CHAPTE R
6
Equations
The
cconnectED.mcgraw-hill.com
onn
BIG Idea
Investigate
When solving an equation,
why is it necessary to
perform the same
operation on both sides
of an equal sign?
Animations
Vocabulary
Multilingual
eGlossary
Learn
Personal Tutor
Virtual
Manipulatives
n and
Additioction
Subtrations
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to help you organize
your notes.
n
licatio
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Practice
Self-Check Practice
Worksheets
Assessment
ry
Review Vocabulanes
the
de las operacio
order of operations orden
tion to perform first when
rules that tell which opera
used in an expression
more than one operation is
inside grouping
1. Simplify the expressions
es.
hes
symbols, like parent
ers.
2. Find the value of all pow
er from left to right.
3. Multiply and divide in ord
er from left to right.
4. Add and subtract in ord
Key Vocabulary
English
equation
solution
solve
two-step equation
310
Equations
Español
ecuación
solución
resolver
ecuación de dos pasos
When Will I Use This?
Your Tthuiis rn!
You will sollve
ter.
problem in the chap
Equations
311
Are You Ready
for the Chapter?
Text Option
Y have two options for checking
You
pprerequisite skills for this chapter.
Take the Quick Check below. Refer to the Quick Review for help.
Add. Write in simplest form.
1
2
3
1
1. _ + _
2. _ + _
6
8
3
2
3. _ + _
4
3
5
2
7
5
4. _ + _
9
6
EXAMPLE 1
_ _
Find 3 + 3 .
7
4
3
3
The LCD of _
and _
is 28.
7
Write the
problem.
Rename using
the LCD, 28.
of dog food in the morning and
_3
3×4
_
=
7×4
_3 cup in the evening. How much
4
dog food did Logan feed his dog
altogether?
Subtract. Write in simplest form.
7
1
6. _ - _
8
4
8.
5
1
7. _ - _
6
2
_2 - _4
3
9.
9
_3 - _2
5
7
7
10. RUNNING Pamela ran _
mile on
10
3
Tuesday and _
mile on Thursday.
8
How much farther did she run on
Tuesday?
7
3
+_
4
3×7
_
4×7
28
21
= +_
28
12
_
28
21
+_
28
5
33
_
or 1_
28
28
_ _
4
9
3
5
The LCD of _
and _
is 36.
9
4
Write the
problem.
Rename using
the LCD, 36.
_3
3×9
_
=
4×9
4
5
-_
9
5×4
_
9×4
5
6
dishes. Which chore did he spend
more time completing and by how
much?
Equations
12
_
Find 3 - 5 .
1
the lawn and _
hour washing
Online Option
Add the
numerators.
EXAMPLE 2
4
11. CHORES Howie spent _
hour mowing
312
4
7
5. ANIMALS Logan fed his dog _
cup
8
Take the Online Readiness Quiz.
Subtract the
numerators.
27
_
36
20
= -_
36
27
_
36
20
-_
36
7
_
36
Simplify the Problem
Step 1
Have you ever tried to
solve a long word problem
and didn’t know where to
start? Here’s what I do.
Read the problem.
Kylie wants to order
several pairs of running
shorts from an online
store. They cost $14 each,
and there is a one-time
shipping fee of $7. What
is the total cost of
buying any number of
pairs of shorts?
Step 2
Rewrite the problem to make it
simpler. Keep all of the important
information but use fewer words.
Step 3
Rewrite the problem using even
fewer words. Write a variable for
the unknown.
The total cost of x shorts
is $14x plus $7.
Step 4
Translate the words into an
expression.
14x + 7
Kylie wants to buy some
shorts that cost $14 each
plus a shipping fee of $7.
What is the total cost for
any number of pairs of
shorts?
Practice
Use the method above to write an expression for each problem.
1. MONEY Akira is saving money to
buy a bicycle. He has already
saved $80 and plans to save an
additional $5 each week. Find
the total amount he has saved
after any number of weeks.
2. TRAVEL A taxi company charges
$1.50 per mile plus a $10 fee.
What is the total cost of a taxi
ride for any number of miles?
GLE 0606.1.6 Read and interpret the language of mathematics and use written/oral communication to
express mathematical ideas precisely.
Studying Math
313
Multi-Part
Lesson
1
PART
Addition and Subtraction Equations
A
Main Idea
Solve equations by
using mental math
and the guess, check,
and revise strategy.
Vocabulary
V
e
equation
equals sign
solve
solution
Get ConnectED
GLE 0606.3.4
Use expressions, equations
and formulas to solve
problems. Also addresses
SPI 0606.3.3, GLE 0606.3.2.
B
C
D
E
F
Equations
When the amounts on each side
of a balance are equal, we say it
is balanced.
Place four centimeter
cubes and a paper bag
on one side of a balance.
Place seven centimeter
cubes on the other side of the balance.
1. Replace the bag with centimeter cubes until the two sides are
balanced. How many centimeter cubes did you need?
Let x represent the bag. Model each sentence on a balance. Find
the number of centimeter cubes needed to balance the two sides.
2. x + 2 = 5
3. x + 5 = 7
4. x + 3 = 4
5. x + 6 = 6
An equation is a mathematical sentence showing two expressions
are equal. An equation contains an equals sign, =. A few examples
are shown below.
2+7=9
10 - 6 = 4
14 = 2 · 7
Some equations contain variables.
2+x=9
4=k-6
15 ÷ m = 3
When you replace a variable with a value that results in a true
sentence, you solve the equation. That value for the variable is the
solution of the equation.
2+x=9
2+7 =9
9 = 9 This sentence is true.
The value for the
variable that results in
a true sentence is 7.
So, 7 is the solution.
314
Equations
Solve an Equation Mentally
Is 3, 4, or 5 the solution of the equation a + 7 = 11?
Value of a
a + 7 11
Are Both Sides Equal?
3
3 + 7 = 11
10 ≠ 11
no
4
4 + 7 = 11
11 = 11
yes
5
5 + 7 = 11
12 ≠ 11
no
The solution is 4.
Solve 15 = 3h mentally.
15 = 3h
Think 15 equals 3 times what number?
15 = 3 · 5 You know that 15 = 3 · 5.
15 = 15
The solution is 5.
a. Is 2, 3, or 4 the solution of the equation 4n = 16?
b. Solve 24 ÷ w = 8 mentally.
SKATING The total cost of a pair of skates and kneepads is $63.
The skates cost $45. Use the guess, check, and revise strategy to
solve the equation 45 + k = 63 to find k, the cost of the
kneepads.
Use the guess, check, and revise strategy.
Try 14.
Try 16.
Try 18.
45 + k = 63
45 + k = 63
45 + k = 63
45 + 14 63
45 + 16 63
45 + 18 63
59 ≠ 63
61 ≠ 63
63 = 63 So, the kneepads cost $18.
Real-World Link
R
An ostrich has the
A
largest eye of any
land animal. It is
about 2 inches
(5 centimeters) across.
c. ANIMALS The difference between an ostrich’s speed and a
chicken’s speed is 31 miles per hour. An ostrich can run at a
speed of 40 miles per hour. Use the mental math or the guess,
check, and revise strategy to solve the equation 40 - c = 31 to
find c, the speed a chicken can run.
Lesson 1A Addition and Subtraction Equations
315
Example 1
Example 2
Example 3
Identify the solution of each equation from the list given.
1. 9 + w = 17; 7, 8, 9
2. d - 11 = 5; 14, 15, 16
3. 4 = 2y; 2, 3, 4
4. 8 ÷ c = 8; 0, 1, 2
Solve each equation mentally.
5. x + 6 = 18
6. n - 10 = 30
8. 4x = 32
9. x - 11 = 23
7. 15k = 30
10. w + 15 = 45
11. CIVICS Mississippi and Georgia have a total of 21 electoral votes.
Mississippi has 6 electoral votes. Use mental math or the guess, check,
and revise strategy to solve the equation 6 + g = 21 to find g, the
number of electoral votes Georgia has.
= Step-by-Step Solutions begin on page R1.
Extra Practice begins on page EP2.
Example 1
Example 2
Example 3
Identify the solution of each equation from the list given.
12. a + 15 = 23; 6, 7, 8
13 29 + d = 54; 24, 25, 26
14. 35 = 45 - n; 10, 11, 12
15. 19 = p - 12; 29, 30, 31
16. 6w = 30; 5, 6, 7
17. 63 = 9k; 6, 7, 8
18. 36 ÷ s = 4; 9, 10, 11
19. x ÷ 7 = 3; 20, 21, 22
Solve each equation mentally.
20. j + 7 = 13
21. m + 4 = 17
22. 22 = 30 - m
23. 12 = 24 - y
24. 15 - b = 12
25. 25 - k = 20
26. 5m = 25
27. 10t = 90
28. 22 ÷ y = 2
29. d ÷ 3 = 6
30. 54 = 6b
31. 24 = 12k
For Exercises 32–35, solve using mental math or the guess, check, and
revise strategy.
32. BASKETBALL One season, the Cougars won 20 games. They played a
total of 25 games. Use the equation 20 + g = 25 to find g, the number
of games the team lost.
33. MONEY Five friends earn a total of $50 doing yard work in their
neighborhood. Each friend earns the same amount. Use the equation
5f = 50 to find f, the amount that each friend earns.
B
34. ANIMALS A bottlenose dolphin is 96 inches long. There are 12 inches
in 1 foot. Use the equation 12d = 96 to find d, the length of the
bottlenose dolphin in feet.
35 SCHOOL Last year, 700 students attended Walnut Springs Middle
School. This year, there are 665 students. Use the equation
700 - d = 665 to find d, the decrease in the number of students
from last year to this year.
316
Equations
C
36. NUMBER SENSE What 3 consecutive even numbers added together
equal 42? Use the equation n + (n + 2) + (n + 4) = 42 to help you
solve.
37. OPEN ENDED Give an example of an equation that has a solution of 5.
38. REASONING Tell whether the statement below is always, sometimes,
or never true.
Equations like a + 4 = 8 and 4 - m = 2 have exactly one solution.
CHALLENGE Tell whether each statement is true or false. Then explain
your reasoning.
39. In m + 8, the variable m can have any value.
40. In m + 8 = 12, the variable m can have any value and be a solution.
41. E WRITE MATH Write a real-world problem in which you would
solve the equation a + 12 = 30.
TTest Practice
42. The graph shows the life expectancy of
certain mammals. The difference d
between the number of years a blue
whale lives and the number of years a
gorilla lives is 45. Which equation has
a solution of 45?
