Looking Ahead to Chapter 3 - somersetcentral.enschool.org
Transcript of Looking Ahead to Chapter 3 - somersetcentral.enschool.org
Looking Ahead to Chapter 3Focus In Chapter 3, you will learn how to solve one-and two-step equations and use them
in real-life situations. You will also work with integers and graph equations in the
coordinate plane using the Cartesian coordinate system.
Chapter Warm-upAnswer these questions to help you review skills that you will need in Chapter 3.
Perform the indicated operation.
1. 2. 3.
4. 5. 6.
Use mental math to find the value of x.
7. 8. 9.
Read the problem scenario below.
You start a new job and make $44 each week. Your goal is to save enough money to buy a
new television set that costs $352.
10. Can you buy the television after 6 weeks of working?
11. How many weeks do you need to work so that you have enough money to buy the
television?
12. Suppose that you have saved 75% of the money needed to buy the television.
How much money have you saved?
x100
� 0.2330x � 120x6
� 8
324 � 45(8)12 � 10
78 � 3989
�34
48 � 18
92 Chapter 3 ■ Solving Linear Equations
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
solve an equation ■ p. 96
equivalent equations ■ p. 97
one-step equation ■ p. 98
inverse operations ■ p. 98
algebraic check of a
solution ■ p. 99
graphical check of a
solution ■ p. 100
two-step equation ■ p. 102
commission ■ p. 107
commission rate ■ p. 107
percent equation ■ p. 108
integer ■ p. 131
negative integer ■ p. 131
positive integer ■ p. 131
number line ■ p. 131
quotient ■ p. 136
Cartesian coordinate
system ■ p. 140
x-axis ■ p. 140
y-axis ■ p. 140
origin ■ p. 140
coordinate plane ■ p. 140
ordered pair ■ p. 141
x-coordinate ■ p. 141
y-coordinate ■ p. 141
estimate ■ p. 147
Key Terms
Chapter 3 ■ Solving Linear Equations 93
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Solving Linear Equations
3.1 Collecting Road Tolls
Solving One-Step Equations ■ p. 95
3.2 Decorating the Math Lab
Solving Two-Step Equations ■ p. 101
3.3 Earning Sales Commissions
Using the Percent Equation ■ p. 107
3.4 Rent a Car from Go-Go Car
Rentals, Wreckem Rentals, and
Good Rents Rentals
Using Two-Step Equations,
Part 1 ■ p. 113
3.5 Plastic Containers
Using Two-Step Equations,
Part 2 ■ p. 125
3.6 Brrr! It’s Cold Out There!
Integers and Integer Operations ■ p. 131
3.7 Shipwreck at the Bottom of the Sea
The Coordinate Plane ■ p. 139
3.8 Engineering a Highway
Using a Graph of an Equation ■ p. 143
C H A P T E R
3
Most plastic containers are made using polyethylene, which was discovered in 1933.
Polyethylene is also used to make soda bottles and milk jugs. In Lesson 3.5, you will
design shipping boxes for plastic containers.
94 Chapter 3 ■ Solving Linear Equations
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Lesson 3.1 ■ Solving One-Step Equations 95
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Collecting Road TollsSolving One-Step Equations
ObjectivesIn this lesson,
you will:
■ Write one-step
equations.
■ Solve one-step
equations.
■ Check solutions
algebraically.
■ Check solutions
graphically.
Key Terms■ solve an equation
■ equivalent equations
■ one-step equation
■ inverse operations
■ algebraic check of a
solution
■ graphical check of a
solution
3.1
SCENARIO When you travel on certain bridges or roadways in
the United States, you are charged a toll or a fee to use the bridge or
roadway. The tolls are used to pay for road maintenance and safety.
Problem 1 Paying a Toll
In Virginia, travelers who cross the George P. Coleman Bridge going
northbound on Route 17 pay a toll. A two-axle vehicle pays $2 to
cross the bridge.
A. How much money is collected in tolls if 40 two-axle vehicles
cross the bridge? Use a complete sentence in your answer.
How much money is collected in tolls if 55 two-axle vehicles
cross the bridge? Use a complete sentence in your answer.
How much money is collected in tolls if 59 two-axle vehicles
cross the bridge? Use a complete sentence in your answer.
How much money is collected in tolls if 65 two-axle vehicles
cross the bridge? Use a complete sentence in your answer.
Use complete sentences to explain how you found your answers
in part (A).
B. Let v represent the number of two-axle vehicles that cross the
bridge. Write an algebraic expression that represents the amount
of money collected in tolls.
C. How many two-axle vehicles crossed the bridge if $20 in tolls
were collected from two-axle vehicles? Use a complete
sentence in your answer.
Investigate Problem 11. Just the Math: Solutions of Equations In part (C),
you were actually writing and solving equations. To solve an
equation means to find the value or values of the variable that
make the equation true. Suppose that $800 were collected in
tolls from two-axle vehicles and you want to know how many
two-axle vehicles crossed the bridge. From part (B), you know
that an expression for the amount collected in tolls from two-axle
vehicles is given by 2v.
To determine v, the number of two-axle vehicles that crossed the
bridge when $800 were collected, you can write and solve the
equation .
What is the value of v that makes the equation true?
Use a complete sentence in your answer.
800 � 2v
800 � 2v
96 Chapter 3 ■ Solving Linear Equations
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Problem 1 Paying a Toll
How many two-axle vehicles crossed the bridge if $50 in tolls
were collected from two-axle vehicles? Use a complete
sentence in your answer.
How many two-axle vehicles crossed the bridge if $190 in tolls
were collected from two-axle vehicles? Use a complete
sentence in your answer.
How many two-axle vehicles crossed the bridge if $220 in tolls
were collected from two-axle vehicles? Use a complete
sentence in your answer.
Use complete sentences to explain how you found your answers
in part (C).
Lesson 3.1 ■ Solving One-Step Equations 97
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Investigate Problem 12. Just the Math: Solving One-Step Equations You may
have used mental math to solve the equation in Question 1, but
in some cases you will want to solve an equation by writing an
equivalent equation that has the variable by itself on one side of
the equation. Two equations are equivalent if they have the
same solution or solutions. There are many ways to write
equivalent equations.
Solve by writing an equivalent equation. You can do
this by adding 5 to each side of the equation.
Original equation
Add 5 to each side.
Equivalent equation
The solution of is
Solve by writing an equivalent equation. You can do
this by subtracting 7 from each side of the equation.
Original equation
Subtract 7 from each side.
Equivalent equation
The solution of is
Solve by writing an equivalent equation. You can do this
by multiplying each side of the equation by 4.
