ateg
yT
ry t
o tr
ace
over
eac
h o
f th
e fi
gure
s be
low
wit
hou
t tr
acin
g th
e sa
me
segm
ent
twic
e.
Th
e fi
gure
at
the
left
can
not
be
trac
ed,b
ut
the
one
at t
he
righ
t ca
n.T
he
rule
is
that
a
figu
re i
s tr
acea
ble
if i
t h
as n
o m
ore
than
tw
o po
ints
wh
ere
an o
dd n
um
ber
of
segm
ents
mee
t.T
he
figu
re a
t th
e le
ft h
as t
hre
e se
gmen
ts m
eeti
ng
at e
ach
of
the
fou
r co
rner
s.H
owev
er,t
he
figu
re a
t th
e ri
ght
has
on
ly t
wo
poin
ts,L
and
Q,w
her
e an
odd
nu
mbe
r of
seg
men
ts m
eet.
Det
erm
ine
if e
ach
fig
ure
can
be
trac
ed w
ith
out
trac
ing
the
sam
e se
gmen
t tw
ice.
If i
t ca
n,t
hen
nam
e th
e st
arti
ng
poi
nt
and
nam
e th
e se
gmen
ts i
n
the
ord
er t
hey
sh
ould
be
trac
ed.
1.2.
yes;
X;
X�Y�
,Y�F�,
F�X�,X�
G�,G�
F�,ye
s;E
;E�
D�,D�
A�,A�
E�,E�
B�,B�
F�,F�C�
,C�K�
,F �E�
,E�H�
,H�X�
,X�W�
,W�H�
,H�G�
K�F�,
F�J�,J�
H�,H�
F�,F�E�
,E�H�
,H�G�
,G�E�
3.
no
t tr
acea
ble
T SRU
PV
A
EF
DK
BC
GJ
H
EF
HG
XW
Y
K M
JL
PQ
AB
DC
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-3
3-3
© Glencoe/McGraw-Hill A11 Glencoe Algebra 2
An
swer
s
Answers (Lesson 3-4)
Stu
dy G
uid
e a
nd I
nte
rven
tion
Lin
ear
Pro
gra
mm
ing
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-4
3-4
©G
lenc
oe/M
cGra
w-H
ill13
7G
lenc
oe A
lgeb
ra 2
Lesson 3-4
Max
imu
m a
nd
Min
imu
m V
alu
esW
hen
a s
yste
m o
f li
nea
r in
equ
alit
ies
prod
uce
s a
bou
nde
d po
lygo
nal
reg
ion
,th
e m
axim
um
or m
inim
um
valu
e of
a r
elat
ed f
un
ctio
n w
ill
occu
rat
a v
erte
x of
th
e re
gion
.
Gra
ph
th
e sy
stem
of
ineq
ual
itie
s.N
ame
the
coor
din
ates
of
the
vert
ices
of
the
feas
ible
reg
ion
.Fin
d t
he
max
imu
m a
nd
min
imu
m v
alu
es o
f th
efu
nct
ion
f(x
,y)
�3x
�2y
for
this
pol
ygon
al r
egio
n.
y�
4y
��
x�
6
y�
x�
y�
6x�
4F
irst
fin
d th
e ve
rtic
es o
f th
e bo
un
ded
regi
on.G
raph
th
e in
equ
alit
ies.
Th
e po
lygo
n f
orm
ed i
s a
quad
rila
tera
l w
ith
ver
tice
s at
(0
,4),
(2,4
),(5
,1),
and
(�1,
�2)
.Use
th
e ta
ble
to f
ind
the
max
imu
m a
nd
min
imu
m v
alu
es o
f f(
x,y)
�3x
�2y
.
Th
e m
axim
um
val
ue
is 1
7 at
(5,
1).T
he
min
imu
m v
alu
e is
�7
at (
�1,
�2)
.
Gra
ph
eac
h s
yste
m o
f in
equ
alit
ies.
Nam
e th
e co
ord
inat
es o
f th
e ve
rtic
es o
f th
efe
asib
le r
egio
n.F
ind
th
e m
axim
um
an
d m
inim
um
val
ues
of
the
give
n f
un
ctio
n f
orth
is r
egio
n.
1.y
�2
2.y
��
23.
x�
y�
21
�x
�5
y�
2x�
44y
�x
�8
y�
x�
3x
�2y
��
1y
�2x
�5
f(x,
y) �
3x�
2yf(
x,y)
�4x
�y
f(x,
y) �
4x�
3y
vert
ices
:(1
,2),
(1,4
),ve
rtic
es:
(�5,
�2)
,ve
rtic
es (
0,2)
,(4,
3),
(5,8
),(5
,2);
max
:11
;(3
,2),
(1,�
2);
�,�
�;max
:25
;m
in:
6
min
;�
5m
ax:
10;
min
:�
181 � 3
7 � 3
x
y
Ox
y
O
x
y
O
(x,y
)3x
�2y
f(x,
y)
(0,
4)3(
0) �
2(4)
8
(2,
4)3(
2) �
2(4)
14
(5,
1)3(
5) �
2(1)
17
(�1,
�2)
3(�
1) �
2(�
2)�
7
x
y O
3 � 21 � 2
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill13
8G
lenc
oe A
lgeb
ra 2
Rea
l-W
orl
d P
rob
lem
sW
hen
sol
vin
g li
nea
r p
rogr
amm
ing
prob
lem
s,u
se t
he
foll
owin
g pr
oced
ure
.
1.D
efin
e va
riab
les.
2.W
rite
a s
yste
m o
f in
equ
alit
ies.
3.G
raph
th
e sy
stem
of
ineq
ual
itie
s.4.
Fin
d th
e co
ordi
nat
es o
f th
e ve
rtic
es o
f th
e fe
asib
le r
egio
n.
5.W
rite
an
exp
ress
ion
to
be m
axim
ized
or
min
imiz
ed.
6.S
ubs
titu
te t
he
coor
din
ates
of
the
vert
ices
in
th
e ex
pres
sion
.7.
Sel
ect
the
grea
test
or
leas
t re
sult
to
answ
er t
he
prob
lem
.
A p
ain
ter
has
exa
ctly
32
un
its
of y
ello
w d
ye a
nd
54
un
its
of g
reen
dye
.He
pla
ns
to m
ix a
s m
any
gall
ons
as p
ossi
ble
of
colo
r A
an
d c
olor
B.E
ach
gall
on o
f co
lor
A r
equ
ires
4 u
nit
s of
yel
low
dye
an
d 1
un
it o
f gr
een
dye
.Eac
hga
llon
of
colo
r B
req
uir
es 1
un
it o
f ye
llow
dye
an
d 6
un
its
of g
reen
dye
.Fin
d t
he
max
imu
m n
um
ber
of
gall
ons
he
can
mix
.
