3
Equations
•
Select and use appropriate
operations to solve problems and
justify solutions.
•
Use graphs, tables, and algebraic
representations to make
predictions and solve problems.
Key Vocabulary
area (p. 163)
formula (p. 162)
like terms (p. 129)
sequence (p. 158)
simplest form (p. 130)
Real-World Link
Skyscrapers Rising 630 feet to the top, the Gateway
Arch in St. Louis, Missouri, is 130 feet higher than Mount
Rushmore in Black Hills, South Dakota, 75 feet higher
than the Washington Monument, and 25 higher than
the Seattle Space Needle.
Equations Make this Foldable to help you organize information about expressions and equations. Begin
with five sheets of 812’’ × 11’’ paper.
1 Stack 5 sheets of paper
3
4 inch apart.
2 Roll up the bottom
edges. All tabs should
be the same size.
3 Crease and staple
4 Label the tabs with
along the fold.
topics from the
chapter.
122 Chapter 3 Equations
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GET READY for Chapter 3
Diagnose Readiness You have two options for checking Prerequisite Skills.
Option 2
Take the Online Readiness Quiz at pre-alg.com.
Option 1
Take the Quick Check below. Refer to the Quick Review for help.
Find each product. (Lesson 2-4)
1. 2(-3)
2. -4(3)
3. -5(-2)
4. -4 · 6
5. -11 · -8
6. 9 · (-4)
7. STOCK MARKET The price of a stock
decreased $2.05 each day for five
consecutive days. What was the total
change in value of the stock over the
five-day period? (Lesson 2-4)
Write each subtraction expression as an
addition expression. (Lessons 2-3)
8. 5 - 7
9. 6 - 10
10. -13 - 9
11. 11 - 10
12. 15 - 6
13. -19 - 10
Example 1
Find 7(-2).
7(-2) = -14
The factors have different signs,
so the product is negative.
Example 2
Find -5 · -9.
-5 · -9 = 45
The factors have the same sign,
so the product is positive.
Example 3
Write 8 - 12 as an addition expression.
8 - 12 = 8 + (-12) To subtract 12, add -12.
= -4
Simplify.
14. MONEY Student Council spent $178 on
decorations and $110 on snacks for the
dance. Write an addition expression for
the amount remaining in the dance budget
if Student Council initially had $593.
(Lessons 2-3)
Find each sum. (Lesson 2-2)
15. 6 + (-9)
16. -8 + 4
17. 4 + (-4)
18. 7 + (-10)
19. -13 + (-8)
20. -11 + 12
Example 4
Find -5 + 7.
-5 + 7 = 2 Subtract | -5 | from | 7 |. The sum
is positive because | 7 | > | -5 |.
21. CAVERNS A tour group began 26 feet
underground. During their tour,
they descended 15 feet more and then
ascended 19 feet. Express their current
depth as an integer. (Lesson 2-2)
Chapter 3 Get Ready for Chapter 3
123
3-1
The Distributive Property
BrainPOP® at pre-alg.com
Main Ideas
• Use the Distributive
Property to write
equivalent numerical
expressions.
• Use the Distributive
Property to write
equivalent algebraic
expressions.
To find the area of a rectangle, multiply the length and width. You can
find the total area of the blue and yellow rectangles in two ways.
Method 1
Method 2
Put them together. Add the
lengths. Then multiply.
4
+
Separate them. Multiply to find
each area. Then add.
4
2
2
New Vocabulary
equivalent expressions
3
3
3(4 + 2) = 3 · 6 Add.
= 18
Multiply.
+
3
3 · 4 + 3 · 2 = 12 + 6
= 18
Multiply.
Add.
a. Draw a 2-by-5 and a 2-by-4 rectangle. Find the total area in
two ways.
b. Draw a 4-by-4 and a 4-by-1 rectangle. Find the total area in
two ways.
c. Draw any two rectangles that have the same width. Find the total
area in two ways.
d. What did you notice about the total area in each case?
Distributive Property The expressions 3(4 + 2) and 3 · 4 + 3 · 2 are
equivalent expressions because they have the same value, 18. This
example shows how the Distributive Property combines addition and
multiplication.
Vocabulary Link
Distribute
Everyday Use to deliver
to each member of a
group
Distributive
Math Use property that
allows you to multiply a
number by a sum
Distributive Property
Words
To multiply a number by a sum, multiply each number inside the
parentheses by the number outside the parentheses.
Symbols
a(b + c) = ab + ac
Examples 3(4 + 2) = 3 · 4 + 3 · 2
(b + c)a = ba + ca
(5 + 3)2 = 5 · 2 + 3 · 2
You can use the Distributive Property to evaluate numerical or algebraic
expressions.
124 Chapter 3 Equations
EXAMPLE
Use the Distributive Property
Use the Distributive Property to write each expression as an equivalent
expression. Then evaluate the expression.
a. 2(6 + 4)
b. (8 + 3)5
2(6 + 4) = 2 · 6 + 2 · 4
= 12 + 8 Multiply.
= 20
Add.
(8 + 3)5 = 8 · 5 + 3 · 5
= 40 + 15 Multiply.
= 55
Add.
1A. (6 + 3)4
1B. 4(2 + 9)
AMUSEMENT PARKS A one-day pass to an amusement park costs $40.
A round-trip bus ticket to the park costs $5.
a. Write two equivalent expressions to find the total cost of a one-day pass
and a bus ticket for 15 students.
Method 1 Find the cost for 1 person, then multiply by 15.
15($40 + $5) 15 times the cost for 1 person
Method 2 Find the cost of 15 passes and 15 tickets. Then add.
15($40) + 15($5) cost of 15 passes + cost of 15 tickets
b. Find the total cost.
Real-World Link
Attendance at U.S.
amusement parks
increased 22% in the
1990s. In 2004, about
328 million people
attended these parks.
15($40 + $5) = 15($40) + 15($5) Distributive Property
= $600 + $75
Multiply.
= $675
Add.
The total cost is $675. You can check your results by evaluating 15($45).
2. FOOD A spaghetti dinner costs $10 and a slice of pie costs $2. Write two
equivalent expressions to find the total cost of a spaghetti dinner and
a slice of pie for each member of a family of 4. Then find the total cost.
Source: International
Association of Amusement
Parks and Attractions
Personal Tutor at pre-alg.com
Algebraic Expressions You can also model the Distributive Property by using
algebra tiles and variables.
4HEMODELSHOWSX
4HEREAREGROUPSOFX
3EPARATETHETILESINTO
GROUPSOFXANDGROUPSOF
XÊÎ
Ó
X
X
£
£
X
£
£
£
£
Ó
X
X
Î
Ó
£
£
£
£
£
£
2(x + 3) = 2x + 2 · 3
= 2x + 6
The expressions 2(x + 3) and 2x + 6 are equivalent expressions because for
every value of x, these expressions have the same value.
Extra Examples at pre-alg.com
AP/Wide World Photos
Lesson 3-1 The Distributive Property
125
EXAMPLE
Simplify Algebraic Expressions
Use the Distributive Property to write each expression as an equivalent
algebraic expression.
a. 3(x + 1)
b. (y + 4)5
3(x + 1) = 3x + 3 · 1
(y + 4)5 = y · 5 + 4 · 5
= 3x + 3 Simplify.
3A. 2(a + 5)
EXAMPLE
= 5y + 20 Simplify.
3B. (b + 6)3
Simplify Expressions with Subtraction
Use the Distributive Property to write each expression as an equivalent
algebraic expression.
a. 2(x - 1)
2(x - 1) = 2[x + (-1)]
Look Back
To review subtraction
expressions, see
Lesson 2–3.
Rewrite x - 1 as x + (-1).
= 2x + 2(-1)
Distributive Property
= 2x + (-2)
Simplify.
= 2x - 2
Definition of subtraction
b. -3(n - 5)
-3(n - 5) = -3[n + (-5)]
Rewrite n - 5 as n + (-5).
= -3n + (-3)(-5) Distributive Property
= -3n + 15
4A. 4(d - 3)
Example 1
(p. 125)
(p. 125)
Examples 3, 4
(p. 126)
4B. -7(e - 4)
Use the Distributive Property to write each expression as an equivalent
expression. Then evaluate it.
1. 5(7 + 8)
Example 2
Simplify.
2. 2(9 + 1)
3. (2 + 4)6
4. (3 + 6)4
MONEY For Exercises 5 and 6, use the following information.
Suppose you work in a grocery store 4 hours on Friday and 5 hours on
Saturday. You earn $6.25 an hour.
5. Write two different expressions to find your wages.
6. Find the total wages for that weekend.
ALGEBRA Use the Distributive Property to write each expression as an
equivalent algebraic expression.
7. 4(x + 3)
11. 8(y - 2)
126 Chapter 3 Equations
8. 8(m + 4)
12. 9(a - 10)
9. (n + 2)3
13. -6(x - 5)
10. (p + 4)5
14. -3(s - 7)
HOMEWORK
HELP
For
See
Exercises Examples
15–26
1
27, 28
2
29–36
3
37–44
4
Use the Distributive Property to write each expression as an equivalent
expression. Then evaluate it.
15. 2(6 + 1)
19. (9 + 2)4
23. -3(9 - 2)
16. 5(7 + 3)
20. (8 + 8)2
24. -2(8 - 4)
17. (4 + 6)9
21. 7(3 - 2)
25. -5(8 - 4)
18. (4 + 3)3
22. 6(8 - 5)
26. -5(10 - 3)
2 7. MOVIES One movie ticket costs $7, and one small bag of popcorn costs $3.