44. Which of the following statements is
true for the equation 6x = 78?
F. To find the value of x, subtract 6
from 78.
G. To find the value of x, multiply 6
by 78.
Life Expectancy of Mammals
100
90
80
76
Years
80
H. To find the value of x, add 6 and 78.
I. To find the value of x, divide 78
by 6.
70
60
35
40
20
45.
SHORT RESPONSE The perimeter of
the square shown is 56 units.
a
ril
l
Go
an
m
Hu
El Afr
ep ic
ha an
nt
Bl
W Kill
ha er
le
ue
W
ha
le
0
Mammal
43.
A. d + 35 = 80
C. 80 + 35 = d
B. d - 35 = 80
D. d - 80 = 35
GRIDDED RESPONSE Felipe biked
21 miles in 3 hours. He can find his
mean time per mile b by solving the
equation 3b = 21. What is the value
of b?
x
The equation 4x = 56 can be used to
find the length of each side. Explain
how to solve the equation mentally.
Lesson 1A Addition and Subtraction Equations
317
Multi-Part
Lesson
1
PART
Addition and Subtraction Equations
A
B
C
D
E
F
Problem-Solving Investigation
Main Idea Solve problems by working backward.
Work Backward
LOURDES: Two friends and I went to the movies. We
shared a large tub of popcorn for $8.99. We spent
$22.49 in all. How much did each ticket cost?
YOUR MISSION: Work backward to find the cost of
each ticket.
Understand
You know that three people went to the movies, they shared a tub of popcorn
for $8.99, and $22.49 was spent in all. You need to find the cost of each ticket.
Plan
Solve by working backward. Subtract the cost of the popcorn from the total
spent. Then divide by the number of tickets.
Solve
$8.99
cost of tub of popcorn
$22.49 - $8.99 = $13.50 remainder spent on tickets
$13.50 ÷ 3 = $4.50
divide by number of tickets
The cost of each ticket is $13.50 ÷ 3 or $4.50.
Check
Begin with the three tickets purchased. $4.50 × 3 = $13.50. Add the cost of the
popcorn to find the total spent. $13.50 + $8.99 = $22.49. 1. What is the best way to check your solution when using the work
backward strategy?
2. E WRITE MATH Explain when you would use the work backward strategy
to solve a problem.
GLE 0606.1.2 Apply and adapt a variety of appropriate strategies to problem solving, including estimation,
and reasonableness of the solution. Also addresses GLE 0606.3.2.
318
Equations
= Step-by-Step Solutions begin on page R1.
Extra Practice begins on page EP2.
•
•
•
•
Work backward.
Make a table.
Act it out.
Choose an operation.
8. NUMBER SENSE What is the least
positive number that you can divide by
7 and get a remainder of 4, divide by
8 and get a remainder of 0, and divide
by 9 and get a remainder of 5?
Use the work backward strategy to solve
Exercises 3–5.
3. EXERCISE The table shows the amount of
time it takes Henry to do different
activities before going to school.
Time
(minutes)
Activity
Getting to the gym from home
15
Exercising
35
Getting ready for school
10
Getting to school from the gym
20
9. LADDERS You are standing on the middle
rung of a ladder. If you first climb up
3 rungs, then down 5 rungs, and then up
10 rungs to get onto the top rung, how
many rungs are on the ladder?
10. COMETS Halley’s comet is visible from
Earth approximately every 76 years. If
the next scheduled appearance is in 2062,
when was the last visit by Halley’s
comet?
If he needs to be at school at 8:15 a.m.,
what time should he leave his house in
the morning to get to the gym?
4. AGE Celeste is five years younger than
her sister, Lynn. Lynn is half as old as her
Aunt Mallory. If Mallory is 28, how old is
Celeste?
11. NUMBER SENSE A number is subtracted
from 20. If the difference is divided by 3
and the result is 4, what is the number?
12. COMPUTERS Use the table below to find
how many bits are in a megabyte.
5 NUMBER SENSE A number is multiplied
by 4, and then 6 is added to the product.
The result is 18. What is the number?
1 byte = 8 bits
1 kilobyte = 1,024 bytes
1 megabyte = 1,000 kilobytes
Use any strategy to solve Exercises 6–13.
6. ROLLER COASTERS The list shows how
many times each of 20 students rode a
roller coaster at an amusement park
one day.
Number of Times Rode Roller Coaster
5
10
0
12
8
7
2
6
4
1
0
14
3
11
5
9
13
8
6
3
How many more students rode a roller
coaster 5 to 9 times than 10 to 14 times?
7. CAMERAS Adamo saved 13 pictures on his
digital camera, and deleted 32 pictures. If
there are now 108 pictures, how many
pictures had he saved originally?
13. PARKING The diagram shows the number
of parking spaces available in several
different parking lots on a college campus.
The campus has a total of 310 parking
spaces. Find x, the number of parking
spaces in Lot C.
Lot B
60 spaces
Lot A
86 spaces
Lot C
x spaces
Lot E
52 spaces
Lot D
75 spaces
Lesson 1B Addition and Subtraction Equations
319
Multi-Part
Lesson
1
PART
Addition and Subtraction Equations
A
B
C
D
E
F
Addition Equations
Main Idea
Solve addition
equations using
models.
BASEBALL Bryan played two baseball games last weekend. He got
5 hits in all. He had 1 hit in the first game. How many hits did he
get in the second game?
Model and Solve Addition Equations
Get ConnectED
What do you need to find? the number of hits he had in the
second game
GLE 0606.3.2
Interpret and represent
algebraic relationships with
variables in expressions,
simple equations and
inequalities. GLE 0606.3.4
Use expressions, equations
and formulas to solve
problems. Also addresses
GLE 0606.1.3, SPI 0606.3.5.
Define a variable.
hits in second game, s
Use a bar diagram
to help write
an equation.
5
hits in second game, s
1
s + 1= 5
Work backward.
Since s + 1 = 5, 5 - 1 = s.
So, Bryan had 5 - 1 or 4 hits in the second game.
and Apply
Solve each problem using a bar diagram.
1. SPORTS In the 2008 Summer Olympics, the United States won
11 more medals in swimming than Australia. The United States
won a total of 31 medals. Find how many medals Australia won.
2. ANIMALS A lion can run 50 miles per hour. This is 20 miles per
hour faster than a house cat. Find the speed of a house cat.
3. SHOPPING Terrell bought an MP3 player. He spent the rest of his
money on an Internet music subscription for $25.95. If he started
with $135, how much was the MP3 player?
Write a real-world addition problem for each equation modeled
below. Then write an equation and solve.
4.
length of vacation, v
320
Equations
5.
6 weeks
2 weeks
14 pounds
weight of rabbit, w
5 pounds
An equation is like a balance. The quantity on the
left side of the equals sign is balanced with the
quantity on the right. When you solve an equation,
you need to keep the equation balanced.
To solve an addition equation using cups and
counters, subtract the same number of counters
from each side of the mat so that the equation
remains balanced.
Solve x + 1 = 5 using cups and counters.
Model the equation. Use a cup to
represent x.
=
x+1
=
5
Remove 1 counter from each side
to get the cup by itself.
=
x+1-1
There are 4 counters remaining on
the right side, so x = 4.
=
5-1
=
x
=
4
The solution is 4.
Check
x+1=5
Write the original equation.
4+15
Replace x with 4. Is this sentence true?
5=5
The sentence is true. and Apply
Solve each equation using cups and counters.
6. 1 + x = 8
7. x + 2 = 7
8. 9 = x + 3
the Results
9. MAKE A CONJECTURE Write a rule that you can use to solve an addition
equation without using models.
Lesson 1C Addition and Subtraction Equations
321
Multi-Part
Lesson
1
Addition and Subtraction Equations
PART
A
Main Idea
Solve and write
addition equations.
Vocabulary
V
i
inverse
operations
Subtraction Property
of Equality
B
D
E
F
Solve and Write
Addition Equations
MINIATURE GOLF On the second hole of miniature golf, it took
Anne 3 putts to sink the golf ball. Her score is now 5.
Her score on
the first hole
is unknown.
Get ConnectED
GLE 0606.3.4
Use expressions, equations
and formulas to solve
problems. SPI 0606.3.3
Write equations that
correspond to given
situations or represent a
given mathematical
relationship. Also addresses
GLE 0606.3.2, SPI 0606.3.5.
C
=
Her score is
now 5.
She scored a 3 on
the second hole.
1.
2.
3.
4.
What does the cup represent in the figure?
What addition equation is shown in the figure?
Explain how to solve the equation.
What was Anne’s score on the first hole?
In Lesson 1A, you mentally solved equations. Another way is to
use inverse operations, which undo each other. For example, to
solve an addition equation, use subtraction.
Solve an Equation By Subtracting
Solve 8 = x + 3. Check your solution.
Method 1 Use models.
Method 2 Use symbols.
8 = x + 3 Write the equation.
- 3 = - 3 Subtract 3 from
each side to “undo”
5=x
=
the addition of 3
on the right.
Remove 3 counters from
each side.
Check
8 = x + 3 Write the equation.
8 5 + 3 Replace x with 5.
8=8
This sentence is true.
The solution is 5.
Solve each equation. Use models if necessary.
a. c + 2 = 5
322
Equations
b. 6 = x + 5
c. 3 + y = 12
When you solve an equation by subtracting the same number from
each side of the equation, you are using the Subtraction Property
of Equality.
Subtraction Property of Equality
Words
If you subtract the same number from each side of an
equation, the two sides remain equal.
Examples
Numbers
Algebra
5= 5
-3=-3
2= 2
x+2 = 3
-2=-2
x
= 1
MUSIC Ruben and Tariq have 245 downloaded songs. If Ruben
has 132, how many belong to Tariq? Write and solve an
addition equation to find how many songs belong to Tariq.
Words
Variable
Ruben and Tariq have 245 songs.
Let t represent the number of songs that belong to Tariq.
Tariq’s songs, t
Bar Diagram
245 songs
Tariq’s songs, t
132
Equation
132
+
t
= 245.
132 + t = 245 Write the equation.
- 132
= - 132 Subtract 132 from each side.
t = 113 Simplify.
So, 113 songs belong to Tariq.
Checking Solutions You
should always check your
solution. You will know
immediately whether your
solution is correct or not.
Check
132 + 113 = 245 d. MUSIC Suppose Ruben had 147 songs of the 245 that were
downloaded. Write and solve an addition equation to find
how many songs belong to Tariq.