Original equation
Multiply each side by 4.
Equivalent equation
The solution of is
Solve by writing an equivalent equation. You can do this
by dividing each side of the equation by 3.
Original equation
Divide each side by 3.
Equivalent equation
The solution of is x � _____.3x � 24
� � �
3x
��
24
�
3x � 24
3x � 24
x � _____.x4
� 6
� � �
x4
� � � 6 � �
x4
� 6
x4
� 6
x � _____.x � 7 � 10
� � �
x � 7 � � � 10 � �
x � 7 � 10
x � 7 � 10
x � _____.x � 5 � 8
� � �
x � 5 � � � 8 � �
x � 5 � 8
x � 5 � 8
98 Chapter 3 ■ Solving Linear Equations
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Investigate Problem 13. The equations in Question 2 are one-step equations. Write a
complete sentence to explain why you think this is so.
4. In Question 2, you solved the equation by using the
operation of addition, which undoes the operation of subtraction.
Operations that undo each other are inverse operations. For
each equation below, use a complete sentence to identify the
operation you would use to get the variable by itself on one side
of the equation. Then solve the equation.
5. Complete the table of values that shows the relationship between
the amount collected in tolls and the number of two-axle vehicles
that cross the bridge.
Quantity Name
Unit
Expression
2x � 450x � 15 � 45
x � 9 � 14x6
� 5.5
x � 5 � 8
Number of two-axle vehicles Tolls collected
vehicles dollars
40
200
350
480
600
Lesson 3.1 ■ Solving One-Step Equations 99
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Investigate Problem 16. Use the grid below to create a graph of the data from the table in
Question 4. First, choose your bounds and intervals. Be sure to
label your graph clearly.
7. Just the Math: Algebraic Check of a SolutionA toll taker collected $250 in tolls. The toll taker thinks he
counted 125 two-axle vehicles crossing the bridge. Did he
count correctly? The answer can be determined by checking
the solution of an equation. To algebraically check whether 125
is a solution of the equation , substitute the value for
the variable in the equation.
Substitute 125 for v:
Because both sides of the equations are equal, 125 is a solution.
Algebraically determine whether 169 is a solution of the equation
. Then, algebraically determine whether 179 is a solu-
tion of the equation . Show all your work.2v � 3582v � 358
250 � 250 2(125) � 250
2v � 250
Variable quantity Lower bound Upper bound Interval
(label) (units)
(lab
el)
(units)
100 Chapter 3 ■ Solving Linear Equations
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Investigate Problem 18. Just the Math: Graphical Check of a Solution
To graphically check whether 125 is a solution of the equation
, find the point on your line that corresponds to
125 vehicles. This point also corresponds to 250 dollars in tolls.
So, 125 is a solution of the equation because the tolls
collected for 125 two-axle vehicles is $250.
Graphically determine whether 250 is a solution of the equation
. Use a complete sentence to explain your reasoning.
Graphically determine whether 375 is a solution of the equation
. Use a complete sentence to explain.
9. Use complete sentences to explain which method of checking
the solution of an equation is better, algebraic or graphical.
2v � 800
2v � 500
2v � 250
2v � 250
Take NoteDraw straight lines to help
you determine whether a
value is a solution of an
equation. Draw a vertical
line up from the value of the
number of two-axle vehicles
until it intersects the line.
Then draw a horizontal line
over from the amount of tolls
collected until it intersects
the line. If you intersect the
line at the same place, then
the value is a solution of
the equation.
t
v50 100 150 200 2500 300 350 400 450 500 550 750600 650 700
100
200
300
400
0
500
600
700
800
900
1000
1100
1200
1300
1400
1500Collecting Road Tolls
Two-axle vehicles (vehicles)
Tolls
co
llecte
d (d
olla
rs) t = 2v
125
250
Take NoteWhenever you see the
share with the class icon,
your group should prepare
a short presentation to share
with the class that describes
how you solved the problem.
Be prepared to ask questions
during other groups’
presentations and to
answer questions during
your presentation.
Lesson 3.2 ■ Solving Two-Step Equations 101
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Decorating the Math LabSolving Two-Step Equations
ObjectivesIn this lesson,
you will:
■ Solve two-step
equations.
■ Check solutions
algebraically.
■ Check solutions
graphically.
Key Terms■ two-step equation
■ inverse operations
■ algebraic check of a
solution
■ graphical check of a
solution
SCENARIO Your class will decorate the walls of your math
computer lab with posters of great mathematicians. Your teacher
finds a catalog that sells posters for $2 each. There is an additional
charge of $4 to ship an order.
Problem 1 Finding the Total Cost
A. What will the cost be if your class orders 3 posters? Use a
complete sentence in your answer.
What will the cost be if your class orders 5 posters? Use a
complete sentence in your answer.
What will the cost be if your class orders 8 posters? Use a
complete sentence in your answer.
What will the cost be if your class orders 12 posters? Use a
complete sentence in your answer.
Use complete sentences to explain how you found your answers
in part (A).
B. Let p represent the number of posters ordered. Write an
algebraic expression that represents the total cost of the order.
3.2
Investigate Problem 11. Just the Math: Solving Two-Step Equations When
you were finding the number of posters ordered in part (C), you
were solving a two-step equation. Suppose that your class paid
$54 for posters and you want to know how many posters were
ordered. From part (B), you know that an expression for the total
cost of an order is .
To determine p, the number of posters ordered when the total
cost was $54, you can write and solve the two-step equation
.
This equation is a two-step equation because it requires two
steps in order to solve it. To solve a two-step equation, you need
to perform operations together that “undo” each other. What
operations do you need to undo in the equation to
get the variable by itself on the right side? What are the inverse
operations of these operations?
2p � 4 � 54
2p � 4 � 54
2p � 4
102 Chapter 3 ■ Solving Linear Equations
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Problem 1 Finding the Total Cost
C. How many posters were ordered if the cost of the order was $8?
Use a complete sentence in your answer.
How many posters were ordered if the cost of the order was
$16? Use a complete sentence in your answer.
How many posters were ordered if the cost of the order was
$32? Use a complete sentence in your answer.
How many posters were ordered if the cost of the order was
$44? Use a complete sentence in your answer.
Use complete sentences to explain how you found your answers
in part (C).
Lesson 3.2 ■ Solving Two-Step Equations 103
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Investigate Problem 1The first step is to undo the addition. Write the equivalent
equation that is created when you undo the addition with the
inverse operation of subtraction. Then simplify each side of
the equation.
The second step is to undo the multiplication. Write the
equivalent equation created when you undo the multiplication
with the inverse operation of division. Then simplify each side of
the equation.