Write two equivalent expressions for the total cost of four movie tickets
and four bags of popcorn. Then find the cost.
28. SPORTS A volleyball uniform costs $15 for the shirt, $10 for the pants,
and $8 for the socks. Write two equivalent expressions for the total cost
of 12 uniforms. Then find the cost.
ALGEBRA Use the Distributive Property to write each expression as an
equivalent algebraic expression.
29.
33.
3 7.
41.
2(x + 3)
(x + 3)4
3(x - 2)
(r - 5)6
30.
34.
38.
42.
3 1.
35.
39.
43.
5(y + 6)
(y + 2)10
9(m - 2)
(x - 3)12
ANALYZE GRAPHS For Exercises
45–47, use the double bar graph.
7(y + 8)
(2 + x)5
15(s - 3)
(a - 6)(-5)
Annual Fashion Spending
$2200
$1964
$2000
Average Spending per Teen
45. Find the total amount spent on
average by two male teenagers
and two female teenagers on
fashion products in 2003.
46. Find the total amount spent
on average by three female
teenagers in both 2002 and
2003 on fashion products.
47. Did teen spending on fashion
products increase? How do
you know? Explain.
32.
36.
40.
44.
3(n + 1)
(3 + y)6
8(z - 3)
-2(z - 4)
$1800
$1600
$1400
$1342
$1200
$890
$1000
$834
$800
$600
$400
$200
0
2002
Female
2003
Male
Source: www.piperjaffray.com
ALGEBRA Use the Distributive Property to write each expression as an
equivalent algebraic expression.
48. 2(x + y)
5 1. 4(j - k)
EXTRA
PRACTICE
See pages 765, 796.
Self-Check Quiz at
pre-alg.com
49. 3(a + b)
52. 10(r - s)
50. (e + f)(-5)
53. (u - w)(-8)
MENTAL MATH Find each product mentally.
Example 15 · 12 = 15(10 + 2)
Think: 12 is 10 + 2.
= 150 + 30 or 180 Distributive Property
54. 7 · 14
55. 8 · 23
56. 9 · 32
57. 16 · 11
Lesson 3-1 The Distributive Property
127
58. THEME PARKS Admission to an amusement park is $41.99 for adults and
$26.99 for children. The Diego family has a coupon for $10 off each ticket.
Write an expression for the cost for x adults and y children.
H.O.T. Problems
59. OPEN ENDED Write an equation using three integers that is an example of
the Distributive Property.
60. FIND THE ERROR Julia and Catelyn are using the Distributive Property to
simplify 3(x + 2). Who is correct? Explain your reasoning.
Catelyn
3(x + 2) = 3x + 6
Julia
3(x + 2) = 3x + 2
61. CHALLENGE Is 3 + (x · y) = (3 + x) · (3 + y) a true statement? If so, explain
your reasoning. If not, give a counterexample.
62.
Writing in Math
Explain how rectangles can be used to show the
Distributive Property.
63. A ticket to a baseball game costs t
dollars. A soft drink costs s dollars.
Which expression represents the total
cost of a ticket and soft drink for p
people?
A pst
C t(p + s)
B p + (ts)
D p(t + s)
64. Which equation is always true?
F 5(a + b) = 5a + b
G 5(ab) = (5a)(5b)
H 5(a + b) = 5(b + a)
J 5(a + 0) = 5a + 5
ALGEBRA The table shows several solutions of the
equation x + y = 4. (Lesson 2-6)
65. Graph the ordered pairs on a coordinate plane.
66. Describe the graph.
-4y
x+y=4
x
y
(x, y)
-1
5
(-1, 5)
1
2
3
2
(1, 3)
(2, 2)
67. ALGEBRA Evaluate x if x = 2 and y = -3. (Lesson 2-5)
68. FITNESS Jake ran x miles on Monday, y miles on Tuesday, and z miles
on Wednesday. Write an expression for the average number of miles
Jake ran. (Lesson 1-2)
PREREQUISITE SKILL Write each subtraction expression as an addition
expression. (Lesson 2-3)
69. 5 - 3
128 Chapter 3 Equations
70. -8 - 4
71. 10 - 14
72. 8 - (-6)
3-2
Simplifying Algebraic
Expressions
Main Idea
• Use the Distributive
Property to simplify
algebraic expressions.
New Vocabulary
term
coefficient
like terms
constant
simplest form
simplifying an
expression
In a set of algebra tiles, X represents the variable x,
represents the
£
integer 1, and £ represents the integer -1. You can use algebra tiles
to represent expressions. You can also sort algebra tiles by their shapes
and group them.
The tiles below represent the expression 2x + 3 + 3x + 1. On the right,
the algebra tiles have been sorted and combined.
XTILES
£
X
X
£
£
X
X
X
X
£
X
X
X
X
£
£
£
£
TILES
ÓX
Î ÎX
xX
£
{
Therefore, 2x + 3 + 3x + 1 = 5x + 4.
Model each expression with algebra tiles or a drawing. Then sort
them by shape and write an expression represented by the tiles.
a. 3x + 2 + 4x + 3
b. 2x + 5 + x
c. 4x + 5 + 3
d. x + 2x + 4x
Simplify Expressions When plus or minus signs separate an algebraic
expression into parts, each part is a term. The numerical part of a term
that contains a variable is called the coefficient of the variable.
Four terms
2x + 8 + x + 8
2 is the coefficient of 2x.
Vocabulary Link
Constant
Everyday Use unchanging
Math Use a fixed value in
an expression
1 is the coefficient of x because
x = 1x.
Like terms are terms that contain the same variables, such as 2n and 5n
or 6xy and 4xy. A term without a variable is called a constant. Constant
terms are also like terms.
Like terms
5y + 3 + 2y + 8y
Constant
Lesson 3-2 Simplifying Algebraic Expressions
129
Rewriting a subtraction expression using addition will help you identify the
terms of an expression.
EXAMPLE
Identify Parts of Expressions
Identify the terms, like terms, coefficients, and constants in the
expression 3x - 4x + y - 2.
3x - 4x + y - 2 = 3x + (– 4x) + y + (–2)
= 3x + (– 4x) + 1y + (–2)
Definition of subtraction
Identity Property
The terms are 3x, – 4x, y, and –2. The like terms are 3x and – 4x.
The coefficients are 3, –4, and 1. The constant is –2.
1. Identify the terms, like terms, coefficients, and constants in the expression
9a – 2a + 3b – 5.
An algebraic expression is in simplest form if it has no like terms and no
parentheses. When you use the Distributive Property to combine like terms,
you are simplifying the expression.
EXAMPLE
Simplify Algebraic Expressions
Simplify each expression.
a. 6n + 3 + 2n
Equivalent
Expressions
To check whether
6n + 2n and 8n are
equivalent expressions,
substitute any value
for n and see whether
the expressions have
the same value.
6n and 2n are like terms.
6n + 3 + 2n = 6n + 2n + 3
Commutative Property
= (6 + 2)n + 3 Distributive Property
= 8n + 3
Simplify.
b. 3x - 5 - 8x + 6
3x and -8x are like terms. -5 and 6 are also like terms.
3x - 5 - 8x + 6 = 3x + (-5) + (-8x) + 6
Definition of subtraction
= 3x + (-8x) + (-5) + 6
Commutative Property
= [3 + (-8)]x + (-5) + 6
Distributive Property
= -5x + 1
Simplify.
c. m + 3(n + 4m)
m + 3(n + 4m) = m + 3n + 3(4m) Distributive Property
2A. 4x + 6 - 3x
130 Chapter 3 Equations
= m + 3n + 12m
Associative Property
= 1m + 3n + 12m
Identity Property
= 1m + 12m + 3n
Commutative Property
= (1 + 12)m + 3n
Distributive Property
= 13m + 3n
Simplify.
2B. 2m + 3 - 7m - 4
2C. 4(q + 8p) + p
Extra Examples at pre-alg.com
BASEBALL CARDS Suppose your brother has 15 more baseball cards in
his collection than you have. Write an expression in simplest form that
represents the total number of cards in both collections.
Words
Variables
Expression
Source: CMG Worldwide
plus
number of your brother’s cards
Let x = number of cards you have.
Let x + 15 = number of cards your brother has.
x + (x + 15)
x + (x + 15) = (x + x) + 15
= (1x + 1x) + 15
= (1 + 1)x + 15
= 2 x + 15
Real-World Link
Honus Wagner is
considered by many to
be baseball’s greatest
all-around player.
In July, 2000, one of his
baseball cards sold for
$1.1 million.
number of your cards
Associative Property
Identity Property
Distributive Property
Simplify.
The expression 2x + 15 represents the total number of cards, where x is the
number of cards you have.
3. STAMPS Matt and Lola both collect stamps. Lola has 25 more stamps in
her collection than Matt has. Write an expression in simplest form that
represents the total number of stamps in both collections.
Personal Tutor at pre-alg.com
Example 1
(p. 130)
Identify the terms, like terms, coefficients, and constants in each
expression.
1. 4x + 3 + 5x + y
Example 2
(p. 130)
3. 4y - 2x - 7
4. 6a + 4 + 2a
5. x + 9x + 3
6. 9y + 8 - 8
7. 3x + 2y + 4y
8. 6c + 4 + c + 8
9. 2x - 5 - 4x + 8
10. x + 3(x + 4y)
11. 8e - 4(2f + 5e)
Simplify each expression.
12. 5 - 3(y + 7)
(p. 131)
13. MONEY You have saved some money. Your friend has saved $20 more than
you. Write an expression in simplest form that represents the total amount
of money you and your friend have saved.