Lesson 1D Addition and Subtraction Equations
323
ANIMALS A male gorilla weighs 379 pounds on average. This is
181 pounds more than the weight of the average female gorilla.
Write and solve an addition equation to find the weight of an
average female gorilla.
Words
181 pounds plus
the weight of an
is 379 pounds.
average female gorilla
Variable
Let w represent the weight of an average female gorilla.
Bar Diagram
379 pounds
weight, w
Equation
Real-World Link
R
Gorillas typically travel
G
no more than 20 feet
on two legs. The twolegged upright stance
is generally used for
chest-beating or to
observe or reach
something.
181
181 pounds
w
+
= 379
181 + w = 379 Write the equation.
- 181
= - 181 Subtract 181 from each side.
w = 198 379 - 181 = 198
So, an average female gorilla weighs 198 pounds.
Check
181 + 198 = 379 e. INTERNET In a recent year, there were 171 million home
Internet users in the United States and Japan. Of that total,
36 million users were in Japan. Write and solve an addition
equation to find the number of Internet users in the
United States.
Example 1
Solve each equation. Check your solution.
1. x + 3 = 5
2. 2 + m = 7
c+3=6
4. 10 = 6 + e
3
324
Example 2
5. CARPENTRY A board that measures 19 meters in length is cut into two
pieces. One piece measures 7 meters. Write and solve an equation to
find the length of the other piece.
Example 3
6. MUSCLES It takes 43 facial muscles to frown. This is 26 more muscles
than it takes to smile. Write and solve an equation to find the number
of muscles it takes to smile.
Equations
= Step-by-Step Solutions begin on page R1.
Extra Practice begins on page EP2.
Example 1
Solve each equation. Check your solution.
7. y + 7 = 10
8. x + 5 = 11
9. 9 = 2 + x
10. 7 = 4 + y
11. 7 + a = 9
12. 5 + g = 6
13. d + 3 = 8
14. x + 4 = 6
15. 3 + f = 8
Example 2
16. MONEY Zacarias and Paz together have $756. If Zacarias has $489, how
much does Paz have? Write and solve an addition equation to find
how much money belongs to Paz.
Example 3
17 SNAKES The average length of a King Cobra is 118 inches, which is
22 inches longer than a Black Mamba. Write and solve an addition
equation to find the average length of a Black Mamba.
18. GAMES Louise had a score of 7 in the second round of a certain game.
This made her total score after two rounds equal to 11. Write and solve
an addition equation to find her score in the first round.
19. TRUCKS The table shows the heights
of three monster trucks. Bigfoot 5
is 4.9 feet taller than Bigfoot 2. Write
and solve an addition equation to
find the height of Bigfoot 2.
B
Truck
Height (ft)
Bigfoot 5
15.4
Swamp Thing
12.2
Bigfoot 2
Solve each equation. Check your solution.
20. t + 1.9 = 3.8
21. 1.8 + n = 2.1
22. a + 6.1 = 8.4
23 7.8 = x + 1.5
1 _
24. m + _
=2
1 _
25. t + _
=
3
3
4
3
4
26. GRAPHIC NOVEL Refer to the graphic novel frame below for Exercises a–b.
Remember, I
need 50
points for the
pizza party.
a. If Mei has already earned 30 points, write and solve an addition
equation to find the number of points she still needs.
b. Suppose Julie has already earned 36 points. Write and solve an
addition equation to find the number of points she still needs to
earn the pizza party.
Lesson 1D Addition and Subtraction Equations
325
C 27. OPEN ENDED Write two different addition equations that have 12 as
the solution.
28. CHALLENGE In the equation x + y = 5, the value for x is a whole
number greater than 2 but less than 6. Determine the possible
solutions for y.
29. E WRITE MATH Write a real-world problem that can be represented
by the equation x + 5 = 87.
TTest Practice
30. The model represents the equation
x + 4 = 7.
32. Refer to Exercise 31. How much
money does Niko still need to save?
=
33.
What is the first step in finding the
value of x?
A. Add 4 counters to each side.
B. Subtract 7 counters from each side.
A. $100
C. $65
B. $70
D. $60
EXTENDED RESPONSE The table
shows the point values from a bag
toss game.
Scoring Toss
Points
went through hole
3
landed on board
1
C. Add 7 counters to each side.
D. Subtract 4 counters from each side.
31. Niko wants to buy a skateboard that
costs $85. He has already saved $15.
Which equation represents the amount
of money Niko needs to buy the
skateboard?
F. t - 15 = 85
H. 15 - t = 85
G. t + 15 = 85
I. t = 15 + 85
Part A Before Wes’s last toss, he had
a score of 15 points. After tossing the
bag five more times, he had a score of
24 points. Write and solve an equation
to show how many points Wes scored
on the five tosses.
Part B How could Wes have scored
the points in the five tosses if none of his
tosses missed the board? Explain.
34. MUSIC Trent downloaded 3 times as many songs as Harriet last month.
Harriet downloaded 5 fewer than Barry, but 3 more than Morgan. Barry
downloaded 10 songs. How many songs did each person download?
(Lesson 1B)
Solve each equation mentally.
(Lesson 1A)
35. 16 + h = 24
36. 15 = 50 - m
326
Equations
37. 14t = 42
Multi-Part
Lesson
1
PART
Addition and Subtraction Equations
A
B
C
D
E
F
Subtraction Equations
Main Idea
Solve subtraction
equations using
models.
Get ConnectED
GLE 0606.3.2
Interpret and represent
algebraic relationships with
variables in expressions,
simple equations and
inequalities. GLE 0606.3.4
Use expressions, equations
and formulas to solve
problems. Also addresses
GLE 0606.1.3, SPI 0606.3.5.
BASEBALL CARDS Zack gave 5 baseball cards to his sister. Now he
has 41 cards. How many did he have originally?
What do you need to find? the number of cards he had
originally
Define a variable.
original number of cards, c
Write an equation.
original number of cards, c
c - 5 = 41
41 cards
5 cards
Work backward.
Since c - 5 = 41, 41 + 5 = c.
So, Zack originally had 46 baseball cards.
and Apply
Solve using bar diagrams.
1. Suppose Zack gave his younger brother 8 cards and was left
with 37 cards. How many did he have originally?
2. SHOPPING Clinton has $12 after buying an item at the mall. The
item cost $5. How much money did Clinton have originally?
3. Write a real-world subtraction problem for the equation
modeled below. Then write an equation and solve.
miles driven, m
128 miles
67 miles
the Results
4. Why is it important to define the variable?
5. MAKE A CONJECTURE Write a rule for solving equations like
x - 4 = 7.
Lesson 1E Addition and Subtraction Equations
327
Multi-Part
Lesson
1
Addition and Subtraction Equations
PART
A
Main Idea
Solve and write
subtraction equations.
Vocabulary
V
Addition Property of
A
Equality
Get ConnectED
GLE 0606.3.4
Use expressions, equations
and formulas to solve
problems. SPI 0606.3.3
Write equations that
correspond to given
situations or represent a
given mathematical
relationship. Also addresses
GLE 0606.1.4, GLE 0606.3.2,
SPI 0606.3.5.
B
C
D
F
E
Solve and Write
Subtraction Equations
BOWLING Meghan’s bowling score
wass 36 po
points less than Charmaine’s.
Meghan’s score was 109.
1. Let s represent Charmaine’s score.
Write an equation for 36 points less
than Charmaine’s score is equal to 109.
2. Find Charmaine’s score by counting forward. What operation
does counting forward suggest?
Because addition and subtraction are inverse operations, you can
solve a subtraction equation by adding.
Solve an Equation by Adding
Solve x - 2 = 3.
Method 1 Use models.
Method 2 Use symbols.
Model the equation.
x - 2 = 3 Write the equation.
+ 2 = + 2 Add 2 to each side.
x
= 5 Simplify.
x
3
2
Solve the equation. Work
backward. Find 3 + 2.
Check
x-2=3
5-2=3
3=3
Write the equation.
Replace x with 5.
This sentence is true.
The solution is 5.
Solve. Use models if necessary.
a. x - 7 = 4
b. y - 6 = 8
c. 9 = a - 5
When you solve an equation by adding the same number to each side
of the equation, you are using the Addition Property of Equality.
328
Equations
Addition Property of Equality
Words
If you add the same number to each side of an equation,
the two sides remain equal.
Examples
Numbers
Algebra
5= 5
+3=+3
8= 8
x-2= 3
+2=+2
x
= 5
SPACE FLIGHT At age 25, Gherman Titov of Russia was the
youngest person to travel into space. This is 52 years less than
the oldest person to travel in space, John Glenn. How old was
John Glenn? Write and solve a subtraction equation.
Real-World Link
R
TTo prepare for space
missions, astronauts
train in the Neutral
Buoyancy Laboratory.
It is 202 feet long,
102 feet wide, and
40 feet deep (20 ft
above ground level
and 20 ft below). It
holds 6.2 million
gallons of water.
Words
Oldest age
minus
is 52 years.
Let a represent the oldest age in space.
Variable
age, a
Bar
Diagram
a years
25 years
Equation
youngest age
a
-
52 years
25
= 52
a - 25 = 52 Write the equation.
+ 25 = + 25 Add 25 to each side.
a
= 77 Simplify.
John Glenn was 77 years old.
Check
77 - 25 = 52 d. ROLLERBLADES Raheem’s rollerblades cost $70 less than
his bicycle. His rollerblades cost $45. Write and solve a
subtraction equation to find how much his bicycle cost.
e. GROWTH Georgia’s height is 4 inches less than Sienna’s
height. Georgia is 58 inches tall. Write and solve a subtraction
equation to find Sienna’s height.
Lesson 1F Addition and Subtraction Equations
329
Example 1
Solve each equation. Check your solution.
1. a - 5 = 9
Example 2
2. b - 3 = 7
3. 4 = y - 8
4. AGES Devon is 13 years old. This is 4 years younger than his older
brother, Todd. Write and solve a subtraction equation to find
Todd’s age.
5. STUDYING Catherine studied for 45 minutes for her science test. This was
20 minutes less than she studied for her algebra test. Write and solve a
subtraction equation to find how long she studied for her algebra test.
= Step-by-Step Solutions begin on page R1.
Extra Practice begins on page EP2.
Example 1
Solve each equation. Check your solution.