Ste
p 1
Def
ine
the
vari
able
s.x
�th
e n
um
ber
of g
allo
ns
of c
olor
A m
ade
y�
the
nu
mbe
r of
gal
lon
s of
col
or B
mad
e
Ste
p 2
Wri
te a
sys
tem
of
ineq
ual
itie
s.S
ince
th
e n
um
ber
of g
allo
ns
mad
e ca
nn
ot b
en
egat
ive,
x�
0 an
d y
�0.
Th
ere
are
32 u
nit
s of
yel
low
dye
;eac
h g
allo
n o
f co
lor
A r
equ
ires
4 u
nit
s,an
d ea
ch g
allo
n o
f co
lor
B r
equ
ires
1 u
nit
.S
o 4x
�y
�32
.S
imil
arly
for
th
e gr
een
dye
,x�
6y�
54.
Ste
ps
3 an
d 4
Gra
ph t
he
syst
em o
f in
equ
alit
ies
and
fin
d th
e co
ordi
nat
es o
f th
e ve
rtic
es o
f th
e fe
asib
le r
egio
n.
Th
e ve
rtic
es o
f th
e fe
asib
le r
egio
n a
re (
0,0)
,(0,
9),(
6,8)
,an
d (8
,0).
Ste
ps
5–7
Fin
d th
e m
axim
um
nu
mbe
r of
gal
lon
s,x
�y,
that
he
can
mak
e.T
he
max
imu
m n
um
ber
of g
allo
ns
the
pain
ter
can
mak
e is
14,
6 ga
llon
s of
col
or A
an
d 8
gall
ons
of c
olor
B.
1.FO
OD
A d
elic
ates
sen
has
8 p
oun
ds o
f pl
ain
sau
sage
an
d 10
pou
nds
of
garl
ic-f
lavo
red
sau
sage
.Th
e de
li w
ants
to
mak
e as
mu
ch b
ratw
urs
t as
pos
sibl
e.E
ach
pou
nd
of
brat
wu
rst
requ
ires
po
un
d of
pla
in s
ausa
ge a
nd
pou
nd
of g
arli
c-fl
avor
ed s
ausa
ge.
Fin
d th
e m
axim
um
nu
mbe
r of
pou
nds
of
brat
wu
rst
that
can
be
mad
e.10
lb
2.M
AN
UFA
CTU
RIN
GM
ach
ine
A c
an p
rodu
ce 3
0 st
eeri
ng
wh
eels
per
hou
r at
a c
ost
of $
16pe
r h
our.
Mac
hin
e B
can
pro
duce
40
stee
rin
g w
hee
ls p
er h
our
at a
cos
t of
$22
per
hou
r.A
t le
ast
360
stee
rin
g w
hee
ls m
ust
be
mad
e in
eac
h 8
-hou
r sh
ift.
Wh
at i
s th
e le
ast
cost
invo
lved
in
mak
ing
360
stee
rin
g w
hee
ls i
n o
ne
shif
t?$1
94
2 � 3
1 � 43 � 4
(x,y
)x
�y
f(x,
y)
(0,
0)0
�0
0
(0,
9)0
�9
9
(6,
8)6
�8
14
(8,
0)8
�0
8
Co
lor
A (
gal
lon
s)
Color B (gallons)
510
1520
2530
3540
4550
550
40 35 30 25 20 15 10 5
( 6, 8
)
( 8, 0
)( 0
, 9)
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
Lin
ear
Pro
gra
mm
ing
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-4
3-4
Exam
ple
Exam
ple
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A12 Glencoe Algebra 2
Answers (Lesson 3-4)
Skil
ls P
ract
ice
Lin
ear
Pro
gra
mm
ing
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-4
3-4
©G
lenc
oe/M
cGra
w-H
ill13
9G
lenc
oe A
lgeb
ra 2
Lesson 3-4
Gra
ph
eac
h s
yste
m o
f in
equ
alit
ies.
Nam
e th
e co
ord
inat
es o
f th
e ve
rtic
es o
f th
efe
asib
le r
egio
n.F
ind
th
e m
axim
um
an
d m
inim
um
val
ues
of
the
give
n f
un
ctio
n f
orth
is r
egio
n.
1.x
�2
2.x
�1
3.x
�0
x�
5y
�6
y�
0y
�1
y�
x�
2y
�7
�x
y�
4f(
x,y)
�x
�y
f(x,
y) �
3x�
yf(
x,y)
�x
�y
max
.:9,
min
.:3
max
.:2,
min
.:�
5m
ax.:
21,m
in.:
0
4.x
��
15.
y�
2x6.
y�
�x
�2
x�
y�
6y
�6
�x
y�
3x�
2f(
x,y)
�x
�2y
y�
6y
�x
�4
f(x,
y) �
4x�
3yf(
x,y)
��
3x�
5y
max
.:13
,no
min
.n
o m
ax.,
min
.:20
max
.:22
,min
.:�
2
7.M
AN
UFA
CTU
RIN
GA
bac
kpac
k m
anu
fact
ure
r pr
odu
ces
an i
nte
rnal
fra
me
pack
an
d an
exte
rnal
fra
me
pack
.Let
xre
pres
ent
the
nu
mbe
r of
in
tern
al f
ram
e pa
cks
prod
uce
d in
one
hou
r an
d le
t y
repr
esen
t th
e n
um
ber
of e
xter
nal
fra
me
pack
s pr
odu
ced
in o
ne
hou
r.T
hen
th
e in
equ
alit
ies
x�
3y�
18,2
x�
y�
16,x
�0,
and
y�
0 de
scri
be t
he
con
stra
ints
for
man
ufa
ctu
rin
g bo
th p
acks
.Use
th
e pr
ofit
fu
nct
ion
f(x
) �
50x
�80
yan
d th
eco
nst
rain
ts g
iven
to
dete
rmin
e th
e m
axim
um
pro
fit
for
man
ufa
ctu
rin
g bo
th b
ackp
acks
for
the
give
n c
onst
rain
ts.
$620
( 1, 5
)
( –1,
–1)
( –3,
1)
x
y
O
( 3, 6
)
( 2, 4
)
x
y
O
( –1,
7)
x
y
O
x
y
O
( 0, 7
)
( 0, 0
)( 7
, 0)
x
y
O
( 1, 6
)
( 1, –
1)
( 8, 6
)
x
y
O
( 2, 4
)
( 2, 1
)( 5
, 1)
( 5, 4
)
©G
lenc
oe/M
cGra
w-H
ill14
0G
lenc
oe A
lgeb
ra 2
Gra
ph
eac
h s
yste
m o
f in
equ
alit
ies.
Nam
e th
e co
ord
inat
es o
f th
e ve
rtic
es o
f th
efe
asib
le r
egio
n.F
ind
th
e m
axim
um
an
d m
inim
um
val
ues
of
the
give
n f
un
ctio
n f
orth
is r
egio
n.