So, _____ posters were ordered.
2. For each equation below, use a complete sentence to identify the
operations that you would use to get the variable by itself on one
side of the equation and the order in which you would perform
these operations. Then solve the equation.
x3
� 5 � 1x3
� 2 � 4
1 � 2x � 75x � 3 � 2
Take NoteWhen you undo operations
to solve an equation, you
must follow the order of
operations in reverse order.
This is why you first undo the
addition and then you undo
the multiplication to solve
the equation .2p � 4 � 54
104 Chapter 3 ■ Solving Linear Equations
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Investigate Problem 13. Complete the table of values that shows the relationship between
the number of posters ordered and the total cost of the order.
Quantity Name
Unit
Expression
4. Use the grid below to create a graph of the data from the table
above. First, choose your bounds and intervals. Be sure to label
your graph clearly.
Number of posters ordered Total cost
posters dollars
14
18
22
30
40
Variable quantity Lower bound Upper bound Interval
(label) (units)
(lab
el)
(units)
Lesson 3.2 ■ Solving Two-Step Equations 105
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Investigate Problem 15. Your teacher tells you that it cost $70 for the posters. One of
your fellow classmates claims that 32 posters were ordered.
Another classmate claims that 35 posters were ordered.
You think that 33 posters were ordered.
Algebraically determine whether 32 and 33 are solutions of the
equation . Use complete sentences to explain.
Graphically determine whether 35 is a solution of the equation
. Use complete sentences to explain.
Graphically determine whether 50 is a solution of the equation
Use complete sentences to explain.
6. Determine whether 11 is a solution of the equation
graphically or algebraically. Use complete sentences to explain
the reasons for the choice of your method.
2p � 4 � 26
2p � 4 � 50.
2p � 4 � 70
2p � 4 � 70
106 Chapter 3 ■ Solving Linear Equations
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Lesson 3.3 ■ Using the Percent Equation 107
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Earning Sales CommissionsUsing the Percent Equation
ObjectivesIn this lesson,
you will:
■ Find commissions.
■ Solve percent equations.
■ Solve two-step
equations.
Key Terms■ commission
■ commission rate
■ percent equation
SCENARIO Often, part of a salesperson’s pay is made up of a
commission. A commission is an amount based on a percent of the
salesperson’s total sales. The percent is called the commission rate.
You have started working at a car dealership as a salesperson.
You earn a commission that is 2% of your total sales.
Problem 1 What Is My Commission?
A. During your first week, your total sales are $10,000.
Use a proportion to find your commission for your first week.
Show your work and use a complete sentence in your answer.
B. During your second week, your total sales are $15,000.
Use a proportion to find your commission for your second week.
Show your work and use a complete sentence in your answer.
C. During your third week, your total sales are $25,000.
Use a proportion to find your commission for your third week.
Show your work and use a complete sentence in your answer.
D. During your fourth week, your total sales are $32,000.
Use a proportion to find your commission for your fourth week.
Show your work and use a complete sentence in your answer.
3.3
Investigate Problem 11. Write a proportion that relates the commission, the total sales,
and the commission rate. Use the variable t to represent your
total sales and c to represent the amount of money that you
earn in commission.
Now, use means and extremes to solve the proportion.
Then, write the equation that you can use to find the commission
earned c on the total sales for any value of t. Show all
your work.
2. Just the Math: The Percent Equation The equation you
wrote in Question 1 is a percent equation. Recall that a percent
is a ratio that compares two numbers. A percent equation is an
equation of the form where p is the percent and the
numbers being compared are a and b. Use complete sentences
to explain how the percent equation is equivalent to the
proportion .
3. In each statement below, identify a, b, and p. Then write a
percent equation that represents the situation.
18 is 40% of 45.
Sixty percent of 35 is 21.
ab
�p
100
a �p
100b
108 Chapter 3 ■ Solving Linear Equations
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Lesson 3.3 ■ Using the Percent Equation 109
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Investigate Problem 14. Write and solve a percent equation to answer each question.
Use a complete sentence in your answer.
Fifty is what percent of 200? What is 20% of 160?
Eight is 5% of what number? What is 40% of 85?
Problem 2 How Much Money Did I Earn?
You are working at a car dealership as a salesperson. In addition
to your 2% commission, you also earn a base salary of $215
each week.
A. How much money will you earn if your total sales for a week are
$10,000? Show all your work and use a complete sentence in
your answer.
B. How much money will you earn if your total sales for a week are
$25,000? Show all your work and use a complete sentence in
your answer.
C. How much money will you earn if your total sales for a week are
$100,000? Show all your work and use a complete sentence in
your answer.
D. How much money will you earn if your total sales for a week are
$200,000? Show all your work and use a complete sentence in
your answer.
E. Let t represent your total sales for a week and let E represent
your earnings. Write an equation for your earnings in terms of
your total sales.
Take NoteIt is common to write the
expression as a decimal
when using the percent
equation. For instance,
when the percent is 35%,
the percent equation can
be written as a � 0.35b.
p100
110 Chapter 3 ■ Solving Linear Equations
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Investigate Problem 21. Use your equation to find your total sales for a week if your
earnings are $300. Show all your work and use a complete
sentence in your answer.
What are your total sales for a week if your earnings are $215?
Show all your work and use a complete sentence in your answer.
What are your total sales for a week if your earnings are $220?
Show all your work and use a complete sentence in your answer.
What are your total sales for a week if your earnings are $400?
Show all your work and use a complete sentence in your answer.
What are your total sales for a week if your earnings are $1000?
Show all your work and use a complete sentence in your answer.
Use complete sentences to explain how you found your answers
in Question 1.
2. What is the independent variable quantity in your equation?
What is the dependent variable quantity in your equation?
Lesson 3.3 ■ Using the Percent Equation 111
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Investigate Problem 23. Complete the table of values that shows the relationship between
the earnings and the total sales for a week.
Quantity Name
Unit
Expression
4. Use the grid below to create a graph of the data from the table
above. First, choose your bounds and intervals. Be sure to label
your graph clearly.
Variable quantity Lower bound Upper bound Interval
Total sales Earnings
0
235
5000
12,000
635
(label) (units)
(lab
el)
(units)
112 Chapter 3 ■ Solving Linear Equations
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Investigate Problem 25. You have heard that a different sales job offers a base salary that
is lower than your base salary, but the commission rate is higher.
Yet another sales job offers a base salary that is higher than your
base salary, but the commission rate is lower. A third sales job
offers only a straight salary. Under which conditions will each
amount of pay be the best? Would you stay at the dealership
that you are at, or try to get a job at a different dealership?