6. c - 1 = 8
7. f - 1 = 5
8. 2 = e - 1
1=g-3
10. r - 3 = 1
11. t - 7 = 2
9
Example 2
12. AGES Pete is 15 years old. This is 6 years younger than his sister
Victoria. Write and solve a subtraction equation to find Victoria’s age.
13. SOCIAL STUDIES The difference between the
number of electoral votes for Florida and
North Carolina is 12 votes. Write and solve
a subtraction equation to find the number
of electoral votes for Florida.
B
Electoral Votes
Number
of Votes
State
Florida
North Carolina
15
14. ALGEBRA Find the value of t if t - 7 = 12.
15. ALGEBRA If b - 10 = 5, what is the value of b + 6?
Solve each equation. Check your solution.
16. a - 2.1 = 5.8
17. a - 1.1 = 2.3
18. 4.6 = e - 3.2
1 _
19. m - _
=2
1 _
20. n - _
=
1 _
21. s - _
=7
3
22.
3
4
3
4
MULTIPLE REPRESENTATIONS The bar
diagram represents a subtraction
equation.
3
9
x°F
74°F
13°F
a. WORDS Write a real-world problem that can be represented by the
bar diagram.
b. ALGEBRA Write a subtraction equation that can be represented by
the bar diagram.
c. NUMBERS Solve the equation you wrote in part b.
23 SHOPPING Alejandra spent her birthday money on a video game that
cost $24, a controller for $13, and a memory card for $16. The total tax
was $3. Write and solve a subtraction equation to find how much money
Alejandra gave the cashier if she received $4 in change.
330
Equations
C 24. FIND THE ERROR Elisa is explaining how to solve
the equation d - 6 = 4. Find her mistake and
correct it.
Subtract 6 from
each side.
25. E WRITE MATH Write a real-world problem that could be represented
by d - 32 = 64.
TTest Practice
26. Which of the following statements is
true concerning the equation x - 5 = 13?
27. Arizona became a state 96 years later
than Indiana. Which equation can be
used to find the year y Arizona became
a state?
A. To find the value of x, add 5 to
each side.
B. To find the value of x, subtract 5
from each side.
C. To find the value of x, add 13 to
each side.
D. To find the value of x, subtract 13
from each side.
29. TIME After school, you spent
1
1_
hours at practice and then
Year It Became
a State
Arizona
Indiana
1816
F. y = 1816 - 96 H. y - 1816 = 96
G. y + 96 = 1816 I. 1816 - y = 96
Major League Baseball Stadiums
60,000
57,396
,
56,000
56,000
Capacity
28. BASEBALL Refer to the graph.
Write and solve an addition
equation to find how many
fewer people can be seated
at Dodger Stadium than at
Shea Stadium. (Lesson 1D)
State
52,000
50,516
50,445
,
50,096
Rogers
Centre
Coors
Field
Turner
Field
48,000
2
45 minutes at the library. It took
you 10 minutes to get home
at 5:30 p.m. What time did
practice start? (Lesson 1B)
44,000
Shea
Stadium
Dodger
Stadium
Stadium
Lesson 1F Addition and Subtraction Equations
331
Mid-Chapter
Check
Solve each equation mentally.
1. 7n = 84
2. d ÷ 9 = 3
(Lesson 1A)
3. p - 25 = 40
4. d + 19 = 25 5. w ÷ 4 = 11 6. 8x = 56
7. ENTERTAINMENT Cassie is going to dinner
and a movie with some friends. The
movie starts at 7:30 p.m. The restaurant
is 15 minutes from the theater, and
dinner will take one hour. If Cassie
lives 20 minutes from the restaurant,
what time should she leave her house?
(Lesson 1B)
8. MULTIPLE CHOICE Fonzi spent a total of
90 minutes completing his chores this
week. Which of the following equations
represents the number of minutes Fonzi
spent washing the dishes? (Lesson 1D)
14. BAKING Colin used 3_ cups of flour for
3
4
a bread recipe and 2 cups for a cookie
1
recipe. After baking, he had 4_
cups
2
of flour left. Write and solve an equation
to find the amount of flour Colin had
originally. (Lesson 1F)
15. MULTIPLE CHOICE Mrs. Sanchez spent
$45.67 at the grocery store. After
shopping, she had $32.17 left. Which
equation could be used to find how
much money Mrs. Sanchez had before
she went shopping? (Lesson 1F)
F. x + 45.67 = 32.17
G. x + 32.17 = 45.67
H. 32.17 - x = 45.67
I. x - 45.67 = 32.17
Chore
Time
(min)
Vacuuming
42
(Lesson 1F)
Dishes
3
7
16. t - _
= 2_
Solve each equation. Check your solution.
8
8
A. m = 42 + 90
C. m + 42 = 90
17. s - 19.4 = 22
B. 42 - m = 90
D. 90 = m - 42
18. h - 6.2 = 17.1
9. KITES A kite is 70 feet in the air. A few
minutes later it ascends to 120 feet. Write
and solve an addition equation to find
the kite’s change in altitude. (Lesson 1B)
Solve each equation. Check your solution.
1
= 10
19. r - 5_
6
20. TIDES The difference between the water
levels for high and low tide was 3.6 feet.
Write and solve an equation to find the
water level at high tide. (Lesson 1F)
(Lesson 1D)
3
1
10. y + _
= 9_
4
2
2
11. 15_
= x + 12_
3
12. h + 7.9 = 13
13. a + 1.6 = 2.1
332
Equations
5
6
Tide Level at the Lake
Worth Pier
High
Low
0.2 foot
Multi-Part
Lesson
2
Multiplication and Division Equations
PART
A
B
C
D
Multiplication Equations
Main Idea
Solve multiplication
equations using
models.
Get ConnectED
GLE 0606.3.2
Interpret and represent
algebraic relationships with
variables in expressions,
simple equations and
inequalities. GLE 0606.3.4
Use expressions, equations
and formulas to solve
problems. Also addresses
GLE 0606.1.3, SPI 0606.3.5.
RUNNING In 5 days, Nicole ran a total of 10 miles. She ran the same
amount each day. How much did she run each day?
What do you need to find? the distance Nicole ran in one day
Define a variable.
Let d represent the distance run in one day.
d
Write an equation.
10 miles
d
d
d
d
d
5d = 10
Work backward.
Since 5d = 10, 10 ÷ 5 = d. So, d = 2.
Nicole ran 2 miles each day.
and Apply
Solve each problem by drawing a bar diagram.
1. RUNNING Suppose Nicole ran 12 miles in four days. If she ran
the same amount each day, how much did she run in one day?
2. CELL PHONES Krista has owned her cell phone for 8 months,
which is twice as long as her sister Allie has owned her cell
phone. How long has Allie had her cell phone?
3. ANIMALS The average lifespan of a horse is 40 years, which
is five times longer than the average lifespan of a guinea
pig. Find the average lifespan of a guinea pig.
4. SAVINGS Kosumi is saving an equal amount each week for
4 weeks to buy a $40 video game. Find how much he is saving
each week.
Lesson 2A Multiplication and Division Equations
333
Solve 3x = 12. Check your solution.
Model the equation.
=
33x
=
Divide the 12
counters equally
into 3 groups. There
are 4 in each group.
12
=
3
3x
3
=
12
3
x
=
4
3x = 12 Write the original equation.
Check
3(4) = 12
Replace x with 4.
12 = 12 This sentence is true. The solution is 4.
and Apply
Solve each equation using cups and counters.
5. 4n = 8
6. 3x = 9
7. 10 = 5x
8. FOOD Two friends agree to split the cost of nachos equally. If the nachos
cost $6, how much will each friend pay?
9. GYMNASTICS Madison is taking gymnastic classes that last 11 weeks. If
the total cost is $121, find the cost for each week.
the Results
10. MAKE A CONJECTURE Write a rule for solving equations like 2x = 24
without using models.
For Exercises 11–12, write and solve an equation to represent each situation.
11.
c
334
12.
$12
Equations
c
c
c
=
Multi-Part
Lesson
2
PART
Multiplication and Division Equations
A
Main Idea
Solve and write
multiplication
equations.
Vocabulary
V
c
coefficient
Division Property of
Equality
Get ConnectED
GLE 0606.3.4
Use expressions, equations
and formulas to solve
problems. SPI 0606.3.3
Write equations that
correspond to given
situations or represent a
given mathematical
relationship. Also addresses
GLE 0606.3.2, SPI 0606.3.5.
B
C
D
Solve and Write
Multiplication Equations
CELL PHONES Matthew is
downloading ringtones on his cell
down
phone. The cost to download each
ringtone is $2. When Matthew is
finished, he has spent a total of $10.
1. Let x represent the number of
ringtones. What does the
expression 2x represent?
2. Explain how the equation 2x = 10
represents the situation.
3. Suppose Matthew spent $12. How
would the equation change?
The equation 2x = 10 is a multiplication equation. In 2x, 2 is the
coefficient of x because it is the number by which x is multiplied.
Multiplication and division are inverse operations. So, to solve a
multiplication equation, use division.
Solve a Multiplication Equation
Solve 2x = 10. Check your solution.
2x = 10
Write the equation.
10
2x
_
=_
Divide each side by 2.
2
2
x=5
Check
2x = 10
2(5) 10
10 = 10
Write the original equation.
Replace x with 5.
This sentence is true.
Solve each equation. Use models if necessary.
a. 3x = 15
b. 8 = 4x
c. 2x = 14
Lesson 2B Multiplication and Division Equations
335
When you solve an equation by dividing both sides of the
equation by the same number, you are using the Division
Property of Equality.
Division Property of Equality
Words
If you divide each side of an equation by the same nonzero
number, the two sides remain equal.
Examples
Numbers
Algebra
18 _
_
= 18
18 = 18
3x = 12
3 =3
x =4
6
3x _
_
= 12
3
6
3
SHOPPING Vicente and some friends shared the cost of a
package of blank CDs. The package cost $23.28 and each
person contributed $5.82. How many people shared the cost
of the CDs?
Words
Amount
each
contributed
Variables
times
number
of
people
Bar Diagram
The number of
sections is
unknown, but
each section
represents $5.82.
....
$5.82
Equation
5.82
·
5.82x = 23.28
Write the equation.
5.82x
23.28
_
=_
Divide each side by 5.82.
5.82
x=4
Check
cost of
CDs.
Let x represent the number of people that contributed
money.
$23.28
5.82
equals
x
=
23.28
Simplify.
5.82 × 4 = 23.28 There were 4 people who split the cost of the CDs.