1.2x
�4
�y
2.3x
�y
�7
3.x
�0
�2x
�4
�y
2x�
y�
3y
�0
y�
2y
�x
�3
y�
6f(
x,y)
��
2x�
yf(
x,y)
�x
�4y
y�
�3x
�15
f(x,
y) �
3x�
y
max
.:8,
min
.:�
4m
ax.:
12,m
in.:
�16
max
.:15
,min
.:0
4.x
�0
5.y
�3x
�6
6.2x
�3y
�6
y�
04y
�3x
�3
2x�
y�
24x
�y
��
7x
��
2x
�0
f(x,
y) �
�x
�4y
f(x,
y) �
�x
�3y
y�
0f(
x,y)
�x
�4y
�3
max
.:28
,min
.:0
max
.:,n
o m
in.
no
max
.,m
in.:
PRO
DU
CTI
ON
For
Exe
rcis
es 7
–9,u
se t
he
foll
owin
g in
form
atio
n.
A g
lass
blo
wer
can
for
m 8
sim
ple
vase
s or
2 e
labo
rate
vas
es i
n a
n h
our.
In a
wor
k sh
ift
of n
om
ore
than
8 h
ours
,th
e w
orke
r m
ust
for
m a
t le
ast
40 v
ases
.
7.L
et s
repr
esen
t th
e h
ours
for
min
g si
mpl
e va
ses
and
eth
e h
ours
for
min
g el
abor
ate
vase
s.W
rite
a s
yste
m o
f in
equ
alit
ies
invo
lvin
g th
e ti
me
spen
t on
eac
h t
ype
of v
ase.
s
0,e
0,
s �
e�
8,8s
�2e
40
8.If
th
e gl
ass
blow
er m
akes
a p
rofi
t of
$30
per
hou
r w
orke
d on
th
e si
mpl
e va
ses
and
$35
per
hou
r w
orke
d on
th
e el
abor
ate
vase
s,w
rite
a f
un
ctio
n f
or t
he
tota
l pr
ofit
on
th
e va
ses.
f(s,
e) �
30s
�35
e
9.F
ind
the
nu
mbe
r of
hou
rs t
he
wor
ker
shou
ld s
pen
d on
eac
h t
ype
of v
ase
to m
axim
ize
prof
it.W
hat
is
that
pro
fit?
4 h
on
eac
h;
$260
17 � 234 � 5
x
y
(3 – 2, 1)
( 0, 2
) O
(–7 – 5, 9 – 5)
( –2,
0)
x
y O
x
y ( 0, –
7)
(–7 – 4, 0)
( 0, 0
)O
( 3, 6
)
( 5, 0
)
( 0, 6
)
( 0, 0
)x
y
O
( 2, –
1)
( 4, 5
)
( 0, –
3)
x
y
O
( –3,
2)
( 3, 2
)
( 0, –
4)
x
y
O
Pra
ctic
e (
Ave
rag
e)
Lin
ear
Pro
gra
mm
ing
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-4
3-4
© Glencoe/McGraw-Hill A13 Glencoe Algebra 2
An
swer
s
Answers (Lesson 3-4)
Readin
g t
o L
earn
Math
em
ati
csL
inea
r P
rog
ram
min
g
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-4
3-4
©G
lenc
oe/M
cGra
w-H
ill14
1G
lenc
oe A
lgeb
ra 2
Lesson 3-4
Pre-
Act
ivit
yH
ow i
s li
nea
r p
rogr
amm
ing
use
d i
n s
ched
uli
ng
wor
k?
Rea
d th
e in
trod
uct
ion
to
Les
son
3-4
at
the
top
of p
age
129
in y
our
text
book
.
Nam
e tw
o or
mor
e fa
cts
that
in
dica
te t
hat
you
wil
l n
eed
to u
se i
neq
ual
itie
sto
mod
el t
his
sit
uat
ion
.S
amp
le a
nsw
er:T
he
buoy
ten
der
can
car
ry u
p t
o 8
new
bu
oys.
Th
ere
seem
s to
be
a lim
it o
f 24
ho
urs
on
th
e ti
me
the
crew
has
at s
ea.T
he
crew
will
wan
t to
rep
air
or
rep
lace
th
e m
axim
um
nu
mb
er o
f bu
oys
po
ssib
le.
Rea
din
g t
he
Less
on
1.C
ompl
ete
each
sen
ten
ce.
a.W
hen
you
fin
d th
e fe
asib
le r
egio
n f
or a
lin
ear
prog
ram
min
g pr
oble
m,y
ou a
re s
olvi
ng
a sy
stem
of
lin
ear
call
ed
.Th
e po
ints
in t
he
feas
ible
reg
ion
are
of
th
e sy
stem
.
b.
Th
e co
rner
poi
nts
of
a po
lygo
nal
reg
ion
are
th
e of
th
e fe
asib
le r
egio
n.
2.A
pol
ygon
al r
egio
n a
lway
s ta
kes
up
only
a l
imit
ed p
art
of t
he
coor
din
ate
plan
e.O
ne
way
to t
hin
k of
th
is i
s to
im
agin
e a
circ
le o
r re
ctan
gle
that
th
e re
gion
wou
ld f
it i
nsi
de.I
n t
he
case
of
a po
lygo
nal
reg
ion
,you
can
alw
ays
fin
d a
circ
le o
r re
ctan
gle
that
is
larg
e en
ough
to c
onta
in a
ll t
he
poin
ts o
f th
e po
lygo
nal
reg
ion
.Wh
at w
ord
is u
sed
to d
escr
ibe
a re
gion
that
can
be
encl
osed
in
th
is w
ay?
Wh
at w
ord
is u
sed
to d
escr
ibe
a re
gion
th
at i
s to
o la
rge
to b
e en
clos
ed i
n t
his
way
?b
ou
nd
ed;
un
bo
un
ded
3.H
ow d
o yo
u f
ind
the
corn
er p
oin
ts o
f th
e po
lygo
nal
reg
ion
in
a l
inea
r pr
ogra
mm
ing
prob
lem
?Yo
u s
olv
e a
syst
em o
f tw
o li
nea
r eq
uat
ion
s.
4.W
hat
are
som
e ev
eryd
ay m
ean
ings
of
the
wor
d fe
asib
leth
at r
emin
d yo
u o
f th
em
ath
emat
ical
mea
nin
g of
th
e te
rm f
easi
ble
regi
on?
Sam
ple
an
swer
:p
oss
ible
or
ach
ieva
ble
Hel
pin
g Y
ou
Rem
emb
er
5.L
ook
up
the
wor
d co
nst
rain
tin
a d
icti
onar
y.If
mor
e th
an o
ne
defi
nit
ion
is
give
n,c
hoo
seth
e on
e th
at s
eem
s cl
oses
t to
th
e id
ea o
f a
con
stra
int
in a
lin
ear
prog
ram
min
g pr
oble
m.
How
can
th
is d
efin
itio
n h
elp
you
to
rem
embe
r th
e m
ean
ing
of c
onst
rain
tas
it
is u
sed
inth
is l
esso
n?