HELP
Identify the terms, like terms, coefficients, and constants in each
expression.
Example 3
HOMEWORK
2. 2m - n + 6m
For
See
Exercises Examples
14–19
1
20–34
2
35–38
3
14. 3 + 7x + 3x + x
15. y + 3y + 8y + 2
16. 2a + 5c - a + 6a
17. 5c - 2d + 3d - d
18. 6m - 2n + 7
19. 7x - 3y + 3z - 2
Lesson 3-2 Simplifying Algebraic Expressions
Kit Kittle/CORBIS
131
Simplify each expression.
20. 2x + 5x
21. 7b + 2b
22. y + 10y
23. 5y + y
24. 2a + 3 + 5a
25. 4 + 2m + m
26. 2y + 8 + 5y + 1
27. 8x + 5 + 7 + 2x
28. 5x - 3x
29. 10b - 2b
30. 4y - 5y
31. r - 3r
32. 8 + x - 5x
33. 6x + 4 - 7x
34. 2x + 3 - 3x + 9
For Exercises 35–38, write an expression in simplest form that represents
the total amount in each situation.
35. SCHOOL SUPPLIES You bought 5 folders that each cost x dollars, a calculator
for $45, and a set of pens for $3.
36. SHOPPING Suppose you buy 3 shirts that each cost s dollars, a pair of shoes
for $50, and jeans for $30.
37. FASHION Your friend Natasha has y pairs of shoes. Her sister has 5 fewer
pairs.
38. BABY-SITTING Alicia earned d dollars baby-sitting. Her friend earned twice
as much. You earned $2 less than Alicia’s friend earned.
Simplify each expression.
Real-World Link
In a recent survey, 10%
of students in grades
6–12 reported that most
of their spending
money came from
baby-sitting.
Source: USA WEEKEND
39. 6m + 2n + 10m
40. -2y + x + 3y
41. c + 2(d - 5c)
42. 3(b + 2) + 2b
43. 5(x + 3) + 8x
44. -3(a + 2) - a
45. -2(x + 3) + 2x
46. 4x - 4(2 + x)
47. 8a - 2(a - 7)
GEOMETRY You can find the perimeter of a geometric figure by adding the
measures of its sides. Write an expression in simplest form for the
perimeter of each figure.
2x ⫹ 1
48.
49.
3x
EXTRA
PRACTICE
H.O.T. Problems
x
x
2x ⫹ 1
See pages 765, 796.
Self-Check Quiz at
pre-alg.com
5x
4x
Simplify to make the calculation as easy as possible.
50. 16 · (-31) + 16 · 32
51. 72(38) + (-72)(18)
52. OPEN ENDED Write an expression in simplest form containing three terms.
One of the terms should be a constant.
53. FIND THE ERROR Koko and John are simplifying the expression
5x - 4 + x + 2. Who is correct? Explain your reasoning.
Koko
5x - 4 + x + 2 =
6x - 2
John
5x - 4 + x + 2 =
5x - 2
54. Which One Doesn’t Belong? Identify the algebraic expression that does not
belong with the other three. Explain your reasoning.
-6(x - 2)
132 Chapter 3 Equations
Mary Kate Denny/PhotoEdit
x + 12 - 7x
-x - 5x + 12
-6x - 12
CHALLENGE You use deductive reasoning when you base a conclusion on
mathematical rules or properties. Indicate the property that justifies each
step that was used to simplify 3(x + 4) + 5(x + 1).
55. 3(x + 4) + 5(x + 1) = 3x + 12 + 5x + 5
56.
= 3x + 5x + 12 + 5
57.
= 3x + 5x + 17
58.
= 8x + 17
59.
Writing in Math
Explain how algebra tiles can be used to simplify an
algebraic expression. Illustrate your reasoning with an example.
60. The perimeter of DEF is 4x + 3y.
What is the measure of the third side of
the triangle?
A -2x + 2y
$
xXÊÊÎY
B 2x + 2y
C x-y
XÊÊ{Y
&
%
D -x + 2y
61. You spend x minutes reading a book on
Saturday. On Sunday, you spend 35
more minutes reading than you did on
Saturday. Which expression represents
the total amount of time spent reading
the book on Saturday and Sunday?
F 2x + 35
H 2x - 35
G x + 35
J x - 35
ALGEBRA Use the Distributive Property to write each expression as an
equivalent expression. (Lesson 3-1)
62. 3(a + 5)
63. -2(y - 8)
64. 7(d - 10)
65. -3(x - 1)
66. Name the quadrant in which P(-5, -6) is located. (Lesson 2-6)
67. CRUISES The table shows the number of people who took a cruise in
various years. Make a scatter plot of the data. (Lesson 1-7)
9EAR
.UMBERMILLIONS -œÕÀVi\#RUISE,INES)NTERNATIONAL!SSOCIATION
3(4a - 3b)
b-4
68. ALGEBRA What is the value of _ if a = 6 and b = 7? (Lesson 1-3)
69. DECORATING A wallpaper roll contains a sheet that is 40 feet long and
18 inches wide. What is the minimum number of rolls of wallpaper
needed to cover 500 square feet of wall space? (Lesson 1-1)
PREREQUISITE SKILL Find each sum. (Lesson 2-2)
70. -5 + 4
71. -8 + (-3)
72. 10 + (-1)
Lesson 3-2 Simplifying Algebraic Expressions
133
EXPLORE
3-3 AND 3-4
Algebra Lab
Solving Equations Using
Algebra Tiles
ACTIVITY 1
In a set of algebra tiles, X represents the variable x,
£
represents the
integer 1, and £ represents the integer -1. You can use algebra tiles
and an equation mat to model equations.
£
X
£
£
£
XÊÊÎ
£
£
£
£
X
£
£
£
XÊÊÓ
x
£
When you solve an equation, you are trying to find the value of x that
makes the equation true. The following example shows how to solve
x + 3 = 5 using algebra tiles.
£
X
£
£
XÊÊÎ
£
£
£
X
x
X
Remove the same number of 1-tiles from each side of
the mat until the x-tile is by itself on one side.
£
X
Model the equation.
£
X
£
£
The number of tiles remaining on the right side
of the mat represents the value of x.
Ó
Therefore, x = 2. Since 2 + 3 = 5, the solution is correct.
EXERCISES
Use algebra tiles or a drawing to model and solve each equation.
1. 3 + x = 7
2. x + 4 = 5
3. 6 = x + 4
4. 5 = 1 + x
134 Chapter 3 Equations
Animation at tx.pre-alg.com
ACTIVITY 2
Animation at pre-alg.com
Some equations are solved by using zero pairs. You may add or subtract
a zero pair from either side of an equation mat without changing its value.
The following example shows how to solve x + 2 = -1 by using zero pairs.
£
X
£
XÊÊÓ
£
£
£
£
£
XÊÊÓÊ­Ó®
X
£
X
£
Model the equation. Notice it is not possible to remove
the same kind of tile from each side of the mat.
£
£
£
X
£
£
Add 2 negative 1-tiles to the left side of the mat to make zero
pairs. Add 2 negative 1-tiles to the right side of the mat.
£Ê­Ó®
£
£
£
£
Remove all of the zero pairs from the left side. There are
3 negative 1-tiles on the right side of the mat.
Î
Therefore, x = -3. Since -3 + 2 = -1, the solution is correct.
EXERCISES
Use algebra tiles or a drawing to model and solve each equation.
5. x + 2 = -2
6. x - 3 = 2
7. 0 = x + 3
8. -2 = x + 1
ACTIVITY 3
The equation 2x = -6 is modeled using more than one x-tile. Arrange the tiles
into equal groups to match the number of x-tiles.
X
X
ÓX
£
£
£
£
£
£
È
X
£
£
£
£
£
£
X
X
Î
Therefore, x = -3. Since 2(-3) = -6, the solution is correct.
EXERCISES
Use algebra tiles or a drawing to model and solve each equation.
9. 3x = 3
10. 2x = -8
11. 6 = 3x
12. -4 = 2x
Explore 3-3 and 3-4 Algebra Lab: Solving Equations Using Algebra Tiles
135
3-3
Main Ideas
• Solve equations by
using the Subtraction
Property of Equality.
• Solve equations by
using the Addition
Property of Equality.
New Vocabulary
inverse operation
equivalent equations
Solving Equations by
Adding or Subtracting
On the balance at the right, the
paper bag contains a certain
number of blocks. (Assume that
the paper bag weighs nothing.)
a. Without looking in the bag,
how can you determine the
number of blocks in the bag?
b. Explain why your method works.
Solve Equations by Subtracting The equation x + 4 = 7 is a model of
the situation above. You can use inverse operations to solve the equation.
Inverse operations “undo” each other. For example, to undo the addition
of 4 in the expression x + 4, you would subtract 4.
To solve the equation x + 4 = 7, subtract 4 from each side.
x+4=7
x+4-4=7-4
Subtract 4 from the left
side of the equation to
isolate the variable.
x+0=3
x=3
Subtract 4 from the right
side of the equation to
keep it balanced.
The solution is 3.
You can use the Subtraction Property of Equality to solve any equation
like x + 4 = 7.
Subtraction Property of Equality
Words
If you subtract the same number from each side of an equation,
the two sides remain equal.
Symbols
For any numbers a, b, and c, if a = b, then a - c = b - c.
Examples
5=5
5-3=5-3
2=2
x+2=3
x+2-2=3-2
x=1
READING
in the Content Area
For strategies in reading
this lesson, visit
pre-alg.com.