Real-World Link
R
On their trip to the
O
South Pole, both
Pen Hadow and
Simon Murray started
out pulling a sled with
400 pounds of food
and gear.
336
Equations
d. EXPLORATION In 2004, Pen Hadow and Simon Murray walked
680 miles to the South Pole. The trip took 58 days. Suppose
they traveled the same distance each day. Write and solve a
multiplication equation to find about how many miles they
traveled each day.
Example 1
Solve each equation. Check your solution.
1. 2a = 6
Example 2
2. 20 = 4c
3. 16 = 8b
4. MEASUREMENT The length of an object in feet is equal to 3 times its
length in yards. The length of a waterslide is 48 feet. Write and solve a
multiplication equation to find the length of the waterslide in yards.
5. MUSIC The total time to burn a CD is 18 minutes. Last weekend,
Demitri spent 90 minutes burning CDs. Write and solve a
multiplication equation to find the number of CDs Demitri burned
last weekend. Explain how you can check your solution.
= Step-by-Step Solutions begin on page R1.
Extra Practice begins on page EP2.
Example 1
Example 2
Solve each equation. Check your solution.
6. 5d = 30
7. 4c = 16
8. 5t = 25
9. 5a = 15
10. 3f = 12
11 4g = 24
12. 21 = 3g
13. 6x = 12
14. 36 = 6e
15. JEWELRY A jewelry store is selling a set of 4 pairs of gemstone earrings
for $58, including tax. Neva and three of her friends bought the set so
each could have one pair of earrings. Write and solve a multiplication
equation to find how much each person should pay.
16. TRAVEL The Raimonde family drove 1,764 miles across the United
States on their vacation. If it took a total of 28 hours, what was their
average speed in miles per hour?
B Solve each equation. Check your solution.
17. 1.5x = 3
18. 2.5y = 5
19. 8.1 = 0.9a
3
20. 39 = 1_b
10
1
1
21. _
e=_
2
4
3
2
22. _
g=_
23 FOOTBALL Use the table that shows
football data.
a. George Blanda played in the NFL
for 26 years. Write and solve
an equation to find how many
points he averaged each year.
b. Norm Johnson played in the NFL
for 16 years. Write and solve
an equation to find how many
points he averaged each year.
5
5
Top NFL Kickers
Player
Career
Points
Gary Anderson
2,434
Morten Andersen
2,437
George Blanda
2,002
John Carney
1,749
Norm Johnson
1,736
24. HEARTBEATS An average person’s heart beats about 103,680 times a
day. Write and solve an equation to find about how many times the
average person’s heart beats in one minute.
Lesson 2B Multiplication and Division Equations
337
C 25. Which One Doesn’t Belong? Identify the equation that does not belong
with the other three. Explain your reasoning.
5x = 20
4b = 7
8w = 32
12y = 48
1
26. CHALLENGE Explain how you know that the equations _
= 2x and
4
_1 ÷ x = 2 have the same solution. Then, find the solution.
4
27. E WRITE MATH Write a problem about a real-world situation that
can be represented by the equation 4x = 24.
TTest Practice
28. The Walkers traveled 182 miles in
1
3_
hours. The equation 3.5m = 182
2
can be used to find their mean rate of
travel. What is the value of m?
A. 60
C. 50
B. 52
D. 48
29. If Mr. Solomon bikes at a constant
speed of 12 miles per hour, which
method can be used to find the
number of hours it will take him
to bike 54 miles?
F. Add 12 to 54.
G. Subtract 12 from 54.
H. Multiply 54 by 12.
I. Divide 54 by 12.
Solve each equation.
31. p + 3 = 8
30.
GRIDDED RESPONSE Marguerite’s
bottle of iced tea has this label.
Nutrition
Facts
Servings per container about 2
Calories 80
Total Fat 0g
Sodium 50mg
Total Carbohydrate 64g
Sugars 64g
The equation 2c = 64, where
c represents the amount of sugar
in each serving, can be used to
find the amount of sugar in one
serving. How many grams of
sugar are in each serving?
(Lessons 1D and 1F)
32. 7 + q = 15
33. b - 5 = 2
34. g - 6 = 7
35. CELL PHONES Enrico used 120 minutes of his monthly cell phone minutes
the first week. The second week he used 95 and the third week he used
212. If he had 73 minutes left to use, how many minutes did he have for
the month? Use the work backward strategy. (Lesson 1B)
36. SOCCER Arnold kicked the soccer ball 72 feet. There are 3 feet in 1 yard.
Use the equation 3x = 72 to find x, the length of the kick in yards.
(Lesson 1A)
338
Equations
Multi-Part
Lesson
2
PART
Multiplication and Division Equations
A
B
C
D
Division Equations
Main Idea
Solve division
equations using
models.
Get ConnectED
GLE 0606.3.2
Interpret and represent
algebraic relationships with
variables in expressions,
simple equations and
inequalities. GLE 0606.3.4
Use expressions, equations
and formulas to solve
problems. Also addresses
GLE 0606.1.3, SPI 0606.3.5.
CONCERTS Four friends decided to split the cost of season concert
tickets equally. Each person paid $35. Find the total cost of the
season concert tickets.
What do you need to find? the total cost of the season
concert tickets
Draw a bar diagram.
total cost, c
amount each
person pays
amount each
person pays
amount each
person pays
amount each
person pays
$35
Write an equation.
_c
35
=
4
total cost ÷ 4 friends
cost per person
Work backward.
Since c ÷ 4 = 35, 35 × 4 = c. So, c = 140.
The cost of the season tickets was $140.
and Apply
1. GASOLINE Mrs. Perez, Mrs. Hancock, and Mrs. Escalante went
to a conference. They shared the cost of gasoline. Each teacher
paid $38.50. Draw a bar diagram to find the total cost of gasoline.
1
2. SHOPPING Antonio bought a shirt for _
off. He paid $21.75
2
for the shirt. Draw a bar diagram to find the original cost
of the shirt.
the Results
3. MAKE A CONJECTURE Write a rule that could be used to solve
x
10 = _
.
2
Lesson 2C Multiplication and Division Equations
339
Multi-Part
Lesson
2
PART
Multiplication and Division Equations
A
Main Idea
Solve and write
division equations.
Get ConnectED
GLE 0606.3.4
Use expressions, equations
and formulas to solve
problems. SPI 0606.3.3
Write equations that
correspond to given
situations or represent a
given mathematical
relationship. Also addresses
GLE 0606.3.2, SPI 0606.3.5.
B
D
C
Solve and Write
Division Equations
ALLOWANCES Leslie spends
$5 a month on snacks at
school, which is one fourth
of her monthly allowance.
How much is Leslie’s
monthly allowance?
1. Draw a bar diagram to
represent the problem.
2. What is Leslie’s monthly
allowance?
a
The equation _
= 5, where a represents her monthly allowance,
4
means her monthly allowance divided by 4 equals $5. Since
multiplication and division are inverse operations, use
multiplication to solve division equations.
Solve Division Equations
_
Solve a = 7.
3
Method 1 Use models.
Method 2 Use symbols.
_a = 7
Model the equation.
3
_a (3) = 7(3)
a
3
a = 21
7
Check
Solve the equation. Work
backward.
a
Since _
= 7, 7 × 3 = a.
_a = 7
3
21
_=7
3
7=7
3
Write the equation.
Multiply each side by 3.
Simplify.
Write the original
equation.
Replace a with 21.
This is a true
sentence. The solution is 21.
Solve each equation. Check your solution.
a.
340
Equations
_x = 9
8
b.
_y = 8
4
c.
m
_
=9
5
b
d. 30 = _
2
WEIGHT The weight of an object on the Moon is one sixth that
of its weight on Earth. If an object weighs 35 pounds on the
Moon, write and solve a division equation to find its weight
on Earth.
Words
Real-World Link
R
Because of Jupiter’s
size and gravity,
objects weigh more
on Jupiter than on
Earth. A 210-pound
object would weigh
over 490 pounds on
Jupiter.
Variables
Weight
g of
weight
g on
object on divided by
6
equals
Moon.
Earth
Let w represent the weight of the object on Earth.
Bar Diagram
w
35 lb
w
_
Equation
w
_
= 35
6
w
_
(6) = 35(6)
6
w = 210
=
6
35
Write the equation.
Multiply each side by 6.
6 × 35 = 210
The object weighs 210 pounds on Earth.
1
e. FRUIT Nathan picked a total of 60 apples in _
hour. Write and
3
solve a division equation to find how many apples Nathan
could pick in 1 hour.
Example 1
Solve each equation. Check your solution.
m
1. _
= 10
6
p
4. 4 = _
3
Example 2
k
2. _
= 11
5
5
p
5=_
4
3.
_s = 20
2
6. 16 = _t
2
7. FOOD Mr. Ortega is buying a ham. He wants to divide it into 6-ounce
servings for 12 people. Write and solve a division equation to find
what size ham Mr. Ortega should buy.
8. STICKERS Kerry and Tya are sharing a pack of stickers. Each girl gets
11 stickers. Write and solve a division equation to find how many
total stickers there are.
Lesson 2D Multiplication and Division Equations
341
= Step-by-Step Solutions begin on page R1.
Extra Practice begins on page EP2.
Example 1
Solve each equation. Check your solution.
10. 12 = _
r
11. 18 = _
h
12. _
= 13
13
r
15. 2 = _
j
13. _ = 11
12
w
16. 17 = _
z
14. _ = 8
7
w
17. 7 = _
v
18. _ = 14
13
a
19. _ = 5
20
x
20. _ = 11
33
8
18
Example 2
q
7
r
9. 4 = _
6
2
19
Write and solve a division equation to solve each problem.
21 BAKING Caroline baked 3 dozen oatmeal raisin cookies for the bake
sale at school. This is one fourth the number of dozens of cookies she
baked in all. How many dozens of cookies did she bake in all?
22. BIRDS One third of a bird’s eggs hatched. If 2 eggs hatched, how many
eggs did the bird lay?
23. CRAFTS Blake is cutting a piece of rope into fourths. If each piece is
16 inches long, what is the length of the entire rope?
1
24. TOYS A model plane is _
the size of the actual plane. If the model
48
plane is 28 inches long, how long is the actual plane?
B Solve each equation. Check your solution.
g
g
26. 0.5 = _
25 4.7 = _
3.2
2
d
c
_
_
28. = 80.4
29.