Sam
ple
an
swer
:A
co
nst
rain
tis
a r
estr
icti
on
or
limit
atio
n.T
he
con
stra
ints
in a
lin
ear
pro
gra
mm
ing
pro
ble
m a
re r
estr
icti
on
s o
n t
he
vari
able
s th
at t
ran
slat
e in
to in
equ
alit
y st
atem
ents
.
vert
ices
solu
tio
ns
con
stra
ints
ineq
ual
itie
s
©G
lenc
oe/M
cGra
w-H
ill14
2G
lenc
oe A
lgeb
ra 2
Co
mp
ute
r C
ircu
its
and
Lo
gic
Com
pute
rs o
pera
te a
ccor
din
g to
th
e la
ws
of l
ogic
.Th
e ci
rcu
its
of a
com
pute
r ca
n b
e de
scri
bed
usi
ng
logi
c.
1.W
ith s
witc
h A
open
, no
cur
rent
flo
ws.
The
val
ue 0
is a
ssig
ned
to a
n op
en s
witc
h.
2.W
ith s
witc
h A
clos
ed,
curr
ent
flow
s. T
he v
alue
1 is
ass
igne
d to
a c
lose
d sw
itch.
3.W
ith s
witc
hes
Aan
d B
ope
n, n
o cu
rren
t flo
ws.
Thi
s ci
rcui
t ca
n be
des
crib
ed b
y th
e co
njun
ctio
n, A
B
.
4.In
thi
s ci
rcui
t, cu
rren
t flo
ws
if ei
ther
Aor
B is
clo
sed.
Thi
s ci
rcui
t ca
n be
des
crib
ed b
y th
e di
sjun
ctio
n, A
�B
.
Tru
th t
able
s ar
e u
sed
to d
escr
ibe
the
flow
of
curr
ent
in a
cir
cuit
.T
he
tabl
e at
th
e le
ft d
escr
ibes
th
e ci
rcu
it i
n d
iagr
am 4
.Acc
ordi
ng
to t
he
tabl
e,th
e on
ly t
ime
curr
ent
does
not
flo
w t
hro
ugh
th
e ci
rcu
it i
s w
hen
bot
h s
wit
ches
A a
nd
B a
re o
pen
.
Dra
w a
cir
cuit
dia
gram
for
eac
h o
f th
e fo
llow
ing.
1.(A
B
) �
C2.
(A �
B)
C
3.(A
�B
)
(C �
D)
4.(A
B
) �
(C
D)
5.C
onst
ruct
a t
ruth
tab
le f
or t
he
foll
owin
g ci
rcu
it.
B C
A
A CDB
A BD
C
A
BC
AB
C
A A A B
AB
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-4
3-4
AB
C(B
�C
)A
(B �
C)
00
00
00
01
10
01
11
00
10
10
10
11
11
10
11
11
11
11
00
00
AB
A�
B
00
00
11
10
11
11
© Glencoe/McGraw-Hill A14 Glencoe Algebra 2
Answers (Lesson 3-5)
Stu
dy G
uid
e a
nd I
nte
rven
tion
So
lvin
g S
yste
ms
of
Eq
uat
ion
s in
Th
ree
Var
iab
les
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-5
3-5
©G
lenc
oe/M
cGra
w-H
ill14
3G
lenc
oe A
lgeb
ra 2
Lesson 3-5
Syst
ems
in T
hre
e V
aria
ble
sU
se t
he
met
hod
s u
sed
for
solv
ing
syst
ems
of l
inea
req
uat
ion
s in
tw
o va
riab
les
to s
olve
sys
tem
s of
equ
atio
ns
in t
hre
e va
riab
les.
A s
yste
m o
fth
ree
equ
atio
ns
in t
hre
e va
riab
les
can
hav
e a
un
iqu
e so
luti
on,i
nfi
nit
ely
man
y so
luti
ons,
orn
o so
luti
on.A
sol
uti
on i
s an
ord
ered
tri
ple
.
Sol
ve t
his
sys
tem
of
equ
atio
ns.
3x�
y�
z�
�6
2x�
y�
2z�
84x
�y
�3z
��
21
Ste
p 1
Use
eli
min
atio
n t
o m
ake
a sy
stem
of
two
equ
atio
ns
in t
wo
vari
able
s.3x
�y
�z
��
6F
irst
equa
tion
2x�
y�
2z�
8S
econ
d eq
uatio
n
(�)
2x�
y�
2z�
8S
econ
d eq
uatio
n(�
) 4x
�y
�3z
��
21T
hird
equ
atio
n
5x�
z�
2A
dd t
o el
imin
ate
y.6x
�
z�
�13
Add
to
elim
inat
e y.
Ste
p 2
Sol
ve t
he
syst
em o
f tw
o eq
uat
ion
s.5x
�z
�2
(�)
6x�
z�
�13
11x
��
11A
dd t
o el
imin
ate
z.
x�
�1
Div
ide
both
sid
es b
y 11
.
Su
bsti
tute
�1
for
xin
on
e of
th
e eq
uat
ion
s w
ith
tw
o va
riab
les
and
solv
e fo
r z.
5x�
z�
2E
quat
ion
with
tw
o va
riabl
es
5(�
1) �
z�
2R
epla
ce x
with
�1.
�5
�z
�2
Mul
tiply
.
z�
7A
dd 5
to
both
sid
es.
Th
e re
sult
so
far
is x
��
1 an
d z
�7.
Ste
p 3
Su
bsti
tute
�1
for
xan
d 7
for
zin
on
e of
th
e or
igin
al e
quat
ion
s w
ith
th
ree
vari
able
s.3x
�y
�z
��
6O
rigin
al e
quat
ion
with
thr
ee v
aria
bles
3(�
1) �
y�
7 �
�6
Rep
lace
xw
ith �
1 an
d z
with
7.
�3
�y
�7
��
6M
ultip
ly.
y�
4S
impl
ify.
Th
e so
luti
on i
s (�
1,4,
7).
Sol
ve e
ach
sys
tem
of
equ
atio
ns.
1.2x
�3y
�z
�0
2.2x
�y
�4z
�11
3.x
�2y
�z
�8
x�
2y�
4z�
14x
�2y
�6z
��
112x
�y
�z
�0
3x�
y�
8z�
173x
�2y
�10
z�
113x
�6y
�3z
�24
(4,�
3,�
1)�2,
�5,
�in
fin
itel
y m
any
solu
tio
ns
4.3x
�y
�z
�5
5.2x
�4y
�z
�10
6.x
�6y
�4z
�2
3x�
2y�
z�
114x
�8y
�2z
�16
2x�
4y�
8z�
166x
�3y
�2z
��
123x
�y
�z
�12
x�
2y�
5
�,2
,�5 �
no
so
luti
on
�6,,�
�1 � 4
1 � 22 � 3
1 � 2
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill14
4G
lenc
oe A
lgeb
ra 2
Rea
l-W
orl
d P
rob
lem
s
Th
e L
ared
o S
por
ts S
hop
sol
d 1
0 b
alls
,3 b
ats,
and
2 b
ases
for
$99
on
Mon
day
.On
Tu
esd
ay t
hey
sol
d 4
bal
ls,8
bat
s,an
d 2
bas
es f
or $
78.O
n W
edn
esd
ayth
ey s
old
2 b
alls
,3 b
ats,
and
1 b
ase
for
$33.