136 Chapter 3 Equations
The equations x + 4 = 7 and x = 3 are equivalent equations because
they have the same solution, 3. When you solve an equation, you should
always check to be sure that the first and last equations are equivalent.
EXAMPLE
Solve Equations by Subtracting
Solve x + 8 = -5. Check your solution and graph it on a number line.
x + 8 = -5
x + 8 - 8 = -5 - 8
x + 0 = -13
x = -13
Checking
Equations
It is always wise to
check your solution.
You can often use
arithmetic facts to
check the solutions
of simple equations.
Write the equation.
Subtract 8 from each side.
8 - 8 = 0, -5 - 8 = -13
Identity Property; x + 0 = x
To check your solution, replace x with -13 in the original equation.
CHECK
x + 8 = -5
-13 + 8 -5
-5 = -5 Write the equation.
Check to see whether this sentence is true.
The sentence is true.
The solution is -13. To graph it, draw a dot at -13 on a number line.
£{ £Î £Ó ££ £ä ™ n
Ç
Solve each equation. Check your solution and graph it on a number line.
1A. 4 = x + 10
1B. 16 + z = 14
Solve Equations by Adding Some equations can be solved by adding the
same number to each side. This uses the Addition Property of Equality.
Addition Property of Equality
Words
If you add the same number to each side of an equation, the two sides
remain equal.
Symbols
For any numbers a, b, and c, if a = b, then a + c = b + c.
Examples
6=6
6+3=6+3
9=9
x-2=5
x-2+2=5+2
x=7
If an equation has a subtraction expression, first rewrite the expression as an
addition expression. Then add the additive inverse to each side.
EXAMPLE
Solve Equations by Adding
Solve y - 7 = -25.
y - 7 = -25
y + (-7) = -25
y + (-7) + 7 = -25 + 7
y + 0 = -25 + 7
y = -18
The solution is -18.
Write the equation.
Rewrite y - 7 as y + (-7).
Add 7 to each side.
Additive Inverse Property; (-7) + 7 = 0.
Identity Property; y + 0 = y
Check your solution.
Solve each equation.
2A. -20 = y - 13
Extra Examples at pre-alg.com
2B. -115 + b = -84
Lesson 3-3 Solving Equations by Adding or Subtracting
137
Jessica downloaded 54 songs onto her digital music player. This is 17 less
than the number of songs Kaela downloaded earlier. Which equation can
be used to find the number of songs Kaela downloaded onto her digital
music player?
Key Words When
translating words to
equation, look for key
words that indicate
operations. The
phrase “less than”
can indicate
subtraction or an
inequality.
A x - 17 = 54
B x + 17 = 54
C 17 - x = 54
D -54 = 17 + x
Read the Test Item Translate the verbal sentence into an equation.
Solve the Test Item
Words
Jessica downloaded 17 less songs than Kaela.
Variable
Let x = the number of songs Kaela downloaded.
Equation
54 = x - 17
So, the equation 54 = x - 17 or x - 17 = 54 can be used to find the number
of songs Kaela downloaded. This is choice A.
3. During the night, the temperature dropped 14° to -9°F. Which equation
can be used to find the temperature at the beginning of the night?
F -9 + x = -14
H -9 - x = 14
G 14 - x = -9
J x - 14 = -9
Personal Tutor at pre-alg.com
SLEDDING Use the information at the left. Write and solve an equation to
find the distance of the Northern Route of the Iditarod Trail Sled Dog Race.
The Southern Route is 49 miles longer than the Northern Route.
Real-World Link
There are two different
routes for the Iditarod
Trail Sled Dog Race.
During the odd years,
the race takes place on
the 1161-mile Southern
Route. This is 49 miles
longer than the
Northern Route that
takes place during the
even years.
Source: iditarod.com
138 Chapter 3 Equations
AP/Wide World Photos
Let d = the distance of the Northern Route.
1161= d + 49
Write the equation.
1161 - 49 = d + 49 - 49 Subtract 49 from each side.
1112 = d
Simplify.
CHECK
1161 = d + 49
1161 1112 + 49
1161 = 1161 Write the equation.
Check to see whether this statement is true.
The statement is true.
The Northern Route is 1112 miles long.
4. BUILDINGS The Jefferson Memorial in Washington, D.C., is 129 feet tall.
This is 30 feet taller than the Lincoln Memorial. Write and solve an
equation to find the height of the Lincoln Memorial.
2. w + 4 = -10
3. 16 = y + 20
4. n - 8 = 5
5. k - 25 = 30
6. r - 4 = -18
7. MULTIPLE CHOICE A video store sells a DVD for $12 more than it pays for it. If
the selling price of the DVD is $19, which equation can be used to find how
much the store paid for the DVD?
A x + 19 = -12
For
See
Exercises Examples
9–26
1, 2
27, 28
4
43, 44
3
ALGEBRA Solve each equation. Check your solution and graph it on a
number line.
9. y + 7 = 21
10. x + 5 = 18
1 1. m + 10 = -2
12. x + 5 = -3
13. a + 10 = -4
14. t + 6 = -9
15. y + 8 = 3
16. 9 = 10 + b
17. k - 6 = 13
18. r - 5 = 10
19. 8 = r - 5
20. 19 = g - 5
21. x - 6 = -2
22. y - 49 = -13
23. -15 = x - 16
24. -8 = t - 4
25. 23 + y = 14
26. 59 = s + 90
27. ELECTIONS In the 2004 presidential election, Georgia had 15 electoral votes.
That was 19 votes fewer than the number of electoral votes in Texas. Write
and solve an equation to find the number of electoral votes in Texas.
28. WEATHER The difference between the record high and low temperatures in
Charlotte, North Carolina, is 109˚F. The record low temperature was -5˚F.
Write and solve an equation to find the record high temperature.
29. RESEARCH Use the Internet or another source to find record temperatures in
your state. Use the data to write a problem.
ANALYZE GRAPHS For Exercises 30
and 31, use the graph and the
following information.
Tokyo’s population is 10 million
greater than New York City’s
population. Los Angeles’
population is 2 million less than
New York City’s population.
30. Write two different equations
to find New York City’s
population.
31. Solve the equations to find the
population of New York City.
Most Populous Urban Areas
28
?
18
18
18
16
14
Shanghai
HELP
8. FUND-RAISING Jim sold 43 candles to raise money for a class trip. This is 15
less than the number Diana sold. Write and solve an equation to find the
number of candles Diana sold.
Los Angeles
HOMEWORK
D 12 - x = 19
Sao Paulo
(p. 138)
C x + 12 = 19
Bombay
Example 4
B 19 + x = 12
Mexico City
(p. 138)
1. x + 14 = 25
New York City
Example 3
ALGEBRA Solve each equation. Check your solution and graph it on a
number line.
Tokyo
(p. 137)
Population (millions)
Examples 1, 2
Cities
Source: infoplease.com
Lesson 3-3 Solving Equations by Adding or Subtracting
139
ALGEBRA Solve each equation. Check your solution.
PRACTICE
32. a - 6.1 = 3.4
33. 14.8 + x = - 20.1
34. 17.6 = y + 11.5
See pages 766, 796.
35. p - (-13.35) = -19.72
36. -52.23 + b = 40.04
37. z - 37.98 = 65.21
Self-Check Quiz at
pre-alg.com
38. ALGEBRA If a number x satisfies x + 4 = -2, find the numerical value
of -3x - 2.
EXTRA
H.O.T. Problems
39. OPEN ENDED Write two equations that are equivalent. Then write two
equations that are not equivalent. Justify your reasoning.
40. SELECT A TECHNIQUE Jaime’s golf score was -9 today. She decreased her
score by 5 strokes from yesterday. Which of the following techniques might
you use to determine what her golf score was yesterday? Justify your
selection(s). Then use the technique(s) to solve the problem.
computer
draw a model
real objects
41. CHALLENGE Write two equations in which the solution is -5.
42.
Writing in Math Formulate a problem situation for the equation x + 7 = 20.
The table shows the five nearest stars to
Earth, excluding the Sun.
Star
Distance
(light-years)
Proxima Centauri
4.22
Alpha Centauri A
4.40
Alpha Centauri B
4.40
Barnard’s Star
5.94
Wolf 359
7.79
43. Which equation will best help
you find how much closer
Proxima Centauri is to Earth
than Barnard’s Star?
A x - 5.94 = 4.22 C 5.94 + x = 4.22
B x + 4.22 = 5.94 D 5.94 + 4.22 = x
44. GRIDDABLE How many light years
closer is Alpha Centauri B to Earth
than Wolf 359?
ALGEBRA Simplify each expression. (Lessons 3-1 and 3-2)
45. -2(x + 5)
46. (t + 4)3
47. -4(x - 2)
48. 6z - 3 - 10z + 7
49. 2(x + 6) + 4x
50. 3 - 4(m + 1)
51. GEOLOGY The width of a beach is changing at a rate of -9 inches per year.
How long will it take for the width of the beach to change -4.5 feet? (Lesson 2-5)
52. MONEY Xavier opened a checking account with a deposit of $200. During
the next week, he wrote checks for $65, $83, and $28 and made a deposit of
$50. Write an addition expression for this situation and find the balance in
his account. (Lesson 2-2)
PREREQUISITE SKILL Divide. (Lesson 2–5)
53. -100 ÷ 10
54. 50 ÷ (-2)
140 Chapter 3 Equations
55. -49 ÷ (-7)
72
56. _
-8
3-4
Solving Equations by
Multiplying or Dividing
Main Ideas
• Solve equations by
using the Division
Property of Equality.
• Solve equations by
using the
Multiplication
Property of Equality.