Write a paragraph to answer the questions.
Lesson 3.4 ■ Using Two-Step Equations, Part 1 113
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Rent a Car from Go-Go Car Rentals,Wreckem Rentals, and Good Rents RentalsUsing Two-Step Equations, Part 1
ObjectivesIn this lesson,
you will:
■ Write and solve two-step
equations.
■ Determine the better deal.
■ Compare three problem
situations.
Key Term■ two-step equation
3.4
Investigate Problem 11. Use the equation that you wrote in part (F) to find the number of
miles driven in one day if the total rental cost is $30. Show all
your work and use a complete sentence in your answer.
SCENARIO You are taking a road trip and you need to rent
a car. You look on the Internet to compare car rental prices. The first
company you find, Go-Go Car Rentals, charges $28 each day along
with a charge of $.25 per mile.
Problem 1 Renting from Go-Go Car Rentals
A. What will the cost be for one day if you drive the car 10 miles?
Show your work and use a complete sentence in your answer.
B. What will the cost be for one day if you drive the car 50 miles?
Show your work and use a complete sentence in your answer.
C. What will the cost be for one day if you drive the car 536 miles?
Show your work and use a complete sentence in your answer.
D. What will the cost be for one day if you drive the car 750 miles?
Show your work and use a complete sentence in your answer.
E. Use complete sentences to explain how you determined the cost
of the car rental in parts (A) through (D).
F. Let t represent the total cost of the car rental in dollars and let mrepresent the number of miles driven. Write an equation that
gives the total cost in terms of the number of miles driven.
Investigate Problem 1Find the number of miles driven in one day if the total rental
cost is $45. Show your work and use a complete sentence
in your answer.
Find the number of miles driven in one day if the total rental
cost is $67.50. Show your work and use a complete sentence
in your answer.
Find the number of miles driven in one day if the total rental cost is
$121.25. Show your work and use a complete sentence in your answer.
Use complete sentences to explain how you found your answers
in Question 1.
2. Complete the table of values that shows the relationship between
the total cost and the number of miles driven.
Quantity Name
Unit
Expression
3. Use the grid below to create a graph of the data from the table
above. First, choose your bounds and intervals. Be sure to label
your graph clearly.
114 Chapter 3 ■ Solving Linear Equations
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Take NoteWhen you create your table,
be sure to only consider
distances that are reasonably
traveled in one day.
Miles driven Total cost
miles dollars
Variable quantity Lower bound Upper bound Interval
Lesson 3.4 ■ Using Two-Step Equations, Part 1 115
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Investigate Problem 1
4. Go-Go Car Rentals offers you a special deal. The cost for each
day is $35, but you get 500 miles for free. For any number of
miles driven above 500, you pay $.25 per mile. If you plan to
drive 20 miles in one day, is this deal better? Show all your work
and use complete sentences to explain your reasoning.
If you plan to drive 400 miles in one day, is the special deal
better? Show all your work and use complete sentences
to explain.
If you plan to drive 600 miles in one day, is the special deal
better? Show your work and use complete sentences
to explain.
(label) (units)
(lab
el)
(un
its)
116 Chapter 3 ■ Solving Linear Equations
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Problem 2 Renting from Wreckem Rentals
Another rental company, Wreckem Rentals, charges $29.95 each day
along with a charge of $.23 per mile.
A. What will the cost be for one day if you drive the car 10 miles?
Show your work and use a complete sentence in your answer.
B. What will the cost be for one day if you drive the car 50 miles?
Show your work and use a complete sentence in your answer.
C. What will the cost be for one day if you drive the car 536 miles?
Show your work and use a complete sentence in your answer.
D. What will the cost be for one day if you drive the car 2500 miles?
Show your work and use a complete sentence in your answer.
E. Use complete sentences to explain how you determined the cost
of the car rental in parts (A) through (D).
F. Let t represent the total cost of the car rental in dollars and let mrepresent the number of miles driven. Write an equation that
gives the total cost in terms of the number of miles driven.
Investigate Problem 21. Use the equation that you wrote in part (F) to find the number of
miles you drove in one day if the total cost of the rental is $30.64.
Show all your work and use a complete sentence in your answer.
Find the number of miles driven in one day if the total rental
cost is $33.17. Show your work and use a complete sentence
in your answer.
Lesson 3.4 ■ Using Two-Step Equations, Part 1 117
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Investigate Problem 2Find the number of miles driven in one day if the total rental
cost is $46.28. Show your work and use a complete sentence
in your answer.
Find the number of miles driven in one day if the total rental
cost is $83.77. Show your work and use a complete sentence
in your answer.
Use complete sentences to explain how you found your answers
in Question 1.
2. Complete the table of values that shows the relationship between
the total cost and the number of miles driven.
Quantity Name
Unit
Expression
3. Use the grid on the next page to create a graph of the data
from the table above. First, choose your bounds and intervals.
Be sure to label your graph clearly.
Take NoteRemember that when you
create your table, you should
only consider distances that
are reasonably traveled in
one day.
Variable quantity Lower bound Upper bound Interval
118 Chapter 3 ■ Solving Linear Equations
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Investigate Problem 2
4. Wreckem Rentals offers you a special deal. The cost for each
day is $32, but you get 250 miles for free. For any number of
miles driven above 250, you pay $.23 per mile. If you plan to
drive 20 miles in one day, is this deal better? Show all your work
and use complete sentences to explain your reasoning.
If you plan to drive 400 miles in one day, is the special deal
better? Show your work and use complete sentences to explain.
If you plan to drive 600 miles in one day, is the special deal
better? Show your work and use complete sentences to explain.
(label) (units)
(lab
el)
(un
its)
Lesson 3.4 ■ Using Two-Step Equations, Part 1 119
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Problem 3 Renting from Good Rents Car Rentals
The last rental company you consider, Good Rents Car Rentals,
charges $21.65 each day along with a charge of $.27 per mile.
A. What will the cost be for one day if you drive the car 10 miles?
Show your work and use a complete sentence in your answer.
B. What will the cost be for one day if you drive the car 50 miles?
Show your work and use a complete sentence in your answer.
C. What will the cost be for one day if you drive the car 536 miles?
Show your work and use a complete sentence in your answer.
D. What will the cost be for one day if you drive the car 2500 miles?
Show your work and use a complete sentence in your answer.
E. Use complete sentences to explain how you determined the cost
of the car rental in parts (A) through (D).
F. Let t represent the total cost of the car rental in dollars and let mrepresent the number of miles driven. Write an equation that
gives the total cost in terms of the number of miles driven.