60.W
hat
are
th
e p
rice
s of
1 b
all,
1 b
at,
and
1 b
ase?
Fir
st d
efin
e th
e va
riab
les.
x�
pric
e of
1 b
all
y�
pric
e of
1 b
atz
�pr
ice
of 1
bas
e
Tra
nsl
ate
the
info
rmat
ion
in
th
e pr
oble
m i
nto
th
ree
equ
atio
ns.
10x
�3y
�2z
�99
4x�
8y�
2z�
782x
�3y
�z
�33
.60
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
So
lvin
g S
yste
ms
of
Eq
uat
ion
s in
Th
ree
Var
iab
les
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-5
3-5
Exam
ple
Exam
ple
Su
btra
ct t
he
seco
nd
equ
atio
n f
rom
th
e fi
rst
equ
atio
n t
o el
imin
ate
z.10
x�
3y�
2z�
99(�
) 4x
�8y
�2z
�78
6x�
5y�
21
Mu
ltip
ly t
he
thir
d eq
uat
ion
by
2 an
dsu
btra
ct f
rom
th
e se
con
d eq
uat
ion
.4x
�8y
�2z
�78
(�)
4x�
6y�
2z�
67.2
02y
�10
.80
y�
5.40
Su
bsti
tute
5.4
0 fo
r y
in t
he
equ
atio
n
6x�
5y�
21.
6x�
5(5.
40)
�21
6x�
48x
�8
Su
bsti
tute
8 f
or x
and
5.40
for
yin
on
e of
the
orig
inal
equ
atio
ns
to s
olve
for
z.
10x
�3y
�2z
�99
10(8
) �
3(5.
40)
�2z
�99
80 �
16.2
0 �
2z�
992z
�2.
80z
�1.
40
So
a ba
ll c
osts
$8,
a ba
t $5
.40,
and
a ba
se $
1.40
.
1.FI
TNES
S TR
AIN
ING
Car
ly i
s tr
ain
ing
for
a tr
iath
lon
.In
her
tra
inin
g ro
uti
ne
each
wee
k,sh
e ru
ns
7 ti
mes
as
far
as s
he
swim
s,an
d sh
e bi
kes
3 ti
mes
as
far
as s
he
run
s.O
ne
wee
ksh
e tr
ain
ed a
tot
al o
f 23
2 m
iles
.How
far
did
sh
e ru
n t
hat
wee
k?56
mile
s
2.EN
TERT
AIN
MEN
TA
t th
e ar
cade
,Rya
n,S
ara,
and
Tim
pla
yed
vide
o ra
cing
gam
es,p
inba
ll,
and
air
hoc
key.
Rya
n s
pen
t $6
for
6 r
acin
g ga
mes
,2 p
inba
ll g
ames
,an
d 1
gam
e of
air
hock
ey.S
ara
spen
t $1
2 fo
r 3
raci
ng g
ames
,4 p
inba
ll g
ames
,and
5 g
ames
of
air
hock
ey.T
imsp
ent
$12.
25 f
or 2
rac
ing
gam
es,7
pin
ball
gam
es,a
nd 4
gam
es o
f ai
r ho
ckey
.How
muc
h di
dea
ch o
f th
e ga
mes
cos
t?R
acin
g g
ame:
$0.5
0;p
inb
all:
$0.7
5;ai
r h
ock
ey:
$1.5
0
3.FO
OD
A n
atu
ral
food
sto
re m
akes
its
ow
n b
ran
d of
tra
il m
ix o
ut
of d
ried
app
les,
rais
ins,
and
pean
uts
.On
e po
un
d of
th
e m
ixtu
re c
osts
$3.
18.I
t co
nta
ins
twic
e as
mu
ch p
ean
uts
by w
eigh
t as
app
les.
On
e po
un
d of
dri
ed a
pple
s co
sts
$4.4
8,a
pou
nd
of r
aisi
ns
$2.4
0,an
da
pou
nd
of p
ean
uts
$3.
44.H
ow m
any
oun
ces
of e
ach
in
gred
ien
t ar
e co
nta
ined
in
1 p
oun
dof
th
e tr
ail
mix
?3
oz
of
app
les,
7 o
z o
f ra
isin
s,6
oz
of
pea
nu
ts
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A15 Glencoe Algebra 2
An
swer
s
Answers (Lesson 3-5)
Skil
ls P
ract
ice
So
lvin
g S
yste
ms
of
Eq
uat
ion
s in
Th
ree
Var
iab
les
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-5
3-5
©G
lenc
oe/M
cGra
w-H
ill14
5G
lenc
oe A
lgeb
ra 2
Lesson 3-5
Sol
ve e
ach
sys
tem
of
equ
atio
ns.
1.2a
�c
��
10(5
,�5,
�20
)2.
x�
y�
z�
3(0
,2,1
)b
�c
�15
13x
�2z
�2
a�
2b�
c�
�5
�x
�5z
��
5
3.2x
�5y
�2z
�6
(�3,
2,1)
4.x
�4y
�z
�1
no
so
luti
on
5x�
7y�
�29
3x�
y�
8z�
0z
�1
x�
4y�
z�
10
5.�
2z�
�6
(2,�
1,3)
6.3x
�2y
�2z
��
2(�
2,1,
3)2x
�3y
�z
��
2x
�6y
�2z
��
2x
�2y
�3z
�9
x�
2y�
0
7.�
x�
5z�
�5
(0,0
,1)
8.�
3r�
2t�
1(1
,�6,
2)y
�3x
�0
4r�
s�
2t�
�6
13x
�2z
�2
r�
s�
4t�
3
9.x
�y
�3z
�3
no
so
luti
on
10.5
m�
3n�
p�
4(�
2,3,
5)�
2x�
2y�
6z�
63m
�2n
�0
y�
5z�
�3
2m�
n�
3p�
8
11.2
x�
2y�
2z�
�2
infi
nit
ely
man
y12
.x�
2y�
z�
4(1
,2,1
)2x
�3y
�2z
�4
3x�
y�
2z�
3x
�y
�z
��
1�
x�
3y�
z�
6
13.3
x�
2y�
z�
1(5
,7,0
)14
.3x
�5y
�2z
��
12in
fin
itel
y m
any
�x
�y
�z
�2
x�
4y�
2z�
85x
�2y
�10
z�
39�
3x�
5y�
2z�
12
15.2
x�
y�
3z�
�2
(�1,
3,�
1)16
.2x
�4y
�3z
�0
(3,0
,�2)
x�
y�
z�
�3
x�
2y�
5z�
133x
�2y
�3z
��
125x
�3y
�2z
�19
17.�
2x�
y�
2z�
2(1
,�2,
3)18
.x�
2y�
2z�
�1
infi
nit
ely
man
y3x
�3y
�z
�0
x�
2y�
z�
6x
�y
�z
�2
�3x
�6y
�6z
�3
19.T
he
sum
of
thre
e n
um
bers
is
18.T
he
sum
of
the
firs
t an
d se
con
d n
um
bers
is
15,
and
the
firs
t n
um
ber
is 3
tim
es t
he
thir
d n
um
ber.
