314 Chapter 7 Equations and Inequalities
Writing Equations in
Two Variables
7.4
How can you write an equation in
two variables?
Work with a partner. You earn $8 per hour
working part-time at a store.
a. Complete the table.
Hours
Worked
Money Earned
(dollars)
1
2
3
4
5
b. Use the values from the table to
complete the graph. Then answer
each question below.
● What does the horizontal axis
represent? What variable did
you use to identify it?
● What does the vertical axis
represent? What variable did
you use to identify it?
● How are the ordered pairs in
the graph related to the values
in the table?
● How are the horizontal and vertical distances shown on the graph
related to the values in the table?
c. How can you write an equation that shows how the two variables
are related?
d. What does the green line in the graph represent?
ACTIVITY: Writing an Equation in Two Variables
1
1
1
8
2
3 4 5
16
24
32
40
0
0
COMMON
CORE
Writing Equations
In this lesson, you will
●
identify independent and
dependent variables.
●
write equations in
two variables.
●
use tables and graphs to
analyze the relationship
between two variables.
Learning Standard
6.EE.9
Section 7.4 Writing Equations in Two Variables 315
Work with a partner. Use the equation you wrote in Activity 1.
a. How is this equation different from the equations earlier in this chapter?
b. One of the variables in this equation depends on the other variable.
Determine which variable is which by answering the following questions:
● Does the amount of money you earn depend on the number of hours
you work?
● Does the number of hours you work depend on the amount of money
you earn?
What do you think is the signifi cance of having two types of variables?
How do you think you can use these types of variables in real life?
Work with a partner. Recall that the perimeter of a square is 4 times
its side length.
a. Write the formula for the perimeter of a square.
Tell what each variable represents.
b. Describe how the perimeter of a square changes as
its side length increases by 1 unit. Use a table and
a graph to support your answer.
c. In your formula, which variable depends on which?
ACTIVITY: Describing Variables
2
2
ACTIVITY: Describing a Formula in Two Variables
3
3
4. IN YOUR OWN WORDS How can you write an equation in two variables?
5. The equation y 7.75x shows how the number of movie tickets is related
to the total amount of money spent. Describe what each part of the
equation represents.
6. CHOOSE TOOLS In Activity 1, you want to know the amount of money you
earn after working 30.5 hours during a week. Would you use the table, the
graph, or the equation to fi nd your earnings? What are your earnings?
Explain your reasoning.
7. Give an example of another real-life situation that you can model by an
equation in two variables.
Use what you learned about equations in two variables to
complete Exercises 4 and 5 on page 319.
Look for
Patterns
What pattern do
you notice in the
table for the
perimeter of
the square?
Math
Practice
316 Chapter 7 Equations and Inequalities
Lesson
7.4
EXAMPLE
Identifying Solutions of Equations in Two Variables
1
1
EXAMPLE
Using an Equation in Two Variables
2
2
Tell whether the ordered pair is a solution of the equation.
a. y 2x; (3, 6) b. y 4x 3; (4, 12)
6
?
2(3) Substitute. 12
?
4(4) 3
6 6
✓ Compare. 12 13
✗
So, (3, 6) is a solution. So, (4, 12) is not a solution.
You can use equations in two variables to represent situations involving
two related quantities. The variable representing the quantity that
can change freely is the independent variable. The other variable is
called the dependent variable because its value depends on the
independent variable.
The equation y 128 8x gives the amount y (in fl uid ounces) of milk
remaining in a gallon jug after you pour x cups.
a. Identify the independent and dependent variables.
Because the amount y remaining depends on the number x
of cups you pour, y is the dependent variable and x is the
independent variable.
b. How much milk remains in the jug after you pour 10 cups?
Use the equation to fi nd the value of y when x 10.
y 128 8x
Write the equation.
128 8(10) Substitute 10 for x.
48 Simplify.
There are 48 fl uid ounces remaining.
Tell whether the ordered pair is a solution of the equation.
1. y 7x; (2, 21) 2. y 5x 1; (3, 16)
3. The equation y 10x 25 gives the amount y (in dollars) in
your savings account after x weeks.
a. Identify the independent and dependent variables.
b. How much is in your savings account after 8 weeks?
Exercises 6–11
and 13–17
Lesson Tutorials
Key Vocabulary
equation in two
variables, p. 316
solution of an
equation in two
variables, p. 316
independent variable,
p. 316
dependent variable,
p. 316
An equation in two variables represents two quantities that change in
relationship to one another. A solution of an equation in two variables is
an ordered pair that makes the equation true.