Investigate Problem 31. Use the equation that you wrote in part (F) to find the number of
miles you drove in one day if the total cost of the rental is $29.75.
Show all your work and use a complete sentence in your answer.
Find the number of miles driven in one day if the total rental cost
is $33.26. Show all your work and use a complete sentence in
your answer.
120 Chapter 3 ■ Solving Linear Equations
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Investigate Problem 3Find the number of miles driven in one day if the total rental cost
is $39.74. Show all your work and use a complete sentence in
your answer.
Find the number of miles driven in one day if the total rental cost
is $51.62. Show all your work and use a complete sentence in
your answer.
Use complete sentences to explain how you found your answers
in Question 1.
2. Complete the table of values that shows the relationship between
the total cost and the number of miles driven.
Quantity Name
Unit
Expression
3. Use the grid on the next page to create a graph of the data
from the table above. First, choose your bounds and intervals.
Be sure to label your graph clearly.
Take NoteRemember that when you
create your table, you should
only consider distances that
are reasonably traveled in
one day.
Variable quantity Lower bound Upper bound Interval
Lesson 3.4 ■ Using Two-Step Equations, Part 1 121
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Investigate Problem 3
4. Good Rents Car Rentals offers you a special deal. The cost for
each day is $28, but you get 150 miles for free. For any number
of miles driven above 150, you pay $.27 per mile. If you plan to
drive 20 miles in one day, is this deal better? Show all your
work and use complete sentences to explain your reasoning.
If you plan to drive 400 miles in one day, is the special deal
better? Show all your work and use complete sentences
to explain.
If you plan to drive 600 miles in one day, is the special deal
better? Show all your work and use complete sentences
to explain.
(label) (units)
(lab
el)
(un
its)
122 Chapter 3 ■ Solving Linear Equations
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Investigate Problem 35. Summarize the work that you have done so far in the
table below.
6. Use the bounds and intervals shown below to create graphs of
all three situations on the grid.
Go-Go Rentals Wreckem RentalsGood Rents
Rentals
Quantity Name
Unit
Expression
Variable quantity Lower bound Upper bound Interval
Miles driven 0 300 20
Total cost 0 150 20
(label) (units)
(lab
el)
(units)
Lesson 3.4 ■ Using Two-Step Equations, Part 1 123
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Investigate Problem 37. Compare the three companies’ prices from your Investigate
Problem 3 Question 6 graph. Prepare a written report that
indicates which company will cost the least and for which
numbers of miles. Can you tell whether the cost will ever be
the same for all three companies? If so, for which number
of miles? Use complete sentences to explain.
124 Chapter 3 ■ Solving Linear Equations
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Lesson 3.5 ■ Using Two-Step Equations, Part 2 125
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Plastic ContainersUsing Two-Step Equations, Part 2
ObjectivesIn this lesson,
you will:
■ Write and use two-step
equations.
■ Compare three problem
situations.
Key Term■ two-step equation
SCENARIO Your job at the Nikkerware Company is to create
new packaging and shipping boxes for its plastic containers. The
company makes all different shapes and sizes of plastic containers.
To ship the containers, the lids are removed to allow for stacking.
The company wants to design shipping boxes that will hold two
dozen stacks of the plastic containers without the lids, regardless of
the size or shape of the container.
Problem 1 Different Shapes and Sizes
The table below shows the data gathered from measuring the heights
of different-sized stacks of the various plastic containers.
If you have some stackable plastic containers in your classroom,
you should add the data about them to this table.
A. What is the difference in heights between a stack of two round
containers and a stack of one round container? Use a complete
sentence in your answer.
B. What is the difference in heights between a stack of three
round containers and a stack of two round containers?
Use a complete sentence in your answer.
C. What is the difference in heights between a stack of four
round containers and a stack of three round containers?
Use a complete sentence in your answer.
3.5
Number of
containersStack height (centimeters)
Round Square Rectangular
1 5.4 4.5 14.6
2 5.9 6.2 14.9
3 6.4 7.9 15.2
4 6.9 9.6 15.5
5 7.4 11.3 15.8
Investigate Problem 11. What is the height of 10 round containers? Show all your work
and use a complete sentence in your answer.
What is the height of 15 round containers? Show all your work
and use a complete sentence in your answer.
What is the height of 20 round containers? Show all your work
and use a complete sentence in your answer.
Use complete sentences to explain how you found your answers
in Question 1.
126 Chapter 3 ■ Solving Linear Equations
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Number of
containers
Stack height
(cm)
1 5.4 4.514.6
2 5.9 6.214.9
3 6.4 7.915.2
4 6.9 9.615.5
5 7.4 11.315.8
Problem 1 Different Shapes and Sizes
D. What is the difference in heights between a stack of five
round containers and a stack of four round containers?
Use a complete sentence in your answer.
E. How does the stack height change when one round container is
added to a stack of round containers? Use a complete sentence
in your answer.
F. How does the stack height change when one square container is
added to a stack of square containers? Use complete sentences
to explain how you found your answer.
G. How does the stack height change when one rectangularcontainer is added to a stack of rectangular containers?
Use complete sentences to explain how you found your answer.
Re
cta
ng
ula
r
Sq
ua
re
Ro
un
d
Lesson 3.5 ■ Using Two-Step Equations, Part 2 127
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Investigate Problem 12. If you use c to represent the number of containers in a stack of
round containers, what does the expression represent?
Use a complete sentence to explain.
3. Let h represent the height of a stack. Write an equation that
gives the height of a stack of round containers in terms of the
number of containers c in the stack.
4. Use your equation to find the height of a stack of two dozen
round containers. Show your work and use a complete sentence
in your answer.
5. Let c represent the number of containers in a stack of squarecontainers and let h represent the height of the stack. Write an
equation that gives the height of the stack in terms of the number
of containers in the stack.
6. What is the height of a stack of two dozen square containers?
Show all your work and use a complete sentence to explain
your reasoning.
7. Let c represent the number of containers in a stack of rectangularcontainers and let h represent the height of the stack. Write an
equation that gives the height of the stack in terms of the number
of containers in the stack.
8. What is the height of a stack of two dozen rectangular containers?
Show all your work and use a complete sentence to explain
your reasoning.
c � 1
128 Chapter 3 ■ Solving Linear Equations
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Investigate Problem 19. How are the three equations you wrote similar? Why are these
equations similar? Use complete sentences to explain.
10. How are the three equations you wrote different? Why are these
equations different? Use complete sentences to explain.
11. In all equations, what are the variable quantities? Use a
complete sentence in your answer.