Fin
d th
e n
um
bers
.9,
6,3
©G
lenc
oe/M
cGra
w-H
ill14
6G
lenc
oe A
lgeb
ra 2
Sol
ve e
ach
sys
tem
of
equ
atio
ns.
1.2x
�y
�2z
�15
2.x
�4y
�3z
��
273.
a�
b�
3�
x�
y�
z�
32x
�2y
�3z
�22
�b
�c
�3
3x�
y�
2z�
184z
��
16a
�2c
�10
(3,1
,5)
(1,4
,�4)
(2,1
,4)
4.3m
�2n
�4p
�15
5.2g
�3h
�8j
�10
6.2x
�y
�z
��
8m
�n
�p
�3
g�
4h�
14x
�y
�2z
��
3m
�4n
�5p
�0
�2g
�3h
�8j
�5
�3x
�y
�2z
�5
(3,3
,3)
no
so
luti
on
(�2,
�3,
1)
7.2x
�5y
�z
�5
8.2x
�3y
�4z
�2
9.p
�4r
��
73x
�2y
�z
�17
5x�
2y�
3z�
0p
�3q
��
84x
�3y
�2z
�17
x�
5y�
2z�
�4
q�
r�
1(5
,1,0
)(2
,2,�
2)(1
,3,�
2)
10.4
x�
4y�
2z�
811
.d�
3e�
f�
012
.4x
�y
�5z
��
93x
�5y
�3z
�0
�d
�2e
�f
��
1x
�4y
�2z
��
2 2x
�2y
�z
�4
4d�
e�
f�
12x
�3y
�2z
�21
infi
nit
ely
man
y(1
,�1,
2)(2
,3,�
4)
13.5
x�
9y�
z�
2014
.2x
�y
�3z
��
315
.3x
�3y
�z
�10
2x�
y�
z�
�21
3x�
2y�
4z�
55x
�2y
�2z
�7
5x�
2y�
2z�
�21
�6x
�3y
�9z
�9
3x�
2y�
3z�
�9
(�7,
6,1)
infi
nit
ely
man
y(1
,3,�
2)
16.2
u�
v�
w�
217
.x�
5y�
3z�
�18
18.x
�2y
�z
��
1�
3u�
2v�
3w�
73x
�2y
�5z
�22
�x
�2y
�z
�6
�u
�v
�2w
�7
�2x
�3y
�8z
�28
�4y
�2z
�1
(0,�
1,3)
(1,�
2,3)
no
so
luti
on
19.2
x�
2y�
4z�
�2
20.x
�y
�9z
��
2721
.2x
�5y
�3z
�7
3x�
3y�
6z�
�3
2x�
4y�
z�
�1
�4x
�10
y�
2z�
6�
2x�
3y�
z�
73x
�6y
�3z
�27
6x�
15y
�z
��
19(4
,5,0
)(2
,2,�
3)(1
,2,�
5)
22.T
he
sum
of
thre
e n
um
bers
is
6.T
he
thir
d n
um
ber
is t
he
sum
of
the
firs
t an
d se
con
dn
um
bers
.Th
e fi
rst
nu
mbe
r is
on
e m
ore
than
th
e th
ird
nu
mbe
r.F
ind
the
nu
mbe
rs.
4,�
1,3
23.T
he
sum
of
thre
e n
um
bers
is
�4.
Th
e se
con
d n
um
ber
decr
ease
d by
th
e th
ird
is e
qual
to
the
firs
t.T
he
sum
of
the
firs
t an
d se
con
d n
um
bers
is
�5.
Fin
d th
e n
um
bers
.�
3,�
2,1
24.S
POR
TSA
lexa
ndr
ia H
igh
Sch
ool
scor
ed 3
7 po
ints
in
a f
ootb
all
gam
e.S
ix p
oin
ts a
reaw
arde
d fo
r ea
ch t
ouch
dow
n.A
fter
eac
h t
ouch
dow
n,t
he
team
can
ear
n o
ne
poin
t fo
r th
eex
tra
kick
or
two
poin
ts f
or a
2-p
oin
t co
nve
rsio
n.T
he
team
sco
red
one
few
er 2
-poi
nt
con
vers
ion
s th
an e
xtra
kic
ks.T
he
team
sco
red
10 t
imes
du
rin
g th
e ga
me.
How
man
yto
uch
dow
ns
wer
e m
ade
duri
ng
the
gam
e?5
Pra
ctic
e (
Ave
rag
e)
So
lvin
g S
yste
ms
of
Eq
uat
ion
s in
Th
ree
Var
iab
les
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-5
3-5
© Glencoe/McGraw-Hill A16 Glencoe Algebra 2
Answers (Lesson 3-5)
Readin
g t
o L
earn
Math
em
ati
csS
olv
ing
Sys
tem
s o
f E
qu
atio
ns
in T
hre
e V
aria
ble
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
3-5
3-5
©G
lenc
oe/M
cGra
w-H
ill14
7G
lenc
oe A
lgeb
ra 2
Lesson 3-5
Pre-
Act
ivit
yH
ow c
an y
ou d
eter
min
e th
e n
um
ber
an
d t
ype
of m
edal
s U
.S.
Oly
mp
ian
s w
on?
Rea
d th
e in
trod
uct
ion
to
Les
son
3-5
at
the
top
of p
age
138
in y
our
text
book
.
At
the
1996
Su
mm
er O
lym
pics
in
Atl
anta
,Geo
rgia
,th
e U
nit
ed S
tate
s w
on10
1 m
edal
s.T
he
U.S
.tea
m w
on 1
2 m
ore
gold
med
als
than
sil
ver
and
7fe
wer
bro
nze
med
als
than
sil
ver.
Usi
ng
the
sam
e va
riab
les
as t
hos
e in
th
ein
trod
uct
ion
,wri
te a
sys
tem
of
equ
atio
ns
that
des
crib
es t
he
med
als
won
for
the
1996
Su
mm
er O
lym
pics
.g
�s
�b
�10
1;g
�s
�12
;b
�s
�7
Rea
din
g t
he
Less
on
1.T
he
plan
es f
or t
he
equ
atio
ns
in a
sys
tem
of
thre
e li
nea
r eq
uat
ion
s in
th
ree
vari
able
sde
term
ine
the
nu
mbe
r of
sol
uti
ons.