An exchange rate allows people to
exchange one currency for another.
In Mexico, about 11 pesos can be
exchanged for $1 of U.S. currency,
as shown in the table.
U.S. Value ($)
Number of Pesos
1
11(1) ⴝ 11
2
11(2) ⴝ 22
3
11(3) ⴝ 33
4
11(4) ⴝ 44
In general, if we let d represent
the number of U.S. dollars and p
represent the number of pesos,
then 11d = p.
a. Suppose lunch in Mexico costs 77 pesos. Write an equation to
find the cost in U.S. dollars.
b. How can you find the cost in U.S. dollars?
Solve Equations by Dividing The equation 11x = 77 is a model of the
relationship described above. To undo the multiplication operation in
11x, you would divide by 11.
To solve the equation 11x = 77, divide each side by 11.
11x = 77
Divide the left side of
the equation by 11 to
undo the multiplication
11 · x.
11x
77
_
= _
11
11
1x = 7
x = 7
Divide the right side
of the equation by 11
to keep it balanced.
The solution is 7.
You can use the Division Property of Equality to solve any equation like
11x = 77.
Division Property of Equality
Words
When you divide each side of an equation by the same nonzero
number, the two sides remain equal.
b
_
For any numbers a, b, and c, where c ≠ 0, if a = b then _
c = c.
Examples 14 = 14
3x = -12
a
Symbols
14 _
14
_
=
3x _
-12
_
=
2=2
x = -4
7
7
3
3
Lesson 3-4 Solving Equations by Multiplying or Dividing
141
EXAMPLE
Solve Equations by Dividing
Solve 5x = -30. Check your solution and graph it on a number line.
5x = -30 Write the equation.
5x -30
=
5
5
Divide each side by 5 to undo the multiplication in 5 · x.
1x = -6
5 ÷ 5 = 1, -30 ÷ 5 = -6
x = -6
Identity Property; 1x = x
To check your solution, replace x with -6 in the original equation.
CHECK
5x = -30
5(-6) -30
-30 = -30 Write the equation.
Check to see whether this statement is true.
The statement is true.
The solution is -6. To graph it, draw a dot at -6 on a number line.
⫺7 ⫺6
⫺5
⫺4
⫺3
⫺2 ⫺1
0
1. Solve -48 = 6x. Check your solution and graph it on a number line.
PARKS It costs $3 per car to use the hiking trails along the Columbia
River Highway. If income from the hiking trails totaled $1275 in one
day, how many cars entered the park?
The cost
per car
Words
$3
Equation
The Columbia River
Highway, built in 1913,
is a historic route in
Oregon that curves
around twenty waterfalls
through the Cascade
Mountains.
Source: columbiariverhighway.com
Write the equation.
3x
1275
_
=_
Divide each side by 3.
3
x = 425
CHECK
equals
the total
x
=
$1275
·
3x = 1275
3
the number
of cars
Let x = the number of cars.
Variable
Real-World Link
times
Simplify.
3x = 1275
3(425) 1275
1275 = 1275 Write the equation.
Check to see whether this statement is true.
The statement is true.
Therefore, 425 cars entered the park.
2. PARKS In-state camping permits for New Mexico State Parks cost $180 per
year. If income from the camping permits totaled $8280 during the first
day of sales, how many people bought permits?
Personal Tutor at pre-alg.com
142 Chapter 3 Equations
Stuart Westmorland/CORBIS
Solve Equations by Multiplying Some equations can be solved by multiplying
each side by the same number. This property is called the Multiplication
Property of Equality.
Multiplication Property of Equality
Interactive Lab
pre-alg.com
Words
When you multiply each side of an equation by the same number, the
two sides remain equal.
Symbols
For any numbers a, b, and c, if a = b, then ac = bc.
_x = 7
8=8
Examples
6
_6x 6 = (7)6
8(-2) = 8(-2)
x = 42
-16 = -16
EXAMPLE
Reading Math
Division Expressions
y
Remember, _ means
-4
y divided by -4.
Solve Equations by Multiplying
y
-4
Solve _ = -9. Check your solution and graph it on a number line.
y
_
= -9
-4
y
_
(-4) = -9(-4)
-4
y = 36
Write the equation.
y
-4
Multiply each side by −4 to undo the division in _.
Simplify.
y
CHECK _ = -9
-4
Write the equation.
36
_
-9
Check to see whether this statement is true.
-9 = -9 The statement is true.
-4
The solution is 36. To graph it, draw a dot at 36 on a number line.
ÎÓ
ÎÎ
Î{
Îx
ÎÈ
ÎÇ
În
x
3. Solve 7 = _
. Check your solution and graph it on a number line.
-2
Example 1
(p. 142)
Example 2
(p. 142)
Example 3
(p. 143)
ALGEBRA Solve each equation. Check your solution.
1. 4x = 24
2. -2a = 10
3. -7t = -42
4. TOYS A spiral toy that can bounce down a flight of stairs is made from
80 feet of wire. Write and solve an equation to find how many of these toys
can be made from a spool of wire that contains 4000 feet.
ALGEBRA Solve each equation. Check your solution.
k
=9
5. _
3
Extra Examples at pre-alg.com
y
6. _ = -8
5
n
7. -11 = _
-6
Lesson 3-4 Solving Equations by Multiplying or Dividing
143
HOMEWORK
HELP
For
See
Exercises Examples
8–31
1, 3
32–35
2
ALGEBRA Solve each equation. Check your solution.
8. 3t = 21
h
10. _
=6
9. 8x = 72
4
11. _c = 4
g
12. _ = -7
-2
x
13. -42 = _
14. -32 = 4y
15. 5n = -95
16. -56 = -7p
17. -8j = -64
b
18. 11 = _
h
19. _
= 20
20. 45 = 5x
21. 3u = 51
m
22. _
= -3
24. 86 = -2v
25. -8a = 144
v
27. _
= -132
k
28. -21 = _
30. -116 = -4w
31. -68 = -4m
9
d
= -3
23. _
3
f
26. _ = -10
-13
29. -56 = _t
9
-2
-3
-7
45
-11
8
32. BOATING A forest preserve rents canoes for $12 per hour. Corey has $36.
Write and solve an equation to find how many hours he can rent a canoe.
Indian
Ocean
Pacific
Ocean
Outback
AUSTRALIA
Southern
Ocean
Real-World Link
Some students living in
the Outback are so far
from schools that they
get their education
by special radio
programming. They mail
in their homework and
sometimes talk to
teachers by two-way
radio.
Source: Kids Discover Australia
33. FRUIT Jenny picked a total of 960 strawberries in 1 hour. Write and solve an
equation to find how many strawberries Jenny picked per minute.
34. RANCHING The largest ranch in the world is in the Australian Outback.
It is about 12,000 square miles, which is five times the size of the largest
United States ranch. Write and solve an equation to find the size of the
largest United States ranch.
35. RANCHING In the driest part of an Outback ranch, each cow needs about
40 acres for grazing. Write and solve an equation to find how many cows
can graze on 720 acres of land.
ALGEBRA Graph the solution of each equation on a number line.
36. -6r = -18
37. -42 = -7x
n
38. _
=3
12
MEASUREMENT The chart shows several conversions in
the customary system. Write and solve an equation to
find each quantity.
40. the number of feet in 132 inches
41. the number of yards in 15 feet
42. the number of miles in 10,560 feet
y
39. _ = -1
-4
Customary System
(length)
1 mile = 5280 feet
1 mile = 1760 yards
1 yard = 3 feet
1 foot = 12 inches
1 yard = 36 inches
EXTRA
PRACTICE
43. PAINTING A person-day is a unit of measure that represents one person
working for one day. A painting contractor estimates that it will take
24 person-days to paint a house. Write and solve an equation to find how
many painters the contractor will need to hire to paint the house in 6 days.
See pages 766, 796
Self-Check Quiz at
pre-alg.com
44. FIND THE DATA Refer to the United States Data File on pages 18–21 of your
book. Choose some data and write a real-world problem in which you
would solve an equation by multiplying or dividing.
144 Chapter 3 Equations
H.O.T. Problems
45. OPEN ENDED Write an equation of the form ax = c where a and c are
integers and the solution is 4.
46. NUMBER SENSE Find an equation that is equivalent to -9t = 18.
x = 3, what is the value of 7x + 13?
47. CHALLENGE If 10
48.
Writing in Math
Explain how equations are used to find the U.S.
value of foreign currency. Illustrate your reasoning by finding the cost in
U.S. dollars of a 12-pound bus trip in Egypt, if 6 pounds can be exchanged
for one U.S. dollar.
49. Suppose that one pyramid balances
two cubes and one cylinder balances
three cubes as shown below. Which
statement is NOT true?
50. The solution of which equation is
NOT graphed on the number line
below?
{ Î
A One pyramid and one cube balance
three cubes.
B One pyramid and one cube balance
one cylinder.
Ó
£
ä
£
Ó
Î
F 12 = -6x
H -14 = 7x
G 8x = -16
J -18x = -36
51. During a vacation, the Mulligan
family drove 63 miles in 1 hour. If
they averaged the same speed during
their trip, which equation can be used
to find how far the Mulligan family
drove in 6 hours?
63
A _
x =6
C One cylinder and one pyramid
balance four cubes.