=7
3
0.2
27. 12.6 = _
b
0.8
30.
w
_
= 6.7
8.3
31. GRAPHIC NOVEL Refer to the graphic novel frame below for Exercises a–b.
I need to figure out
how many books I
need to read to
ne
earn the pizza
ea
party.
pa
a. If Mei has earned 30 points, write and solve a multiplication
equation to find how many books she needs to read.
b. Suppose Mei has read 7 books. Write and solve a division equation
to find the number of points she has earned.
342
Equations
C 32. OPEN ENDED Write a division equation that has a solution of 42.
1
33. REASONING True or false: _ is equivalent to _
x. Explain your reasoning.
x
3
3
34. E WRITE MATH When solving an equation, why is it necessary to
perform the same operation on both sides of an equal sign?
TTest Practice
35. Which value of x makes this equation
true?
_x = 14
7
36.
A. 98
C. 2
B. 21
D. 0.5
EXTENDED RESPONSE Shana ran
6 miles in one week. This was one
third of what she ran in the month.
37. Devon is saving his
allowance to purchase
the telescope shown.
If he saves $7 a week
for 14 weeks, which of
the following equations
can be used to find the
total cost of the telescope?
F. 7 + x = 14
G. x - 7 = 14
Part A Write a division equation that
can be used to find how far she ran in
the month.
x
H. _
=7
14
I. 7x = 14
Part B How far did Shana run in the
month?
Solve each equation. Check your solution.
38. 6x = 72
(Lesson 2B)
39. 3m = 30
40. 200 = 4k
41. AGE Xavier’s age is 3 less than Paula’s age. Xavier is 11 years old. Write
and solve a subtraction equation to find Paula’s age. (Lesson 1F)
Solve each equation. Check your solution.
42. y + 5 = 12
(Lesson 1D)
43. 4 + x = 25
44. b + 11 = 16
45. MONEY Joslyn spent $12.75 on a new shirt. Then she spent $4.25 on hair
accessories and 2 times that amount on earrings. She has $3.50 left. How
much money did she have initially? (Lesson 1B)
46. SHOPPING The Card Shop sells thank you cards in
the packages shown. If Traci bought 20 thank
you cards, how many packages of each did
she buy? (Lesson 1A)
Card Packages
u
k yo
than
Size
i
Amount
small
large
7
3
Lesson 2D Multiplication and Division Equations
343
Multi-Part
Lesson
3
PART
Two-Step Equations
A
B
Two-Step Equations
Main Idea
Solve two-step
equations using
models.
Get ConnectED
GLE 0606.3.1
Write and solve two-step
equations and inequalities.
Also addresses SPI 0606.1.5.
MONEY Dario wants to purchase
colore pencils and a pencil case at the
colored
school store. Each pack costs $2 and
pencil cases cost $5. He has $9 to spend.
Use a model to find how many packs of
colored pencils Dario can buy. Use the
equation 2x + 5 = 9.
What do you need to find? how many packs of colored pencils
Dario can buy
Model the equation.
Y
Y
1
1
1
1
=
1
Y
Y
1
1
1
=
1
Divide the
remaining tiles
into two
equal groups.
Y
Y
1
1
1
1
1
1
9
=
1
2x + 5 - 5
1
1
2x + 5
Remove five
1-tiles from each
side of the mat.
1
1
1
1
1
1
1
1
1
1
9-5
=
=
1
1
1
1
There are two 1-tiles in each group, so x = 2. Dario can buy
2 packs of colored pencils.
the Results
1. If Dario has $7, how many packs of colored pencils can he buy?
2. E WRITE MATH Describe how you used inverse operations
when doing Activity 1.
344
Equations
PHONES A popular cell phone costs $140. It costs $20 more than twice the
cost of a less popular cell phone. The cost of the less popular cell phone p
can be represented by 2p + 20 = 140. What is the cost of the less popular
cell phone?
What do you need to find? the cost of the less popular cell phone
Draw a bar diagram.
cost of more popular phone, $140
less popular phone less popular phone $20
p
Subtract $20 from the cost of the more popular cell phone.
$120
less popular phone less popular phone
p
Divide $120 by 2.
$120 ÷ 2 or $60
less popular phone
p
The less popular cell phone costs $60.
the Results
3. If the popular cell phone costs $180, what would be the cost of the less
popular cell phone?
4. How are the order of operations used in Activity 2?
5. E WRITE MATH Explain how you use the work backward strategy to solve
two-step equations.
and Apply
Solve each equation using models.
6. 3x + 1 = 7
7. 2x + 4 = 6
8. 2x + 3 = 7
Lesson 3A Two-Step Equations
345
Multi-Part
Lesson
3
Two-Step Equations
PART
A
Main Idea
Solve and write
two-step equations.
Vocabulary
V
ttwo-step equation
Get ConnectED
GLE 0606.3.1
Write and solve two-step
equations and inequalities.
GLE 0606.3.2 Interpret and
represent algebraic
relationships with variables
in expressions, simple
equations and inequalities.
Also addresses SPI 0606.1.4,
GLE 0606.1.5.
B
Solve and Write Two-Step
Equations
w!
Ne Bak!ed
PotNaetwoBCakh!eipds
PotNaetwoBCakh!eipds
PotNaetwoBCakheipds
SNACKS Lara bought 4 snacks and
3 juice drinks to share with her
friends. The total cost was $8.50.
The juice drinks cost $1.50 each.
Potato Chips
1. Explain how you could use the
work backward strategy to find
the cost of each snack.
2. Find the cost of each snack.
A two-step equation is an equation with two different operations.
To solve a two-step equation, undo the operations in reverse of the
order of operations.
Solve a Two-Step Equation
Solve 2x + 3 = 11. Check your solution.
You can use algebra tiles to solve two-step equations.
Model the equation.
1
Y
Y
1
=
1
2x + 3
Subtract three 1-tiles
from each side.
1
Y
Y
1
2x + 3 - 3
Divide the remaining
1-tiles into two equal
groups.
The solution is 4.
Check
346
Equations
2(4) + 3 = 11 Y
2x
2
1
1
1
1
1
1
1
1
1
1
11
=
1
=
1
1
1
1
1
1
1
1
1
1
1
11 - 3
=
=
Y
1
=
1
1
1
1
1
1
1
1
8
2
a. 3x + 6 = 18
b. 3n + 4 = 13
Solve a Two-Step Equation
Solve each equation. Check your solution.
3y - 8 = 7
Order of Operations
Recall that the order of
operations states addition
and subtraction are done
last. When solving twostep equations, they are
done first.
3y - 8 = 7
+8=+8
3y
= 15
Write the equation.
Undo the subtraction first by adding 8 to each side.
Simplify.
3y _
_
= 15
Divide each side by 3.
3
3
y=5
Simplify.
The solution is 5.
3y - 8 = 7
Check
3(5) - 8 7
15 - 8 7
7=7 Write the original equation.
Replace y with 5.
Simplify.
The sentence is true.
9 + 6b = 51
9 + 6b = 51
-9
= -9
6b = 42
6b _
_
= 42
6
6
b=7
Write the equation.
Subtract 9 from each side.
Simplify.
Divide each side by 6.
Simplify.
The solution is 7.
Check
Since 9 + 6(7) = 51, the solution is correct. c. 5x - 4 = 16
d. 8 + 7m = 50
Solve Two-Step Equations
To solve a two-step equation:
Step 1
Undo the addition or subtraction first.
Step 2
Undo the multiplication or division.
Lesson 3B Two-Step Equations
347
ENTERTAINMENT At the local fun center, a hamburger meal costs
$4.50 and each ride costs $2.50. If you have $22.00, how many
rides can you go on if you buy a meal?
Cost of
hamburger plus
meal
Words
Real-World Link
R
Bumper boats may be
found at family fun
centers. Some
bumper boats can
operate in only
14 inches of water.
number
cost of
equals $22.00.
times
of rides
1 ride
Let r represent the
e number of rides.
Variable
22
Bar Diagram
2.50
4.50
Equation
...
+
2.50
4.50
=
2.50 r
22
4.50 + 2.50r = 22.00 Write the equation.
- 4.50
= - 4.50 Subtract 4.50 from each side.
2.50r = 17.50 Simplify.
2.50r _
_
= 17.50
2.50
2.50
r=7
Check
Divide each side by 2.50.
17.50 ÷ 2.50 = 7
4.50 + 2.50r = 22.00
4.50 + 2.50(7) 22.00
4.50 + 17.50 22.00
Write the original equation
Replace r with 7.
Simplify.
22.00 = 22.00 The sentence is true.
You can ride 7 rides.
e. JEWELRY Ava bought a necklace for $8.50 and some earrings
that cost $3.75 for each pair. The total cost was $19.75. Write
and solve an equation to find how many pairs of earrings
Ava bought.
Examples 1–3
Example 4
348
Equations
Solve each equation. Check your solution.
1. 4x + 7 = 23
2. 5b + 2 = 27
3
9k - 4 = 32
4. 8q - 6 = 50
5. 3 + 5j = 38
6. 2 + 7c = 16
7. JOBS Mrs. Stevens earns $18 an hour at her job. She had $171 after
paying $9.00 for subway fare. Write and solve an equation to find
how many hours Mrs. Stevens worked.
= Step-by-Step Solutions begin on page R1.
Extra Practice begins on page EP2.
Examples 1–3
Example 4
Solve each equation. Check your solution.
8. 5x - 4 = 21
9. 9b + 4 = 76
10. 6m - 3 = 57
11. 4r - 2 = 26
12. 6 + 3p = 39
13. 10 + 7x = 45
14. PIZZA Sadie wants to buy pizza for her friends. Each slice costs $4.
She also wants to order breadsticks for $9. She has $49 to spend.
Write and solve an equation to find how many slices of pizza she
can buy.
15. SKATEBOARDS Pedro is saving money to buy a new skateboard that
costs $149. He has saved $89 so far. He plans on saving $15 each week.
Write and solve an equation to find in how many weeks Pedro will
have enough money to buy a new skateboard.
B
Solve each equation. Check your solution.
16. 4n - 1.6 = 2.4
17. 3x + 5.2 = 14.5
18. 2y - 6.3 = 5.7
19. 5t - 1.7 = 8.2
20. 1.8 + 7s = 15.8
21. 34.3 = 6h + 9.1
22. FINANCIAL LITERACY Nhung
Summer Jobs at the Pool
and Awan are each trying
Job
Hourly Pay ($)
to save $550 to play sports
Lifeguarding
10.00
in the fall. Nhung has saved
Ticket Office
8.00
$100, and takes a job in the
Snack Hut
7.50
Snack Hut at the pool.