Mat
ch e
ach
gra
ph d
escr
ipti
on b
elow
wit
h t
he
desc
ript
ion
of
the
nu
mbe
r of
sol
uti
ons
of t
he
syst
em.(
Som
e of
th
e it
ems
on t
he
righ
t m
aybe
use
d m
ore
than
on
ce,a
nd
not
all
pos
sibl
e ty
pes
of g
raph
s ar
e li
sted
.)
a.th
ree
para
llel
pla
nes
I.
one
solu
tion
b.
thre
e pl
anes
th
at i
nte
rsec
t in
a l
ine
II.
no
solu
tion
s
c.th
ree
plan
es t
hat
in
ters
ect
in o
ne
poin
t II
I.in
fin
ite
solu
tion
s
d.
one
plan
e th
at r
epre
sen
ts a
ll t
hre
e eq
uat
ion
s
2.S
upp
ose
that
th
ree
clas
smat
es,M
oniq
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Josh
,an
d L
illy
,are
stu
dyin
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r a
quiz
on
th
isle
sson
.Th
ey w
ork
toge
ther
on
sol
vin
g a
syst
em o
f eq
uat
ion
s in
th
ree
vari
able
s,x,
y,an
dz,
foll
owin
g th
e al
gebr
aic
met
hod
sh
own
in
you
r te
xtbo
ok.T
hey
fir
st f
ind
that
z�
3,th
enth
at y
��
2,an
d fi
nal
ly t
hat
x�
�1.
Th
e st
ude
nts
agr
ee o
n t
hes
e va
lues
,bu
t di
sagr
eeon
how
to
wri
te t
he
solu
tion
.Her
e ar
e th
eir
answ
ers:
Mon
iqu
e:(3
,�2,
�1)
Josh
:(�
2,�
1,3)
Lil
ly:(
�1,
�2,
3)
a.H
ow d
o yo
u t
hin
k ea
ch s
tude
nt
deci
ded
on t
he
orde
r of
th
e n
um
bers
in
th
e or
dere
dtr
iple
?S
amp
le a
nsw
er:
Mo
niq
ue
arra
ng
ed t
he
valu
es in
th
e o
rder
inw
hic
h s
he
fou
nd
th
em.J
osh
arr
ang
ed t
hem
fro
m s
mal
lest
to
larg
est.
Lill
y ar
ran
ged
th
em in
alp
hab
etic
al o
rder
of
the
vari
able
s.
b.
Wh
ich
stu
den
t is
cor
rect
?L
illy
Hel
pin
g Y
ou
Rem
emb
er
3.H
ow c
an y
ou r
emem
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that
obt
ain
ing
the
equ
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n 0
�0
indi
cate
s a
syst
em w
ith
infi
nit
ely
man
y so
luti
ons,
wh
ile
obta
inin
g an
equ
atio
n s
uch
as
0 �
8 in
dica
tes
a sy
stem
wit
h n
o so
luti
ons?
0 �
0 is
alw
ays
tru
e,w
hile
0 �
8 is
nev
er t
rue.
III
I
III
II
©G
lenc
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lenc
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lgeb
ra 2
Bill
iard
sT
he f
igur
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the
rig
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how
s a
bill
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tab
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The
obj
ect
is t
o us
e a
cue
stic
k to
str
ike
the
ball
at
poi
nt C
so t
hat
the
ball
wil
l hi
t th
e si
des
(or
cush
ions
) of
the
tab
le a
t le
ast
once
bef
ore
hitt
ing
the
ball
loc
ated
at
poin
t A
.In
play
ing
the
gam
e,yo
u ne
ed t
o lo
cate
poi
nt P
.
Ste
p 1
Fin
d po
int
Bso
tha
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and
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.Bis
cal
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the
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in
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w
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dic
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(s)
and
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trik
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e b
all
at p
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t A
.Dra
w a
nd
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el t
he
corr
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pat
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for
the
cue
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ab
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cush
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BH
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NA
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____
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3-5
3-5
© Glencoe/McGraw-Hill A17 Glencoe Algebra 2
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
B:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11. B
D
D
A
C
A
B
D
B
C
B
4 units2
A
B
A
C
B
D
C
B
A
B
B
C
D
A
B
C
A
A
C
D
Chapter 3 Assessment Answer KeyForm 1 Form 2APage 149 Page 150 Page 151
An
swer
s
(continued on the next page)
© Glencoe/McGraw-Hill A18 Glencoe Algebra 2
12.
13.
14.
15.
16.
17.
18.
19.
20.
B:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
B:
C
C
B
A
D
C
C
B
A
C
D
D
D
A
B
C
A
B
A
A
�98
�
A
A
C
D
C
C
D
A
B
Chapter 3 Assessment Answer KeyForm 2A (continued) Form 2BPage 152 Page 153 Page 154
a � 2, b � 4, c � 1,d � 3, f � 0
© Glencoe/McGraw-Hill A19 Glencoe Algebra 2
1.
2.
no solution
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
B: 20 units
(�1, 2, 1)
(0, 1, �1)
50 cars, 10 busses
y
xO
y
xO
(2, �2)
(2, 1)
(0, 4)
(4, 1)
y
xO
y
xO(2, 0)
Chapter 3 Assessment Answer KeyForm 2CPage 155 Page 156
An
swer
s
consistent andindependent
consistent anddependent
(�2, �3), (2, �3),(2, 5)
(�2, 0), (3, �5),(3, 2), (0, 4)
(�2, �3), (�2, 9),(2, 5)
max: f(2, 5) � 1;min: f(�2, 9) � �15
c � 0, b � 0,6c � 30b � 600,c � b � 60
s � m � � � 9 7s � 12m � 15� � 86m � 3�
5 small packages,3 medium packages,1 large package
© Glencoe/McGraw-Hill A20 Glencoe Algebra 2
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
B: (12, �8)
$24
(3, 1, �4)
(�2, 3, 5)
$138
(�5, 7), (1, 7), (�1, 3)
y
xO
y
xO
(2, �2)
(�1, 3)
(�2, �8)
(4, 6)
inconsistent
y
xO(0, �1)
y
xO
(4, 1)
Chapter 3 Assessment Answer KeyForm 2DPage 157 Page 158
consistent and independent
(�3, �2), (0, �2),(�3, 4)
(�1, �3), (�4, 0),(0, 3), (2, 3)
max: f(1, 7) � 10;min: f(�5, 7) � �8
� � 0, v � 4,v � � � 15
w � 2h, t � h � 4,3w � 2h � 5t � 136
© Glencoe/McGraw-Hill A21 Glencoe Algebra 2
1.
2.
3.
4.
5.
6.
7.
8.9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
B: (6, 2)
$90
��15
�, �1, 21�
��12
�, ��23
�, �5�
y
xO
y
xO
(1.5, �2)
��3, �14
��
no solution
(�2, 6)
inconsistent
y
xO (1, �1)
y
xO
(�3, �4)
Chapter 3 Assessment Answer KeyForm 3Page 159 Page 160
An
swer
s
(�6, 3), (2, 3),(2, �4), (1, �4)
(�2, 4), (5, �3),(5, �4), (1, �4),(�2, 2)
(�3, �4), (�3, 1),(0, 4), (3, 2)
max: f(3, 2) � 8;
min: f(�3, 1) � ��129�
60a � 150m � 900,a � 3, m � 4
5 ads and 4 commercial minutes;124,000 people
s � 3a, m � a � 10,15s � 40a � 5m � 2650
consistent anddependent
© Glencoe/McGraw-Hill A22 Glencoe Algebra 2
Chapter 3 Assessment Answer KeyPage 161, Open-Ended Assessment
Scoring Rubric
Score General Description Specific Criteria
• Shows thorough understanding of the concepts of solvingsystems of linear equations by graphing, by substitution,and by elimination; solving systems of inequalities bygraphing; solving problems using linear programming; andsolving systems in three variables.