D One cylinder and one cube balance
two pyramids.
x
B _
= 63
6
C 6x = 63
D 63x = 6
ALGEBRA Solve each equation. Check your solution. (Lesson 3-3)
52. 3 + y = 16
53. 29 = n + 4
54. k - 12 = -40
ALGEBRA Simplify each expression. (Lesson 3-2)
55. 4x + 7x
56. 2y + 6 + 5y
57. 3 - 2(y + 4)
58. AGE Patricia is 12 years old, and her younger sister Renee is 2 years old. How
old will each of them be when Patricia is twice as old as Renee? (Lesson 1-1)
PREREQUISITE SKILL Find each difference. (Lesson 2-3)
59. 8 - (-2)
60. -5 - 5
61. -10 - (-8)
62. -18 - 4
63. -45 - (-9)
64. 33 - (-19)
Lesson 3-4 Solving Equations by Multiplying or Dividing
145
CH
APTER
3
Mid-Chapter Quiz
Lessons 3-1 through 3-4
1. MULTIPLE CHOICE Lucita works at a fitness
center and earns $5.50 per hour. She worked
3 hours on Friday and 7 hours on Saturday.
Which expression does NOT represent her
wages that weekend? (Lesson 3-1)
A 5.50(3 + 7)
B 10(5.50)
9. AVIATION On December 17, 1903, the
Wright brothers made the first flights in
a power-driven airplane. Orville’s flight
covered 120 feet, which was 732 feet
shorter than Wilbur’s. Find the length
of Wilbur Wright’s flight. (Lesson 3-3)
10. WEATHER Before a storm, the barometric
pressure dropped to 29.2, which was 1.3
lower than the pressure earlier in the day.
Write an equation to represent this situation.
C 5.50(3) + 5.50(7)
D 7(5.50 + 3)
2. FUND-RAISING Debbie sold 23 teen
magazines at $3.25 each, 38 sports magazines
at $3.50 each, and 30 computer magazines at
$2.95 each. How much money did Debbie
raise? (Lesson 3-1)
Simplify each expression. (Lessons 3-1 and 3-2)
3. 6(x + 2)
4. 5(x - 7)
5. 6y - 4 + y
6. 2a + 4(a - 9)
7. SCHOOL You spent m minutes studying on
Monday. On Tuesday, you studied 15 more
minutes than you did on Monday. Write an
expression in simplest form that represents
the total amount of time spent studying on
Monday and Tuesday. (Lesson 3-2)
(Lesson 3-3)
ALGEBRA Solve each equation. (Lessons 3-3 and 3-4)
11. 4h = -52
12. y - 5 = -23
x =4
13. -3
14. n + 16 = 44
15. MULTIPLE CHOICE The table shows the five
nearest train stops to Main Street. Which
equation will best help you find how much
further Peach Court is from Main Street than
City Center is from Main Street? (Lesson 3-3)
Train Stop
City Center
14th Street
Grand Hotel
Stadium
Peach Court
8. MULTIPLE CHOICE A paving brick is shown.
Find the perimeter of 5 bricks. (Lesson 3-2)
ÎX
ÓXÓ
Distance to
Main Street
(miles)
4
6
7
12
17
A x - 17 = 4
ÓXÓ
B x + 17 = 4
C x - 4 = 17
xXÎ
D x + 4 = 17
F 12x + 1
G 40x + 10
H 60x + 5
J 50x - 10
146 Chapter 3 Equations
16. MONEY Ricardo spends $3.50 for lunch each
day. Write and solve an equation to find how
long it takes him to spend $21 on lunch.
(Lesson 3-4)
3-5
Solving Two-Step
Equations
Main Idea
• Solve two-step
equations.
The equation 2x + 1 = 9, modeled below, can be solved with algebra tiles.
New Vocabulary
two-step equation
X
X
£
£
£
£
£
£
£
£
£
£
ÓX£
™
Step 1 Remove 1 tile from each side of the mat.
X
X
£
£
£
£
£
£
£
£
£
£
ÓX£Ê£
™Ê£
Step 2 Separate the remaining tiles into two equal groups.
X
£
£
£
£
£
£
£
£
X
ÓX
n
a. What property is shown by removing a tile from each side?
b. What property is shown by separating the tiles into two groups?
c. What is the solution of 2x + 1 = 9?
Solve Two-Step Equations A two-step equation contains two operations.
In the equation 2x + 1 = 9, x is multiplied by 2 and then 1 is added.
To solve two-step equations, use inverse operations to undo each
operation in reverse order.
Step 1 First, undo addition.
2x + 1 = 9
2x + 1 - 1 = 9 - 1 Subtract 1 from each side.
2x = 8
Step 2 Then, undo multiplication.
2x = 8
8
2x
_
=_
2
2
Divide each side by 2.
x=4
The solution is 4.
Lesson 3-5 Solving Two-Step Equations
147
EXAMPLE
Solve Two-Step Equations
a. Solve 5x - 2 = 13. Check your solution.
5x - 2 = 13
Write the equation.
5x - 2 + 2 = 13 + 2 Undo subtraction. Add 2 to each side.
5x = 15
Simplify.
5x
15
_
=_
Undo multiplication. Divide each side by 5.
5
5
x=3
CHECK
Simplify.
5x - 2 = 13
Write the equation.
5(3) - 2 13
Check to see whether this statement is true.
13 = 13 The statement is true.
The solution is 3.
n
+ 11.
b. Solve 4 = _
4
4 - 11
-7
6(-7)
6
n
= _ + 11
6
n
=_
+ 11 - 11
6
n
=_
6
n
=6 _
6
( )
-42 = n
Write the equation.
Undo addition. Subtract 11 from each side.
Simplify.
Undo division. Multiply each side by 6.
Check your solution.
Solve each equation.
n
1B. _
+ 15 = 8
1A. 6x + 1 = 25
3
SALES Liana bought a DVD recorder. If she pays $80 now, her monthly
payments will be $32. The total cost will be $400. Solve 80 + 32x = 400
to find how many months she will make payments.
80 + 32x = 400
Checking Your
Solution
Use estimation to
determine whether
your solution is
reasonable:
80 + 30(10) = 380.
Since $380 is close to
$400, the solution is
reasonable.
Write the equation.
80 - 80 + 32x = 400 - 80 Subtract 80 from each side.
32x = 320
Simplify.
32x
320
_
=_
Divide each side by 32.
32
32
x = 10
Simplify.
Therefore, Liana will make payments for 10 months.
2. COMPUTERS Salvatore purchased a computer for $550. He paid $105
initially, and then he will pay $20 per month until the computer is paid
off. Solve 105 + 20x = 545 to find how many payments he will make.
148 Chapter 3 Equations
EXAMPLE
Equations with Negative Coefficients
Solve 4 - x = 10.
4 - x = 10
Write the equation.
4 - 1x = 10
Identity Property; x = 1x
4 + (-1x) = 10
Definition of subtraction
-4 + 4 + (-1x) = -4 + 10
-1x = 6
Simplify.
6
-1x
_
=_
Divide each side by -1.
-1
-1
x = -6
EXAMPLE
Check your solution.
Solve each equation.
3B. 35 - k = 21
3A. 19 = 9 - y
Mental
Computation
Add -4 to each side.
Combine Like Terms Before Solving
Solve m - 5m + 3 = 47.
You use the Distributive
Property to simplify
1m - 5m.
1m - 5m = (1 - 5)m
= -4m
You can also simplify
the expression mentally.
m - 5m + 3 = 47
1m - 5m + 3 = 47
-4m + 3 = 47
-4m + 3 - 3 = 47 - 3
Write the equation.
Identity Property; m = 1m
Combine like terms, 1m and –5m.
Subtract 3 from each side.
-4m = 44
Simplify.
-4m
44
_
=_
Divide each side by – 4.
-4
-4
m = -11
4A. 4 - 9d + 3d = 58
Simplify.
Solve each equation.
4B. 34 = 4m - 2 + 2m
Personal Tutor at pre-alg.com
Example 1
(p. 148)
Example 2
(p. 148)
Examples 3, 4
(p. 149)
ALGEBRA Solve each equation. Check your solution.
1. 2x - 7 = 9
2. -16 = 6a - 4
y
3. _ + 2 = 10
3
4. MEDICINE For Jillian’s cough, her doctor says that she should take eight
tablets the first day and then four tablets each day until her prescription
runs out. There are 36 tablets. Solve 8 + 4d = 36 to find how many more
days she will take four tablets.
ALGEBRA Solve each equation. Check your solution.
5. -7 - 8d = 17
6. 1 - 2k = -9
8. 2a - 8a = 24
9. -4 = 8y - 9y + 6
Extra Examples at pre-alg.com
-n
-5
7. 8 = _
7
10. -6j + 4 + 3j = -23
Lesson 3-5 Solving Two-Step Equations
149
HOMEWORK
HELP
For
See
Exercises Examples
11–28
1
29–30
2
31–34
3
35–42
4
ALGEBRA Solve each equation. Check your solution.
11. 3x + 1 = 7
12. 5x - 4 = 11
13. 4h + 6 = 22
14. 8n + 3 = -5
15. 37 = 4d + 5
16. 9 = 15 + 2p
17. 2n - 5 = 21
18. 3j - 9 = 12
19. -1 = 2r - 7
20. 12 = 5k - 8
23. 3 + _t = 35
2
w
- 4 = -7
26. _
8
y
21. 10 = 6 + _
7
p
24. 13 + _ = -4
3
c
27. 8 = _
+ 15
-3
n
22. 14 = 6 + _
5
k
25. _
- 10 = 3
5
b
28. -42 = _
+8
-4
29. POOLS There were 640 gallons of water in a 1600-gallon pool. Water is
being pumped into the pool at a rate of 320 gallons per hour. Solve
1600 = 320t + 640 to find how many hours it will take to fill the pool.