What is the relationship between these quantities? In other
words, which variable quantity depends on the other?
12. Does it make sense for the company to have a stack of zero
containers? Why or why not? Use complete sentences to
explain your reasoning.
Use your answer to explain why each of these graphs on the
next page starts at 1.
13. Use the grid on the next page to create graphs for each
container shape’s stack height in terms of the number of
containers used. Use the bounds and intervals indicated in
the table. Be sure to label your graph clearly.
Variable quantity Lower bound Upper bound Interval
Containers 0 75 5
Stack height 0 75 5
Number of
containers
Stack height
(cm)
1 5.4 4.514.6
2 5.9 6.214.9
3 6.4 7.915.2
4 6.9 9.615.5
5 7.4 11.315.8
Re
cta
ng
ula
r
Sq
ua
re
Ro
un
d
Lesson 3.5 ■ Using Two-Step Equations, Part 2 129
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Investigate Problem 1
14. The square container is the shortest container. Find the number(s)
of containers for which the height of a stack of square containers
is the shortest. Use complete sentences to explain why the
square container stack height is not always the shortest.
15. Find the number(s) of containers for which a stack of round
containers is the shortest. Use complete sentences to explain
why a stack of any number of round containers does not remain
the shortest.
(label) (units)
(lab
el)
(un
its)
130 Chapter 3 ■ Solving Linear Equations
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Investigate Problem 116. Find the number(s) of containers for which a stack of rectangular
containers is the shortest. Use complete sentences to explain
why a stack of any number of rectangular containers does not
remain the shortest.
17. Write a memo to your supervisor in which you recommend
the best shipping carton sizes for each type of container.
Use complete sentences to explain your reasoning.
Lesson 3.6 ■ Integers and Integer Operations 131
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Brrr! It’s Cold Out There!Integers and Integer Operations
ObjectivesIn this lesson,
you will:
■ Graph integers on a
number line.
■ Compare integers.
■ Add, subtract, multiply,
and divide integers.
■ Write fractions as
decimals.
■ Write decimals as
fractions.
Key Terms■ integer
■ negative integer
■ positive integer
■ number line
■ quotient
SCENARIO You are working on a report about winter weather
and you are studying the wind-chill temperature. The wind-chill
temperature is a measure of how wind speed affects low tempera-
tures and how the combination of wind and low temperature feels to
a human being.
You find a table of some wind chill temperatures when the wind
speed is 5 miles per hour.
Problem 1 Wind Chill Temperatures
The numbers in the table are integers. The numbers …, –4, –3, –2,
–1, 0, 1, 2, 3, 4, … are integers. Integers include zero, negative
integers (integers less than zero), and positive integers (integers
greater than zero).
A. Use a complete sentence to describe the effect that the wind has
on actual temperatures.
B. One way to show the relationship between integers is to graph
them on a number line. A number line is a line that extends in
both directions forever. One point is assigned a value of zero
and a given length is assigned a length of one unit. Positive
integers are assigned to the units on the right of zero and
negative integers are assigned to the units on the left of zero.
Use the number line below to graph each of the wind chill
temperatures in the table. The first wind chill temperature is
done for you.
C. On the number line, the values of integers increase as you move
from left to right. Write the wind chill temperatures in order from
least to greatest.
1211109876543210−1−2−3−5−6−7−8−9−10 −4
3.6
Actual temperature
(degrees Fahrenheit)15 10 5 0 –5
Wind chill temperature
(degrees Fahrenheit)12 7 0 –5 –10
Take NoteThe symbol … is an ellipsis.
It implies the continuation
of a pattern.
For instance,
1, 2, 3, 4, …
implies that you should
continue counting by 1s.
Take NoteTwo integers are opposites if
they are on opposite sides of
0, but are the same distance
from 0. For example, 4 and
–4 are opposites. You will
learn more about opposites
in Chapter 4.
Investigate Problem 11. When the actual temperature is 10°F and the wind speed is
15 miles per hour, the wind chill temperature is –18°F. When the
wind speed is 30 miles per hour at the same actual temperature,
the wind chill temperature is –33°F. Which wind chill temperature
is higher? Use a complete sentence to explain.
When the actual temperature is 5°F and the wind speed is
15 miles per hour, the wind chill temperature is –13°F. When the
wind speed is 35 miles per hour at the same actual temperature,
the wind chill temperature is –21°F. Which wind chill temperature
is lower? Use a complete sentence to explain.
2. When the actual temperature is 25°F and the wind speed is
15 miles per hour, the wind chill temperature is 2°F. When
the wind speed decreases to 10 miles per hour, the wind chill
temperature increases by 8 degrees. Use a number line to find
the wind chill temperature when the wind speed is 10 miles per
hour. Use a complete sentence to explain.
When the actual temperature is 10°F and the wind speed is
10 miles per hour, the wind chill temperature is –4°F. When the
wind speed decreases to 5 miles per hour, the wind chill
temperature increases by 5 degrees. Use a number line to find
the wind chill temperature when the wind speed is 5 miles per
hour. Use a complete sentence to explain.
−6 −5 −4 −3 −2−7 −1 0 1 2 3 4 85 6 7−9 −8
−1 0 1 2 3−2 4 5 6 7 8 9 1310 11 12−4 −3
−23 −22 −21 −20 −19−24 −18 −17 −16 −15 −14 −13 −9−12 −11 −10−26 −25
−31 −30 −29 −28 −27−32 −26 −25 −24 −23 −22 −21 −17−20 −19 −18−34 −33
132 Chapter 3 ■ Solving Linear Equations
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Lesson 3.6 ■ Integers and Integer Operations 133
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Investigate Problem 13. Just the Math: Adding Negative Integers
In Question 2, you were adding a positive integer to another
integer. Complete the following instructions to add a negative
integer to an integer.
To add 4 and –7, start at 4. Then move 7 units to the left.
Represent this on the number line by graphing 4 on the number
line and then by drawing an arrow that starts at 4 and ends
7 units to the left of 4. Where are you on the number line?
Write an integer addition equation that represents this situation.
To add –2 and –5, start at –2. Then move 5 units to the left.
Represent this on the number line by graphing –2 on the number
line and then by drawing an arrow that starts at –2 and ends
5 units to the left of –2. Where are you on the number line?
Write an integer addition equation that represents this situation.
Complete the rule below for using a number line to add two
integers.
On a number line, move to the ________ when you add a
positive integer, and move to the ________ when you add a
negative integer.