• Uses appropriate strategies to solve problems.• Computations are correct.• Written explanations are exemplary.• Graphs are accurate and appropriate.• Goes beyond requirements of some or all problems.
• Shows an understanding of the concepts of solvingsystems of linear equations by graphing, by substitution,and by elimination; solving systems of inequalities bygraphing; solving problems using linear programming; andsolving systems in three variables.
• Uses appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are effective.• Graphs are mostly accurate and appropriate.• Satisfies all requirements of problems.
• Shows an understanding of most of the concepts ofsolving systems of linear equations by graphing, bysubstitution, and by elimination; solving systems ofinequalities by graphing; solving problems using linearprogramming; and solving systems in three variables.
• May not use appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are satisfactory.• Graphs are mostly accurate.• Satisfies the requirements of most of the problems.
• Final computation is correct.• No written explanations or work is shown to substantiate
the final computation.• Graphs may be accurate but lack detail or explanation.• Satisfies minimal requirements of some of the problems.
• Shows little or no understanding of most of solvingsystems of linear equations by graphing, by substitution,and by elimination; solving systems of inequalities bygraphing; solving problems using linear programming; andsolving systems in three variables.
• 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.
0 UnsatisfactoryAn incorrect solutionindicating no mathematicalunderstanding of theconcept or task, or nosolution is given
1 Nearly Unsatisfactory A correct solution with nosupporting evidence orexplanation
2 Nearly SatisfactoryA partially correctinterpretation and/orsolution to the problem
3 SatisfactoryA generally correct solution,but may contain minor flawsin reasoning or computation
4 SuperiorA correct solution that is supported by well-developed, accurateexplanations
Chapter 3 Assessment Answer KeyPage 161, Open-Ended Assessment
Sample Answers
© Glencoe/McGraw-Hill A23 Glencoe Algebra 2
1. Let J � Janet’s age, K � Kim’s age, andS � Sue’s age.
J2 � (K � S)2 � 400 and K � S � J � 10J2 � (J � 10)2 � 400J2 � J2 � 20J � 100 � 400
20J � 500J � 25
Thus, K � S � 25 � 10 or 15 and J � (K � S) � 25 � 15 or 40. Therefore,(J � K � S)2 � 402 or 1600.
2a. x � 0y � 0
y � ��43�x � 8
2b. Sample answer: Evaluate f(x, y) at eachof the vertices of the region.
3. Student responses should indicate thatthe solution of a system of equations isan ordered pair representing the pointat which the graphs of the equationsintersect. Solutions should include thefact that a system may have no solution,indicating that the graphs of the linesare parallel and that a system may havean infinite number of solutions,indicating that the equations representthe same line.
4. Student responses should indicate thatit is impossible for a system to be both inconsistent (a system of non-intersecting lines) and dependent (a system of two representations of the same line).
5. Students should demonstrate anunderstanding that, if the graphs of thecost and revenue functions are parallel,the cost and revenue have the same rateof change. This means that the companywill never break even on the productionand sale of these products and, therefore,never make a profit. The owner maydecide to increase the price charged forthe product, may look for ways to cutcosts, may put the company resourcesinto the production of other products, ormay even decide to close up shop.
6. Students should indicate that a systemof two linear inequalities has nosolution when there are no orderedpairs which satisfy both inequalities.Graphically, this means that the shadedregions which represent the graphs ofthe inequalities do not intersect.Students’ graphs should show twoparallel lines with shading everywherebut between them.Sample answer: x � �3, x � 2
In addition to the scoring rubric found on page A22, the following sample answers may be used as guidance in evaluating open-ended assessment items.
An
swer
s
© Glencoe/McGraw-Hill A24 Glencoe Algebra 2
Chapter 3 Assessment Answer KeyVocabulary Test/Review Quiz (Lessons 3–1 and 3–2) Quiz (Lesson 3–4)
Page 162 Page 163 Page 164
1. inconsistent system
2. dependent system
3. ordered triple
4. substitution method
5. constraints
6. system ofinequalities
7. elimination method
8. consistent systemindependent system
9. system of equations
10. linear programming
11. Sample answer: In alinear programmingproblem, the regionthat is theintersection of thegraphs of theconstraints is calledthe feasible region.
12. Sample answer:When a system oflinear inequalities isgraphed, if thesolutions form aregion that is not apolygonal region,then we say theregion is anunbounded region.
1.
2.
inconsistent
3.
4.
5.
Quiz (Lesson 3–3)
Page 163
1.
2.
3.
4.
1.
2.
vertices:(�4, 2), (4, 4), (0, �1);max: f(0, �1) � 5;min: f(4, 4) � �16
3.
4.
Quiz (Lesson 3–5)
Page 164
1.
2.
3.
4.
5.
(1, �2, 4)
(2, 3, �1)
(�1, 2, �4)
y
xO
(4, 4)
(�4, 2)
(0, �1)
(0, 0), (0, 4), (2, 0)
y
xO
y
xO
A
(�3, �2)
y
xO
y
xO (3, 1)
no solution;inconsistent system
(1, �2), (2, 0), (3, �1),(3, �2)
max: f(1, 6) � 11;min: f(1, �1) � �3
j � 0, p � 0,j � 2p � 20,4j � 2p � 32
4 jackets and 8 pairs of pants;
$120
x � y � z � 155,x � 9y, y � 3z
135 cars, 15 vans,5 trucks
© Glencoe/McGraw-Hill A25 Glencoe Algebra 2
1.
2.
3.
4.
5.
6.
7.
8.
9.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
max: f(1, 3) � 0;min: f(6, �2) � �20
(1, 3), (6, �2), (1, �2)
consistent andindependent
�387�
No, because x has anexponent other than 1.
No, because the variableappears in adenominator.
�8a3 � 4a � 5
�1�2�3�4�5�6 0 1 2
{x � x � �5 or x � 2} or (�, �5) � (2, �)
y
xO
y
xO
(1, 3)
y
xO
(�2, 2)
A
A
C
C
Chapter 3 Assessment Answer KeyMid-Chapter Test Cumulative ReviewPage 165 Page 166
An
swer
s
consistent anddependent
(�2, 3), (�2, 6),(0, 6), (4, 0)
© Glencoe/McGraw-Hill A26 Glencoe Algebra 2
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11. 12.
13. 14.
15.
16.
17.
18.
19. DCBA
DCBA
DCBA
DCBA
DCBA
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
8
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
9
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
2 0 0
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
2
HGFE
DCBA
HGFE
DCBA
HGFE
DCBA
HGFE
DCBA
HGFE
DCBA
Chapter 3 Assessment Answer KeyStandardized Test Practice
Page 167 Page 168