30. PHONE CARDS A telephone calling card allows for 25¢ per minute plus a
one-time service charge of 75¢. If the total cost of the card is $5, solve
25m + 75 = 500 to find the number of minutes you can use the card.
ALGEBRA Solve each equation. Check your solution.
31. 8 - t = -25
32. 3 - y = 13
33. 8 = -5 - b
34. 10 = -9 - x
35. 2w - 4w = -10
36. 3x - 5x = 22
37. x + 4x + 6 = 31
38. 5r + 3r - 6 = 10
39. 1 - 3y + y = 5
40. 16 = w - 2w + 9
41. 23 = 4t - 7 - t
42. -4 = -a + 8 - 2a
ALGEBRA Find each number.
43. Five more than twice a number is 27. Solve 2n + 5 = 27.
n
- 10 = 5.
44. Ten less than the quotient of a number and 2 is 5. Solve _
2
45. Three less than four times a number is -7. Solve 4n - 3 = -7.
n
+ 6 = -3.
46. Six more than the quotient of a number and 6 is -3. Solve _
6
EXTRA
PRACTICE
See pages 766, 796.
Self-Check Quiz at
pre-alg.com
H.O.T. Problems
47. PERSONAL CARE In nine visits to the styling salon, Andre had spent $169 for
haircuts. Of that amount, $16 was in tips. Write and solve an equation to
find how much Andre pays for each haircut before the tip.
48. BUSINESS Jarret bought old bikes at an auction for $350. He fixed them
and sold them for $50 each. He made a $6200 profit. Write and solve an
equation to determine how many bikes he sold.
49. CHALLENGE The model represents the
equation 6y + 1 = 3x + 1. What is the
value of x?
50. OPEN ENDED Write a two-step equation
that could be solved by using the
Addition and Multiplication Properties of
Equality.
51.
150 Chapter 3 Equations
Writing in Math
Use the information about solving equations
on page 147 to explain how algebra tiles can show the properties of
equality. Illustrate your reasoning by showing how to solve 2x + 3 = 7
using algebra tiles.
52. GRIDDABLE The cost to park at an art
fair is a flat rate plus a per-hour fee.
The graph shows the cost for parking
up to 4 hours. If x represents the
number of hours and y represents the
total cost, what is the cost in dollars
for 7 hours?
4
53. A local health club charges an initial
fee of $45 for the first month and then
a $32 fee each month after that. The
table shows the cost to join the health
club for up to 6 months. What is the
cost to join the health club for
10 months?
y
1
Cost (dollars) 45
Months
3
2
1
O
2
77
3
4
5
6
109 141 173 205
A $215
C $333
B $320
D $450
1 2 3 4 5 6 7 8x
ALGEBRA Solve each equation. Check your solution. (Lessons 3-3 and 3-4)
54. 5y = 60
55. 14 = -2n
x
56. _
= -9
57. x - 4 = -6
58. -13 = y + 5
59. 18 = 20 + x
3
ALGEBRA Simplify each expression. (Lesson 3-1)
60. 4(x + 1)
61. -5(y + 3)
62. 3(k - 10)
63. -9(y - 4)
64. 7(a - 2)
65. -8(r - 5)
66. ANALYZE TABLES The table shows the average game
attendance for three football teams in consecutive
years. What was the total change in attendance
from Year 1 to Year 2 for the Bobcats? (Lesson 2-3)
Team
Year 1
Year 2
Bobcats
Cheetahs
Wildcats
6234
7008
6873
5890
7162
6516
Determine whether a scatter plot of the data for the following might show
a positive, negative, or no relationship. (Lesson 1-7)
67. age and number of siblings
68. temperature and sales of sunscreen
PREREQUISITE SKILL Write an algebraic expression for each verbal expression.
(Lesson 1-3)
69. two times a number less six
70. the quotient of a number and 15
71. the difference between twice a number and 8
72. three times a number increased by 10
73. the sum of 2x, 7x, and 4
Lesson 3-5 Solving Two-Step Equations
151
Translating Verbal Problems into Equations
An important skill in algebra is translating verbal problems into equations.
Consider the following situation.
Jennifer is 6 years older than Akira. The sum of their ages is 20.
You can explore this problem situation by asking and answering questions.
Who is older? Jennifer
How many years older? 6
If Akira is x years old, how old is Jennifer? x + 6
You can summarize this information in an equation.
Words
Jennifer is 6 years older than Akira. The sum of their ages is 20.
Variable
Let x = Akira’s age.
Let x + 6 = Jennifer’s age.
Equation
x + ( x + 6) = 20
Exercises
For each verbal problem, answer the related questions.
1. Lucas is 5 inches taller than Tamika, and the sum of their heights is 137
inches.
a. Who is taller?
b. How many inches taller?
c . If x represents Tamika’s height, how tall is Lucas?
d. What expression represents the sum of their heights?
e. What equation represents the sentence the sum of their heights is 137?
2. There are five times as many students as teachers on the field trip, and the
sum of students and teachers is 132.
a. Are there more students or teachers?
b. How many times more?
c . If x represents the number of teachers, how many students are there?
d. What expression represents the sum of students and teachers?
e. What equation represents the sum of students and teachers is 132?
152 Chapter 3 Equations
3-6
Writing Two-Step Equations
Main Ideas
Logan collected pledges for the charity
walk-a-thon. He is receiving total
contributions of $68 plus $20 for every
mile that he walks. The table shows
how to find the total amount that
Logan could raise.
• Write verbal
sentences as two-step
equations.
• Solve verbal problems
by writing and solving
two-step equations.
Number
of Miles
Total Amount
Raised
0
20(0) + 68 = $68
4
20(4) + 68 = $148
6
20(6) + 68 = $188
10
20(10) + 68 = $268
16
20(16) + 68 = $388
a. Write an expression that
represents the amount Logan
can raise when he walks m miles.
b. Suppose Logan raised $308. Write and solve an equation to find
the number of miles Logan walked.
c. Why is your equation considered to be a two-step equation?
Review
Vocabulary
Expression any
combination of numbers
and operations; Example:
x - 3 (Lesson 1-2)
Write Two-Step Equations In Chapter 1, you learned how to write
verbal phrases as expressions.
Phrase
the sum of 20 times some number and 68
Expression
20n
+ 68
An equation is a statement that two expressions are equal. The
expressions are joined with an equals sign. You can write verbal
sentences as equations.
Sentence
The sum of 20 times some number and 68 is 308.
Equation
EXAMPLE
20n + 68
= 308
Translate Sentences into Equations
Translate each sentence into an equation.
Sentence
a. Six more than twice a number is -20.
Equation
2n + 6 = -20
b. Eighteen is 6 less than four times a number.
18 = 4n - 6
c. The quotient of a number and 5, increased
by 8, is equal to 14.
n
_
+ 8 = 14
5
1A. Four more than three times a number is -26.
1B. Twenty-four is 6 less than twice a number.
1C. The quotient of a number and 7, increased by 6, is equal to 12.
Extra Examples at pre-alg.com
Lesson 3-6 Writing Two-Step Equations
153
EXAMPLE
Translate and Solve an Equation
Seven more than three times a number is 31. Find the number.
Let n = the number.
Equations
Look for the words is,
equals, or is equal to
when you translate
sentences into
equations.
3n + 7 = 31
3n + 7 - 7 = 31 - 7
3n = 24
n=8
Write the equation.
Subtract 7 from each side.
Simplify.
Mentally divide each side by 3.
Therefore, the number is 8.
2. Translate the following sentence into an equation. Then find the number.
Eight less than three times a number is -23.
Two-Step Verbal Problems In some real-world situations you start with a
given amount and then increase it at a certain rate. These situations can be
represented by two-step equations.
CELL PHONES Suppose you are saving money to buy a cell phone that
costs $100. You have already saved $60 and plan to save $5 each week.
How many weeks will you need to save?
Explore
You have already saved $60. You plan to save $5 each week until you
have $100.
Plan
Organize the data for the first few
weeks in a table. Notice the pattern.
Write an equation to represent the
situation.
Let x = the number of weeks.
5x + 60 = 100
Solve
5x + 60 = 100
5x + 60 - 60 = 100 - 60
5x = 40
x=8
Week
Amount
0
1
2
3
5(0) + 60 = 60
5(1) + 60 = 65
5(2) + 60 = 70
5(3) + 60 = 75
Write the equation.
Subtract 60 from each side.
Simplify.
Mentally divide each side by 5.
You need to save $5 each week for 8 weeks.
Real-World Link
About 100 million cell
phones in the United
States are retired each
year.
Source: Inform Inc.
Check
If you save $5 each week for 8 weeks, you’ll have an additional $40.
The answer appears to be reasonable.
3. SHOPPING Jasmine bought 6 CDs, all at the same price. The tax on her
purchase was $7, and the total was $73. What was the price of each CD?
Personal Tutor at pre-alg.com
154 Chapter 3 Equations
Jim West/The Image Works
OLYMPICS In the 2004 Summer Olympics, the United States won 11 more
medals than Russia. Together they won 195 medals. How many medals
did the United States win?
Let x = number of medals won by Russia. Then x + 11 = number of medals
won by the United States.
Alternative
Method
Let x = number of
U.S. medals. Then let
x - 11 = number of
Russian medals.
x + (x - 11) = 195
x + (x + 11) = 195
Write the equation.
(x + x) + 11 = 195
Associative Property
2x + 11 = 195
Combine like terms.
2x + 11 - 11 = 195 - 11
Subtract 11 from each side.
x = 103
2x = 184
Simplify.
In this case, x is the
number of U.S.
medals, 103.