Section 7.4 Writing Equations in Two Variables 317
Tables, Graphs, and Equations
You can use tables and graphs to represent equations in two variables.
The table and graph below represent the equation y x 2.
Independent
Variable, x
Dependent
Variable, y
Ordered Pair,
(x, y)
1 3 (1, 3)
2 4 (2, 4)
3 5 (3, 5)
EXAMPLE
Writing and Graphing an Equation in Two Variables
3
3
An athlete burns 200 calories weight lifting. The athlete then works out
on an elliptical trainer and burns 10 calories for every minute. Write and
graph an equation in two variables that represents the total number of
calories burned during the workout.
Words
The total
number
of calories
burned
equals
calories
burned
weight
lifting
plus
calories
burned per
minute
times
the number
of minutes.
Variables Let c be the total number of calories burned, and let m be the
number of minutes on the elliptical trainer.
Equation c 200 10
m
To graph the equation, fi rst make a table. Then plot the ordered pairs and
draw a line through the points.
4. It costs $25 to rent a kayak plus $8 for each hour. Write
and graph an equation in two variables that represents
the total cost of renting the kayak.
Exercises 22
and 23
Study Tip
When you draw a line
through the points, you
graph all the solutions
of the equation.
Reading
Make sure you read and
understand the context
of the problem. Because
you cannot have a
negative number of
minutes, use only whole
number values of m.
x
y
123
1
2
3
4
5
6
456
(3, 5)
(2, 4)
(1, 3)
0
0
m
c
20
200
40
60
Minutes
Calories
400
600
0
0
(10, 300)
(20, 400)
(30, 500)
Minutes,
m
c 200 10m
Calories,
c
Ordered
Pair, (m, c)
10
c 200 10(10)
300 (10, 300)
20
c 200 10(20)
400 (20, 400)
30
c 200 10(30)
500 (30, 500)
318 Chapter 7 Equations and Inequalities
You can model many rate problems by using the distance formula d rt,
where d is the distance traveled, r is the speed, and t is the time. When
you are given a speed, you can use the formula to write an equation in
two variables that represents the situation.
Distance Formula
Words To fi nd the distance traveled d, multiply the speed r by the
time t.
Algebra d
rt
EXAMPLE
Real-Life Application
4
4
A train averages 40 miles per hour between two cities. Use a graph to
show the relationship between the time and the distance traveled.
Method 1: Use a ratio table.
You can use a ratio table and multiplication to fi nd equivalent rates. Then
plot the ordered pairs (time, distance) from the table and draw a line
through the points.
Time (hours) 1246
Distance (miles) 40 80 160 240
Method 2: Use an equation in two variables.
Use the distance formula to write the equation d 40t. Use the equation
to make a table. Then plot the ordered pairs and draw a line through the
points, as shown in the graph above.
Time (hours), t
d 40t
Distance (miles), d Ordered Pair, (t, d)
1
d 40(1)
40 (1, 40)
2
d 40(2)
80 (2, 80)
4
d 40(4)
160 (4, 160)
6
d 40(6)
240 (6, 240)
5. WHAT IF? The train averages 50 miles per hour. Use a graph to show
the relationship between the time and the distance traveled.
Exercise 25
2 4 6
2 4 6
Remember
Speed is an example of
a rate.
t
d
2
80
4
6
Time (hours)
Distance (miles)
160
240
0
0
(1, 40)
(2, 80)
(4, 160)
(6, 240)
Section 7.4 Writing Equations in Two Variables 319
9
+(-6
)=
3
3
+(-3
)=
4
+(-
9
)=
9
+(-1
)=
1. VOCABULARY How are independent variables and dependent variables different?
2.
PRECISION Explain how to graph an equation in two variables.
3.
WHICH ONE DOESN’T BELONG? Which one does not belong with the other three?
Explain your reasoning.
y 12x 25
c 10t 5
a 7b 11
n 4n 6
Write a formula for the given measure. Tell what each variable represents.
Identify which variable depends on which in the formula.
4. the perimeter of a rectangle with a length of 5 inches
5. the area of a trapezoid with base lengths of 7 feet and 11 feet
Tell whether the ordered pair is a solution of the equation.