Awan has not saved any
money and takes a job lifeguarding.
a. Who do you think will take longer to save enough money? Explain
your reasoning.
b. Write and solve an equation to find how long it will take Nhung to
earn the money.
c. Write and solve an equation to find how long it will take Awan to
earn the money.
d. Who will earn the money quicker? Compare to your answer for
part a.
23 BOWLING Each game of bowling costs $3.00 and shoe rental is $3.50.
You have $15.50 to spend. Write and solve an equation to find how
many games you can bowl.
24. NUTRITION Leah eats 2,000 Calories per day. She has already eaten
1,850 during breakfast, lunch, and dinner. She would like to have
some pretzel sticks that are 50 Calories each for a snack. Write and
solve an equation to find how many pretzel sticks she can have.
Lesson 3B Two-Step Equations
349
25 WEATHER The temperature is 62.8°F. It is expected to rise at a rate of 2.5°
each hour for the next several hours. Write and solve an equation to
find the number of hours until the temperature is 75.3°F. Solve the
equation.
Solve each equation. Check your solution.
x
26. _
+ 2 = 10
2
29.
Real-World Link
R
TThe fastest
recorded
temperature
change occurred
in the Black Hills
in South Dakota.
On January 22,
1943, the
temperature rose
from 4°F below
zero to 45°F
above zero in
two minutes.
x
27. _
- 5 = 10
3
x
28. 4 + _
=8
7
MULTIPLE REPRESENTATIONS Consider the equation 2x + 7 = 15.
a. WORDS Write a real-world problem that could be represented by
the equation.
b. BAR DIAGRAM Draw a bar diagram to represent the equation.
c. ALGEBRA Solve the equation.
d. WORDS Explain why there is only one solution to the equation.
30. MATH IN THE MEDIA Evaluate the use of equations in a media report.
Explain how an equation could be used to solve a problem. Then
write a real-world problem in which you would use an equation to
solve.
C 31. OPEN ENDED If 14 less than 5 times a number is 26, the number is 8.
Write a different sentence where the unknown number is also 8.
32. FIND THE ERROR Noah is solving 5 + 5x = 23.
Find his mistake and correct it.
5 + 5x = 23
+5
=+5
5x = 28
x = 5.6
33. REASONING Tell which operation to undo first in the equation
19 = 4 + 5x. Explain.
34. CHALLENGE Use what you know about the Distributive Property and
solving two-step equations to solve the equation 2(x + 9) = 28.
35. E WRITE MATH A cleaning service charges a flat rate of $35 and $12
for each hour. The Henderson family paid the cleaning service $95.
This can be represented by the equation 12x + 35 = 95. Explain how
each number in the equation relates to the problem.
350
Equations
TTest Practice
36.
GRIDDED RESPONSE The ad below
shows the cost of a popular kind of
computer printer.
38. Dena is ordering DVDs online. She has
saved $61.
Item
Cost ($)
DVD
’s
Mike uteiumr
r
CompEmpo
shipping
$145.00
The equation 14x + 5 = 61 can be used
to find the number of DVDs she can
order with her savings. How many
DVDs can she order?
F. 7
G. 6
37. Which value of x makes this equation
true?
H. 5
I. 4
x - 15 ÷ 3 = 6
A. 1
39.
B. 5
C. 11
SHORT RESPONSE Explain the steps
necessary to solve the equation
below.
3x - 8 = 10
D. 33
Solve each equation. Check your solution.
6
43.
_x = 9
5
5
2010
Printer MP
The cost of a less popular printer x can
be represented by 2x + 25 = 145. What
is the cost, in dollars, of the less
popular printer?
x
40. _
=8
14
41.
(Lesson 2D)
p
_
=7
k
42. 4 = _
3
11
m
44. _
= 12
45.
8
_b = 8
7
46. SPEED If a horse could maintain its average speed for 4 hours, it could
travel 120 miles. Write and solve a multiplication equation to find its
average speed. (Lesson 2B)
47. HAIR Helen had 3 inches of hair cut off when she went to the hair salon.
The length of her hair after she had it cut was 14 inches. Write and solve
a subtraction equation to find the length of Helen’s hair before she had it
cut. (Lesson 1F)
Solve each equation. Check your solution.
(Lesson 1D)
48. b + 7 = 18
49. 11 + p = 29
50. y + 9 = 25
51. 15 + t = 46
52. 18 + x = 32
53. w + 24 = 39
Lesson 3B Two-Step Equations
351
in Music
the
Amping Band
Do you enjoy using electronics to make music sound better? If so, you might
want to explore a career in sound engineering. Sound engineers, or audio
technicians, prepare the sound equipment for recording sessions and live
concert performances. They are responsible for operating consoles and other
equipment to control, replay, and mix sound from various sources. Sound
engineers adjust the microphones, amplifiers, and levels of various instrument
and voice tones so that everything sounds great together.
Are you interested in a career as a sound engineer?
Take some of the following courses in high school.
• Algebra
• Electronic Technology
• Music and Computers
Get ConnectED
352 Equations
• Physics
• Sound Engineering
TPVOETPVSDF
d
NJDSPQIPOFT
GLE 0606.1.7
Use the information in the diagram and
the table to solve each problem.
6 ft
1. In the diagram, the distance between the
microphones is 6 feet. This is 3 times the distance d
from each microphone to the sound source. Write an
equation that represents this situation.
2. Solve the equation that you wrote in Exercise 1.
Explain the solution.
3. The distance from the microphone to the acoustic
guitar sound hole is about 11 inches less than what
it should be. Write an equation that models this
situation.
4. Solve the equation that you wrote in Exercise 3.
Explain the solution.
5. The microphone is about 9 times farther from the
electric guitar amplifier than it should be to produce
a natural, well-balanced sound. Write and solve an
equation to find how far from the amplifier the
microphone should be placed.
Microphone Mistakes
Sound Source
Acoustic guitar
Electric guitar
amplifier
Location of Microphone
3 inches from sound hole
Resulting Sound
very bassy
36 inches from amp
thin, reduced bass
Problem Solving in Music
353
Chapter Study
Guide and Review
Be sure the following
Key Concepts are noted
in your Foldable.
Key Vocabulary
K
coefficient
equals sign
equation
inverse operation
solution
Addition and
Subtraction
Equations
Multiplication
and Division
Equations
solve
two-step equation
Two-Step
Equations
Key Concepts
Equations
(Lesson 1)
• An equation is a mathematical sentence showing
two expressions are equal.
x - 8 = 13
Addition and Subtraction Equations
(Lesson 1)
• Subtraction Property of Equality
If you subtract the same number from each side
of an equation, the two sides remain equal.
8 + 9 - 9 = 17 - 9
• Addition Property of Equality
If you add the same number to each side of an
equation, the two sides remain equal.
15 - 10 + 10 = 5 + 10
Multiplication and Division Equations
(Lesson 2)
• Multiplication and division are inverse operations.
Two-Step Equations
(Lesson 3)
• A two-step equation has two different operations.
• To solve a two-step equation, undo the operations
in reverse of the order of operations.
354
Equations
Vocabulary Check
State whether each sentence is true or false.
If false, replace the underlined word or
number to make a true sentence.
1. The numerical factor of a term that
contains a variable is called a(n) equals
sign.
2. A(n) equation is a sentence that contains
an equals sign.
3. Inverse operations undo each other.
4. To solve the equation 4x + 8 = 16, the
first step is to add 8 to both sides.
5. Division and subtraction are inverse
operations.
6. Addition and multiplication are inverse
operations.
7. In the equation 6x - 8 = 4, the
coefficient is 6.
8. The solution to the equation x + 8 = 11
is 19.
Multi-Part Lesson Review
Lesson 1
Addition and Subtraction Equations
Equations
(Lesson 1A)
Solve each equation mentally.
EXAMPLE 1
Solve x + 9 = 13 mentally.
10. 20 + y = 25
x + 9 = 13
11. 40 = 15 + m
12. n + 10 = 16
4 + 9 = 13 You know 4 + 9 is 13.
13. 27 = x - 3
14. 25 = h + 7
9. p + 2 = 9
15. AGE The equation 18 + p = 34
represents the sum of Reese’s and
Ana’s ages, where p represents Reese’s
age. How old is Reese?
16. INCOME The equation 7h = 63 can be
used to find how many hours h a
person needs to work to earn $63 at
$7 per hour. How many hours does
a person need to work to earn $63?
Think
What number plus 9 is 13?
x=4
The solution is 4.
Check
4 + 9 = 13 EXAMPLE 2
Georgiana has three more
problems left on her math homework.
There are 35 problems in all. Solve the
equation x + 3 = 35 to find how many
problems she has already completed.
x + 3 = 35
Think What plus 3 is 35?
32 + 3 = 35 32 + 3 = 35
x = 32
The solution is 32.
Check
PSI: Work Backward (Lesson 1B)
Solve. Use the work backward strategy.
17. TIME After school, Kyle watched TV for
half an hour, played basketball for 45
minutes, and studied for an hour. If it is
now 7:00 p.m., when did Kyle come
home from school?
18. HEIGHT Fernando is 2 inches taller than
Taye. Taye is 1.5 inches shorter than
Deirdre and 1 inch taller than Pabla.
Hao, who is 5 feet 10 inches tall, is
2.5 inches taller than Fernando. How
tall is each student?
32 + 3 = 35 EXAMPLE 3
A number is divided by 2.
Then 5 is added to the quotient. After
subtracting 7, the result is 28. What is
the number?
Start with the final value and perform the
opposite operation with each resulting
value until you arrive at the starting
value.
28 + 7 = 35 Add 7 to 28.
35 - 5 = 30 Subtract 5 from 35.
30 · 2 = 60
Multiply by 2.
The number is 60.
Check
60 ÷ 2 + 5 - 7 = 28 Chapter Study Guide and Review
355
Chapter Study Guide and Review
Lesson 1
Addition and Subtraction Equations
Solve and Write Addition Equations
(Lesson 1D)
Solve each equation. Check your
solution.
19. c + 8 = 11
20. 54 = m + 9
21. 23 = h + 11
22. w + 18 = 24
23. HEIGHT When Sean stands on a box, he
is 10 feet tall. If the box is 4 feet tall,
write and solve an addition equation to
find Sean’s height.