4. When the actual temperature is 15°F and the wind speed is
10 miles per hour, the wind chill temperature is 3°F. When the
wind speed increases to 20 miles per hour, the wind chill
temperature decreases by 5 degrees. Use a number line to
find the wind chill temperature when the wind speed is 20 miles
per hour. Use complete sentences to explain how you found
your answer.
−6 −5 −4 −3 −2−7 −1 0 1 2 3 4 85 6 7−9 −8
−6 −5 −4 −3 −2−7 −1 0 1 2 3 4 85 6 7−9 −8
−6 −5 −4 −3 −2−7 −1 0 1 2 3 4 85 6 7−9 −8
134 Chapter 3 ■ Solving Linear Equations
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Investigate Problem 1When the actual temperature is 0°F and the wind speed is
5 miles per hour, the wind chill temperature is –11°F. When the
wind speed increases to 10 miles per hour, the wind chill tempera-
ture decreases by 5 degrees. Use a number line to find the
wind chill temperature when the wind speed is 10 miles per hour.
Use complete sentences to explain how you found your answer.
5. Just the Math: Subtracting Negative Integers Recall
that when you add a positive integer on a number line you move
to the right, when you add a negative integer on a number line,
you move to the left. When you subtract a positive integer, which
way do you think that you should move on a number line?
When you subtract a negative integer on a number line, which
way should you move? Use complete sentences to explain
your reasoning.
Complete the following instructions to subtract a negative integer
from an integer.
To subtract –4 from 8, start at 8. Then subtract –4. Where are
you on the number line?
Write an integer subtraction equation that represents this
situation.
To subtract –5 from –3, start at –3. Then subtract –5. Where are
you on the number line?
Write an integer subtraction equation that represents this
situation.
−6 −5 −4 −3 −2−7 −1 0 1 2 3 4 85 6 7−9 −8
3 4 5 6 72 8 9 10 11 12 13 1714 15 160 1
−17 −16 −15 −14 −13−18 −12 −11 −10 −9 −8 −7 −3−6 −5 −4−20 −19
Lesson 3.6 ■ Integers and Integer Operations 135
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Investigate Problem 1Complete the rule below for using a number line to subtract two
integers.
On a number line, move to the ________ when you subtract a
positive integer, and move to the ________ when you subtract a
negative integer.
6. Just the Math: Multiplying Integers Recall that you
can think of multiplication as repeated addition. Complete each
statement below.
What is the effect of multiplying a positive integer by –1?
Use complete sentences in your answer.
What do you think will be the effect of multiplying a negative
integer by –1?
Find each product.
What do you think is the product of –1 and –1? Use a complete
sentence to explain your reasoning.
Find each product.
0(�4) � �
�1(�4) � �1(�4) � �
�2(�4) � �2(�4) � �
�3(�4) � �3(�4) � �
(�1) (4) (�1)(�1) (4)
4(�2) � (�2) � (�2) � (�2) � (�2) � �
(�1) (5) � (�1) � (�1) � (�1) � (�1) � (�1) � �
2(�1) � (�1) � (�1) � �
3(�5) � (�5) � (�5) � (�5) � �
136 Chapter 3 ■ Solving Linear Equations
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Investigate Problem 1Complete the statements below about the product of
two integers.
The product of two positive integers is a __________ integer.
The product of two negative integers is a __________ integer.
The product of a positive integer and a negative integer is a
__________ integer.
The product of a negative integer and a positive integer is a
__________ integer.
7. Just the Math: Dividing Integers You can find the
quotient of two integers by using what you know about multipli-
cation. For instance, if we divide 35 by 5, the result is 7 because
. Find each quotient by writing a related multiplica-
tion problem.
because
because
because
because
Complete the statements below about the quotient of
two integers.
The quotient of two positive integers is a __________ integer.
The quotient of two negative integers is a __________ integer.
The quotient of a positive integer and a negative integer is a
__________ integer.
The quotient of a negative integer and a positive integer is a
__________ integer.
8. Perform the indicated operations.
14 � (�2)�56 � (�7)�18 � 3
(�1)3(�3)24(�9)
�5 � (�6)�3 � (�4)7 � (�12)
�6 � � � 48.48 � (�6) � �
�6 � � � �48.�48 � (�6) � �
6 � � � �48.�48 � 6 � �
6 � � � 48.48 � 6 � �
7 � 5 � 35
Take NoteRecall that the result of
dividing two numbers is
the quotient.
Lesson 3.6 ■ Integers and Integer Operations 137
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Problem 2 Celsius Temperature Scale
As part of your report, you will also write your information about
wind chill temperatures in degrees Celsius. An equation that relates
a temperature C in degrees Celsius to a temperature F in degrees
Fahrenheit is
A. Use the equation to find the temperature in degrees Celsius that
corresponds to a temperature of 23°F. Show all your work and
use a complete sentence in your answer.
B. Use the equation to find the temperature in degrees Fahrenheit
that corresponds to a temperature of –10°C. Show all your work
and use a complete sentence in your answer.
F �95
C � 32.
Investigate Problem 21. Just the Math: Writing a Fraction as a Decimal
When you use the equation above to write 12°C in degrees
Fahrenheit, you get To write this
temperature properly, we need to write it as a decimal instead of
as a fraction. To write a fraction as a decimal, use long division.
53.6
18
3 0
Use long division to write each fraction as a decimal.
45
72
15
255)268.0
F �95
(12) � 32 �2685
.
138 Chapter 3 ■ Solving Linear Equations
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Investigate Problem 22. Just the Math: Writing a Decimal as a Fraction
To write a decimal as a fraction, first determine the name for
the last decimal place of the number that is to the right of the
decimal point. Then use the name to write the decimal as
the appropriate fraction.
hundredths
Write each decimal as a fraction. If possible, simplify
your answer.
0.9 0.08 0.125
3. Write the temperature equation so that the fraction is a decimal.
Is this equation easier to use? Use complete sentences to
explain your reasoning.
4. Convert each temperature to degrees Celsius. Show all
your work.
5ºF –4ºF
Convert each temperature to degrees Fahrenheit. Show all
your work.
8°C –2°C
0.17 �17100
Take NoteRemember that the first
decimal place to the right
of the decimal point is
tenths, the second decimal
place is hundredths, and
the third decimal place is
thousandths.
Lesson 3.7 ■ The Coordinate Plane 139
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Shipwreck at the Bottom of the SeaThe Coordinate Plane
ObjectivesIn this lesson,
you will:
■ Draw a coordinate
system.
■ Write ordered pairs.
■ Plot points in a
coordinate plane.
Key Terms■ Cartesian coordi-
nate system
■ x-axis
■ y-axis
■ origin
■ coordinate plane
■ ordered pair
■ x-coordinate
■ y-coordinate
3.7
SCENARIO You work for a company that salvages shipwrecks.