6. y 4x; (0, 4) 7. y 3x; (2, 6) 8. y 5x 10; (3, 5)
9. y x 7; (1, 6) 10. y 7x 2; (2, 0) 11. y 2x 3; (4, 5)
12. ERROR ANALYSIS Describe and correct
the error in fi nding a solution of the
equation in two variables.
Identify the independent and dependent variables.
13. The equation A 25w gives the area A (in square feet) of a rectangular
dance fl oor with a width of w feet.
14. The equation c 0.09s gives the amount c (in dollars) of commission
a salesperson receives for making a sale of s dollars.
15. The equation t 12p 12 gives the total cost t (in dollars)
of a meal with a tip of p percent (in decimal form).
16. The equation h 60 4m gives the height h (in inches)
of the water in a tank m minutes after it starts to drain.
17.
DRUM SET The equation b 540 30m gives the
balance b (in dollars) that you owe on a drum set
after m monthly payments. What is the balance after
9 monthly payments?
y 3x 2; (5, 1)
5
?
3(1) 2
5 5
So, (5, 1) is a solution.
✗
1
1
2
2
Exercises
7.4
Help with Homework
320 Chapter 7 Equations and Inequalities
OPEN-ENDED Complete the table by describing possible independent or
dependent variables.
Independent Variable Dependent Variable
18. The number of hours you study for a test
19. The speed you are pedaling a bike
20. Your monthly cell phone bill
21. The amount of money you earn
22.
PIZZA A cheese pizza costs $5. Additional toppings cost $1.50 each. Write and
graph an equation in two variables that represents the total cost of a pizza.
23.
GYM MEMBERSHIP It costs $35 to join a gym. The monthly fee is $25.
Write and graph an equation in two variables that represents the
total cost of a gym membership.
24.
TEXTING The maximum size of a text message is 160 characters.
A space counts as one character.
a. Write an equation in two variables that represents the
remaining (unused) characters in a text message as you type.
b. Identify the independent and dependent variables.
c. How many characters remain in the message shown?
25.
CHOOSE TOOLS A car averages 60 miles per hour on a
road trip. Use a graph to show the relationship between
the time and the distance traveled. What method did
you use to create your graph?
Write and graph an equation in two variables that shows
the relationship between the time and the distance traveled.
26.
Moves 2 meters
every 3 hours.
27.
Rises 5 stories
every 6 seconds.
28.
Moves 660 feet
every 10 seconds.
29.
Moves 960 kilometers
every 4 minutes.
3
3
4
4
Jac
kie
Pl
ea
se
c
al
l
me
.
Section 7.4 Writing Equations in Two Variables 321
Fill in the blank so that the ordered pair is a solution of the equation.
30. y 8x 3; (1,
) 31. y 12x 2; (
, 14) 32. y 22 9x; (
, 4)
33.
CRITICAL THINKING Can the dependent variable cause
a change in the independent variable? Explain.
34.
OPEN-ENDED Write an equation in two variables that has
(3, 4) as a solution.
35.
WALKING You walk 5 city blocks in 12 minutes. How
many city blocks can you walk in 2 hours?
36.
ANT How fast should the ant walk to go around the
rectangle in 4 minutes?
37.
LIGHTNING To estimate how far you are from lightning (in miles), count the
number of seconds between a lightning fl ash and the thunder that follows.
Then divide the number of seconds by 5. Use a graph to show the relationship
between the time and the distance. Describe the method you used to create
your graph.
38.
PROBLEM SOLVING You and a friend start biking in opposite directions from the
same point. You travel 108 feet every 8 seconds. Your friend travels 63 feet every
6 seconds.
a. How far apart are you and your friend after 15 minutes?
b. After 20 minutes, you take a 5-minute rest, but your friend does not. How far
apart are you and your friend after 40 minutes? Explain your reasoning.
39.
The graph represents the cost c (in dollars)
of buying n tickets to a baseball game.
a. Should the points be connected with a line to
show all the solutions? Explain your reasoning.
b. Write an equation in two variables that represents
the graph.
Write the fraction as a percent.
(Section 5.5)
40.
3
10
41.
4
5
42.
9
20
43.
17
25
44.
MULTIPLE CHOICE What is the area of the
triangle?
(Section 4.2)
A 36 cm
2
B 68 cm
2
C 72 cm
2
D 76.5 cm
2
16 in.
12 in.
n
c
2
10
4
6
Tickets
Dollars
20
30
0
0
(1, 10)
(2, 20)
(3, 30)
9 cm
17 cm
8 cm