Solve and Write Subtraction Equations
24. z - 7 = 11
25. 14 = m - 5
26. d - 41 = 16
27. 72 = g - 20
EXAMPLE 4
Solve x + 8 = 10.
x + 8 = 10
-8 =-8
x
= 2
Subtract 8 from each side.
Simplify.
Check
x + 8 = 10
Write the equation.
2 + 8 = 10
Replace x with 2.
10 = 10 28. MONEY Felise has $39. She has $8 less
than her brother. Write and solve a
subtraction equation to find how much
money her brother has.
EXAMPLE 5
Solve a - 5 = 3.
a-5 = 3
+5 =+5
a
= 8
Add 5 to each side.
Simplify.
EXAMPLE 6
Solve 4 = m - 8.
4= m-8
+ 8 = + 8 Add 8 to each side.
12 = m
Simplify.
Multiplication and Division Equations
Solve and Write Multiplication Equations
(Lesson 2B)
Solve each equation. Check your
solution.
EXAMPLE 7
29. 4b = 32
30. 3m = 21
31. 60 = 5y
32. 28 = 2d
6y
24
_
=_
33. 18 = 6c
34. 6p = 24
35. CDS A store is selling blank CDs in
packages of 25 for $5. Write and solve a
multiplication equation to find the cost
of one blank CD.
356
This is a true sentence.
(Lesson 1F)
Solve each equation. Check your
solution.
Lesson 2
(continued)
Equations
Solve 6y = 24.
6y = 24 Write the equation.
6
6
y= 4
Check
Divide each side by 6.
Simplify.
6 × 4 = 24 Lesson 2
Multiplication and Division Equations
Solve and Write Division Equations
(Lesson 2D)
_r = 9
5
w
_
38.
=7
10
37.
_
Solve n = 5.
EXAMPLE 8
n
_
=5
6
n
_
(6) = 5(6)
6
Solve each equation. Check your
solution.
36.
(continued)
m
_
=6
8
c
39. _ = 5
12
6
Write the equation.
Multiply each side by 6.
n = 30
40. COOKING Luigi is baking chicken and
the preparation time is 10 minutes,
which is one fourth of the baking time.
Write and solve a division equation to
find the baking time.
41. SPEED LIMITS The speed limit in front
of Meadowbrook Middle School is
15 miles per hour, which is one third
the speed limit of a major street two
blocks away. Write and solve a division
equation to find the speed limit of the
major street.
Multiply.
EXAMPLE 9
The table shows the
number of days of freezing temperatures.
January had one half the number of days
as February. Write and solve a division
equation that could be used to find the
number of days in February that had
freezing temperatures.
Days of Freezing
Temperatures
January
February
12
40°
90°
20°
20
70°
10°
0°
-10°
-20°
C°
_d = 12
110°
30°
50°
30°
10°
-10°
F°
Write the equation.
2
_d (2) = 12(2)
2
d = 24
Multiply each side by 2.
Multiply.
There were 24 days in February that had
freezing temperatures.
Lesson 3
Two-Step Equations
Solve and Write Two-Step Equations
(Lesson 3B)
Solve each equation. Check your
solution.
42. 5y - 14 = 6
43. 7x - 8 = 13
44. 4n + 6 = 30
45. 9m + 11 = 29
46. 6c - 6 = 36
47. 8r + 17 = 57
48. ALGEBRA Twelve less than seven times
a number is 23. Find the number.
EXAMPLE 10
Solve 6p - 8 = 22.
6p - 8 = 22
+ 8 = + 8 Add 8 to each side.
6p = 30
6p
30
_
=_
6
6
Divide each side by 6.
p=5
Check
6(5) - 8 = 22 Chapter Study Guide and Review
357
Practice
Chapter Test
Solve each equation mentally.
1. d + 9 = 14
2. n + 6 = 24
3. 9c = 99
4. y ÷ 4 = 12
5. SCIENCE For a class project, Reynaldo
and Karen are raising tadpoles. They
have 12 tadpoles and some frogs. They
have a total of 20 tadpoles and frogs.
Write and solve an equation to find how
many frogs they have.
6. MULTIPLE CHOICE Reshma drove 385
miles to visit her grandmother. If she
drove at a speed of 55 miles per hour,
which equation represents the time that
it took Reshma to drive 385 miles?
B. t - 55 = 385
C. 385 = 55t
D. 385t = 55
7. CARPENTRY A board that measures
19 feet in length is cut into two pieces.
The shorter of the two pieces measures
7 feet. Write and solve an equation to
find the length of the longer piece.
19 ft
7 ft
8. WEATHER A cold front caused the
temperature to drop 21°F. The
temperature after the drop was 36°F.
Write and solve an equation to find the
temperature before the temperature
dropped.
Equations
Solve each equation.
10. 5b = 65
11. 56 = 7k
s
12. _ = 6
8
13.
_p = 4
6
14. CHICAGO SKYSCRAPERS The Willis Tower
is 1,453 feet tall. It is 323 feet taller than
the John Hancock building. Write and
solve an equation to find the height of
the John Hancock building.
15. FOOD A bag of grapes is being divided
into 24 smaller bags of 4 ounces each for
snacks during a field trip. Write and solve
a division equation to find the number of
ounces in the larger bag.
A. t + 55 = 385
358
9. SCHOOL The number of photos T.J. has
on his laptop is twice as many as the
amount Rolando has on his laptop.
T.J. has 20 photos on his laptop. Write
and solve a multiplication equation to
find how many photos Rolando has on
his laptop.
Solve each equation.
16. 3a + 7 = 34
17. 5d - 9 = 11
18. 7p - 12 = 30
19. 4m + 15 = 43
20.
EXTENDED RESPONSE The table shows
the prices at a carnival.
Carnival
Admission
$16
Ride ticket
$3
If you have $43, how many ride tickets
can you buy?
Part A Write an equation.
Part B Solve the equation to find how
many ride tickets you can buy.
Preparing for
Standardized Tests
Gridded Response: Decimals
When a gridded-response answer is a decimal, write the decimal point in an
answer box at the top of the grid. Then fill in the decimal point bubble.
_
Khamai went down a 61 1 -foot-long water slide in 3 seconds. She can
2
find n, the mean number of feet per second, by solving the equation
shown below. What is the value of n?
61.5 = 3n
61.5 = 3n
Write the equation.
61.5
3n
_
=_
3
Divide each side by 3.
3
20.5 = n
Khamai’s mean distance per second is 20.5 feet. Grid in 20.5.
Correct
NOT Correct
20 . 5
20 . 5
2 0.5
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
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9
9
9
9
or
Find the value of a that makes the equation
below true. Fill in your answer on an answer
grid like the ones above.
Print only one d
iggiitt
or symbol in ea
chh
answer box.
5 × a + 6 = 15
Preparing for Standardized Tests
359
Test Practice
Read each question. Then fill in the correct
answer on the answer sheet provided by
your teacher or on a sheet of paper.
1. The receipt shows the quantity and price
of some clothing. Which equation can be
used to find the price of each sweater?
4. Two thirds of a blueberry pie is left in
the refrigerator. If the leftover pie is cut
in 6 equal-size slices, which fraction
of the original pie is each slice?
Casual Fashions
Quantity
2
Item
Price
1
A. _
pairs of jeans $64.52
3
sweaters
Total
5.
B. 3x = 136.67
C. 64.52 + 3x = 136.67
D. 2x = 64.52
2. A sub shop is keeping track of the number
of pounds of turkey sold each day.
Number of Pounds
of Turkey Sold
6. At a pet store, there are 30 aquariums.
Of the 30, 20 have fresh water with
15 fish each. The other 10 aquariums
have salt water with 6 fish each. Which
equation could be used to find f, the
total number of fish at the pet store?
40.6
Tuesday
25.4
Wednesday
34.7
Thursday
45.2
G. f = 30(15 + 6)
Friday
65.3
H. f = 20(15 + 6)
Saturday
70.4
I. f = (20 × 15) + (10 × 6)
Sunday
50.8
F. about 1.5 times
H. about 3 times
G. about 2 times
I. about 4 times
GRIDDED RESPONSE Bob’s Boot Shop
sold 60% of its stock of winter boots
before the first snow of the year. What
fraction of the stock of winter boots has
not yet been sold?
360
GRIDDED RESPONSE Mikayla bought
a value pack of crackers for $6.72. The
value pack had 24 individually wrapped
cracker packages. Solve 24x = 6.72 to find
the cost per package of crackers.
Monday
The number of pounds of turkey sold
Saturday is equal to about how many
pounds of turkey sold on Wednesday?
3.
4
1
D. _
3
$136.67
A. 64.52 + x = 136.67
Day
1
C. _
9
_
B. 1
6
Equations
F. f = 30 + 20 + 15 + 10 + 6
7.
GRIDDED RESPONSE The ratio of
boys to girls at a rock concert was
about 7 to 6. If there were 1,092 boys
at the rock concert, how many girls
were there?
8. What value of x makes this equation
true?
x - 5 × 4 = 44
A. 11
C. 24
B. 20
D. 64
9.
SHORT RESPONSE During a vacation,
Alice went fly-fishing. She rented a rod
and a driftboat.
12.
SHORT RESPONSE A movie theater has
26 rows of seats with an equal number of
seats in each row. The theater can seat a
total of 390 people. Solve the equation
26x = 390 to find the number of seats x
that are in each row.
Rick’s Rental
fishing rod
driftboat
If she spent a total of $52, write and solve
a two-step equation to find how many
hours she rented the boat.
10. Which of the following diagrams
represents 80%?
F.
H.
EXTENDED RESPONSE Mr. Martin and
his son are planning a fishing trip to
Alaska. The rates of two companies are
shown.
13.
G.
$12
$8/hour
I.
Wild Fishing,
g, Inc.
Per Person Rates
Round-Trip Flight $200 / Daily Rate $25
11. The rates for renting a jet ski are shown.
Fisherman’s Service
Per Person Rates
Round-Trip Flight $150 / Daily Rate $30
East Lake Rentals
Weekday cost per hour
$95
Weekend cost per hour
$110
Part A Which company offers a
better deal for a three-day trip for
Mr. Martin and his son? What will
be the cost?
Which expression could be used to find
the cost of renting a jet ski for 3 hours on
Friday and 2 hours on Saturday?
Part B Which company would charge
$700 for 6 days for both of them?
A. 5(95 + 110)
B. 3(95) + 2(110)
Part C Write an expression to represent
the cost of the trip for Mr. Martin and his
son for each company.
C. 3(95) × 2(110)
D. (3 + 95) × (2 + 110)
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Test Practice
361