Items such as pieces of the ship and cargo from the ship are called
debris. Part of your job is to examine the debris field, or the area in
which items that are related to the shipwreck can be found. Once the
debris field is identified, a grid is placed over the field so that the
relative positions of the items can be recorded.
Problem 1 Searching the Debris Field
The figure below shows the debris field and the accompanying grid at
a wreck that you are exploring.
Use complete sentences to describe the location of the anchor’s
center in relation to the ship’s center.
Use complete sentences to describe the location of the center of the
gold coins in relation to the ship’s center.
Investigate Problem 11. Just the Math: Cartesian Coordinate System In the
previous lesson, you used number lines to graph integers. If we
intersect two number lines at right angles as shown below, we
form the Cartesian coordinate system. The horizontal number
line is the x-axis and the vertical number line is the y-axis.
The point at which the axes cross is the origin. The intersection
of these two number lines forms a coordinate plane. Label the
x-axis, the y-axis, and the origin in the coordinate plane below.
x−6 −5 −4 −3 −2−7 −1 O 1 2 3 4 5 6 7
−3
y
−7
−6
−5
−4
−2
−1
1
2
3
4
5
6
7
140 Chapter 3 ■ Solving Linear Equations
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Problem 1 Searching the Debris Field
Use complete sentences to describe the location of the cannonball’s
center in relation to the ship’s center.
Use complete sentences to describe the location of the flag’s center
in relation to the ship’s center.
Take NoteThe plural of axis is axes.
Lesson 3.7 ■ The Coordinate Plane 141
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Investigate Problem 12. Just the Math: Ordered Pairs Each point on a graph can
be represented by an ordered pair. An ordered pair consists of
two numbers. The first number, the x-coordinate, indicates how
far to the left or right the point is from the (vertical) y-axis. The
second number, the y-coordinate, indicates how far up or down
the point is from the (horizontal) x-axis. An ordered pair is written
as (x-coordinate, y-coordinate), such as (3, –5).
For each item, write the ordered pair that represents the location
of the item’s center.
Draw the x- and y-axes on the grid below so that the origin
is the ship’s center. For each item, plot the ordered pair that
represents the location of the item’s center on the grid.
142 Chapter 3 ■ Solving Linear Equations
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Investigate Problem 13. Plot and label each point in the coordinate plane.
A(5, 2)
B(–3, 0)
C(6, –3)
D(1, 6)
E(–4, 4)
F(–2, –6)
x−6 −5 −4 −3 −2−7 −1 O 1 2 3 4 5 6 7
−3
y
−7
−6
−5
−4
−2
−1
1
2
3
4
5
6
7
Lesson 3.8 ■ Using a Graph of an Equation 143
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Engineering a HighwayUsing a Graph of an Equation
ObjectivesIn this lesson,
you will:
■ Write and use a two-step
equation.
■ Use a graph to estimate
solutions of equations.
Key Term■ estimate
3.8
Investigate Problem 11. Find the number of miles that are completed in 10 days.
Show all your work and use a complete sentence in your answer.
Find the number of miles that are completed in 50 days.
Show all your work and use a complete sentence in your answer.
Find the number of miles that are completed in 1 year.
Show all your work and use a complete sentence in your answer.
Find the number of miles that are completed in 2 years.
Show all your work and use a complete sentence in your answer.
SCENARIO You are a civil engineer who is overseeing a large
interstate-highway construction project that will connect the cities of
Pittsburgh, Pennsylvania, and Cincinnati, Ohio. Construction began
several months ago, and approximately 87 miles of highway have
been completed. The total length of the highway will be 267 miles.
You estimate that new construction can proceed at the rate of
one-fifth mile per day.
Problem 1 Construction Work
A. What are the two quantities that are changing?
B. Write an equation to model the situation.
Investigate Problem 1Use a complete sentence to explain how you found your answers
in Question 1.
2. Find the number of days it will take to complete 100 miles of
highway. Show all your work and use a complete sentence in
your answer.
Find the number of days it will take to complete 150 miles of
highway. Show all your work and use a complete sentence in
your answer.
Find the number of days it will take to complete the entire
highway. Show all your work and use a complete sentence in
your answer.
144 Chapter 3 ■ Solving Linear Equations
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Lesson 3.8 ■ Using a Graph of an Equation 145
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Investigate Problem 1Use complete sentences to explain how you found your answers
in Question 2.
Write an equation that relates the two quantities that are
changing.
3. Use your equation to find the number of days it will take to
complete 50 miles of highway. Show all your work and use
a complete sentence to explain your reasoning.
Use complete sentences to explain the solution of the equation
in terms of the problem situation.
4. How many days before new construction began was the project
started? Show all your work and use complete sentences to
explain your reasoning.
146 Chapter 3 ■ Solving Linear Equations
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Investigate Problem 15. Complete the table of values that shows the relationship between
the number of days and the number of miles that are completed.
Quantity Name
Unit
Expression
6. Use the grid below to create a graph of the data from the table
above. First, choose your bounds and intervals. Be sure to label
your graph clearly.
Take NoteWhen you are choosing
your bounds, remember to
consider when the highway
was started and how many
days it will take to complete
the highway.
Variable quantity Lower bound Upper bound Interval
(label) (units)
(lab
el)
(units)
Lesson 3.8 ■ Using a Graph of an Equation 147
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Investigate Problem 17. Use your graph to estimate when 30 miles of highway were
completed. Use a complete sentence in your answer.
Use your graph to estimate when 60 miles of highway were
completed. Use a complete sentence in your answer.
Use your graph to estimate when 250 miles of highway were
completed. Use a complete sentence in your answer.
Use your graph to estimate when 130 miles of highway were
completed. Use a complete sentence in your answer.
Use your graph to estimate the number of miles that
were completed 50 days before new construction started.
Use a complete sentence in your answer.
Use your graph to estimate the number of miles that
were completed 100 days before new construction started.
Use a complete sentence in your answer.
Use your graph to estimate the number of miles that
were completed 200 days after new construction started.
Use a complete sentence in your answer.
148 Chapter 3 ■ Solving Linear Equations
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3
Investigate Problem 18. Some of the other engineers who are working on this project also
responded to the question and several of them had responses
that were different from yours. Use complete sentences to
explain why this could occur.
9. Prepare an overview of the project that will convince the project
manager that this project is on schedule and that the project is
running smoothly.
Lesson 3.8 ■ Using a Graph of an Equation 149
© 2
008 C
arn
eg
ie L
earn
ing
, In
c.
3