184
2x
_
=_
Divide each side by 2.
2
2
x = 92
Simplify.
Since x represents the number of medals won by Russia, Russia won
92 medals. The United States won 92 + 11 or 103 medals.
4. CAR WASH During the spring car wash, the Activities Club washed 14 fewer
cars than during the summer car wash. They washed a total of 96 cars
during both car washes. How many cars did they wash during the spring?
Examples 1, 2
(pp. 153–154)
Translate each sentence into an equation. Then find each number.
1. Three more than four times a number is 23.
2. Four less than twice a number is -2.
3. The quotient of a number and 3, less 8, is 16.
Solve each problem by writing and solving an equation.
Example 3
(p. 154)
Example 4
(p. 155)
HOMEWORK
HELP
For
See
Exercises Examples
6–11
1, 2
12, 13
3
14, 15
4
4. TEMPERATURE Suppose the current temperature is 17°F. It is expected to
rise 3°F each hour for the next several hours. In how many hours will the
temperature be 32°F?
5. AGES Lawana is five years older than her brother Cole. The sum of their
ages is 37. How old is Lawana?
Translate each sentence into an equation. Then find each number.
6. Seven more than twice a number is 17.
7. Twenty more than three times a number is -4.
8. Four less than three times a number is 20.
9. Eight less than ten times a number is 82.
10. Ten more than the quotient of a number and -2 is three.
11. The quotient of a number and -4, less 8, is -42.
Lesson 3-6 Writing Two-Step Equations
155
For Exercises 12–15, solve each problem by writing and solving an
equation.
12. WILDLIFE Your friend bought 3 bags of wild birdseed and an $18 bird
feeder. Each bag of birdseed costs the same amount. If your friend spent
$45, find the cost of one bag of birdseed.
13. TEMPERATURE The temperature is 8°F. It is expected to fall 5° each hour for
the next several hours. In how many hours will the temperature be -7°F?
14. POPULATION By 2020, California is expected to have 2 million more senior
citizens than Florida, and the sum of the number of senior citizens in the
two states is expected to be 12 million. Find the expected senior citizen
population of Florida in 2020.
Real-World Career
Meteorologist
A meteorologist uses
math to forecast the
weather and analyzes
how weather affects
air pollution and
agriculture.
15. BUILDINGS In New York City,
the Chrysler Building is 320 feet
taller than the Times Square
Tower. The combined height of
both buildings is 1772 feet. How
tall is the Times Square Tower?
Building
Height (ft)
Citigroup Center
915
Chrysler Building
?
Empire State Building
1250
Times Square Tower
?
Woolworth Building
792
Source: emporis.com
Translate each sentence into an equation. Then find each number.
16. If 5 is decreased by 3 times a number, the result is -4.
For more information, go
to pre-alg.com.
17. If 17 is decreased by twice a number, the result is 5.
18. Three times a number plus twice the number plus 1 is - 4.
19. Four times a number plus five more than three times the number is 47.
EXTRA
PRACTICE
See pages 767, 796.
Self-Check Quiz at
pre-alg.com
H.O.T. Problems
20. POPULATIONS Georgia’s Native-American population is 10,000 greater
than Mississippi’s. Mississippi’s Native-American population is 106,000
less than Texas’. If the total population of all three is 149,000, find each
state’s Native-American population.
21. CONSTRUCTION Henry is building a front door. The height of the door
is 1 foot more than twice its width. If the door is 7 feet high, what is
its width?
22. OPEN ENDED Write a two-step equation that has 6 as the solution. Write the
equation using both words and symbols.
23. FIND THE ERROR Alicia and Ben are translating the following sentence into
an equation: Three less than two times a number is 15. Who is correct?
Explain your reasoning.
Alicia
3 - 2x = 15
Ben
2x - 3 = 15
24. CHALLENGE If you begin with an even integer and count by two, you are
counting consecutive even integers. Write and solve an equation to find two
consecutive even integers whose sum is 50.
156 Chapter 3 Equations
Dwayne Newton/PhotoEdit
25. NUMBER SENSE The table shows the expected
population age 65 or older for certain states in 2030.
Use the data to write a problem that can be solved by
using a two-step equation.
26.
Population
(age 65 or older)
Number
State
(millions)
CA
8.3
Writing in Math
Explain how two-step equations
are used to solve real-world problems. Formulate a
problem situation that starts with a given amount
and then increases.
FL
7.8
TX
5.2
NY
3.9
Source: U.S. Census Bureau
27. An electrician charges $35 for a house
call and $80 per hour for each hour
worked. If the total charge was $915
to wire a new house, which equation
would you use to find the number of
hours n that the electrician worked?
28. You and your friend spent a total of
$15 for lunch. Your friend’s lunch cost
$3 more than yours did. How much
did you spend for lunch?
F $6
G $7
A 35n + 2n(80) = 915
H $8
B 80 + 35n = 915
J $9
C 35 + (80 - n) = 915
D 35 + 80n = 915
ALGEBRA Solve each equation. Check your solution. (Lessons 3-3, 3-4, and 3-5)
29. 6 - 2x = 10
30. -4x = -16
31. y - 7 = -3
32. 7y + 3 = -11
33. CONCERTS A concert ticket costs t dollars, a hamburger costs h dollars, and
soda costs s dollars. Write an expression that represents the total cost of a
ticket, hamburger, and soda for n people. (Lesson 3-1)
y
Name the ordered pair for each point graphed on the
coordinate plane at the right. (Lesson 2-6)
34. T
35. C
36. R
T
R
x
O
37. P
38. FOOD The SubShop had 36, 45, 41, and 38 customers
during the lunch hour the last four days. Find the
mean of the number of customers per day. (Lesson 2-5)
P
C
ALGEBRA Evaluate each expression if x = -12, y = 4, and z = -1. (Lesson 2-2)
39. ⎪x⎥ - 7
40.
⎪x⎥
+ ⎪y⎥
41. ⎪z⎥ - ⎪x⎥
42. ⎪y⎥ - ⎪x⎥ + ⎪z⎥
PREREQUISITE SKILL Find the next term in the pattern. (Lesson 1-1)
43. 5, 9, 13, 17, …
44. 326, 344, 362, 380, …
45. 20, 22, 26, 32, …
Lesson 3-6 Writing Two-Step Equations
157
3-7
Sequences and Equations
Main Ideas
• Describe sequences
using words and
symbols.
• Find terms of
arithmetic sequences.
The table shows the distance a
car moves during the time it
takes to apply the brakes and
while braking.
Speed
(mph)
New Vocabulary
a. What is the braking distance
for a car going 70 mph?
sequence
arithmetic sequence
term
common difference
b. What is the difference in
reaction distances for every
10-mph increase in speed?
Reaction
Braking
Distance (ft) Distance (ft)
20
20
20
30
30
45
40
40
80
50
50
125
60
60
180
c. Describe the braking distance
as speed increases.
Describing Sequences A sequence is an ordered list of numbers. An
arithmetic sequence is a sequence in which the difference between any
two consecutive terms is the same. So, you can find the next term in the
sequence by adding the same number to the previous term.
Each number is
called a term of
the sequence.
EXAMPLE
20,
+10
30,
40,
+10
50,
+10
60, … The difference is
called the
common difference.
+10
Describe an Arithmetic Sequence
Describe the sequence 4, 8, 12, 16, … using words and symbols.
+1
Substitute numbers
from the table to check
whether your equation
is true.
158 Chapter 3 Equations
+1
Term Number (n)
1
2
3
4
Term (t )
4
8
12
16
+4
Check Your
Answers
+1
+4
+4
The difference of the term numbers is 1. The terms have a common
difference of 4. Also, a term is 4 times the term number. The equation
t = 4n describes the sequence.
Describe each sequence using words and symbols.
1A. 10, 11, 12, 13, …
1B. 5, 10, 15, 20, …
Finding Terms Once you have described a sequence with a rule or
equation, you can use the rule to extend the pattern and find other terms.
EXAMPLE
Find a Term in an Arithmetic Sequence
Find the 15th term of 7, 10, 13, 16, … .
First write an equation that describes the sequence.
+1
+1
+1
Term Number (n)
1
2
3
4
Term (t )
7
10
13
16
+3
+3
The difference of the term numbers is 1.
The terms have a common difference of 3.
+3
The common difference is 3 times the difference in the term numbers.
This suggests that t = 3n. However, you need to add 4 to get the exact value
of t. Thus, t = 3n + 4.
CHECK If n = 2, then t = 3(2) + 4 or 10.
If n = 4, then t = 3(4) + 4 or 16.
To find the 15th term in the sequence, let n = 15 and solve for t.
t = 3n + 4
= 3(15) + 4 or 49
Write the equation.
So, the 15th term is 49.
2. Find the 20th term of 5, 8, 11, 14, … .
Personal Tutor at pre-alg.com
Real-World Link
The restaurant industry
employs about 12.2
million people, making
it the nation’s largest
employer outside of
government.
Source: restaurant.org
RESTAURANTS The diagram shows the
number of square tables needed to
seat 4, 6, or 8 people at a restaurant.
How many tables are needed to seat
16 people?
Make a table to organize your sequence and find a rule.
Number of Tables (t)
1
2
3
The difference of the term numbers is 1.
Number of People (p)
4
6
8
The terms have a common difference of 2.
The pattern in the table shows the equation p = 2t + 2.
If p = 2t + 2 or 16 = 2t + 2, then t = 7. So, seven tables are needed to seat
a party of 16.
3. RESTAURANTS Suppose the tables are shaped like hexagons. Find how
many tables are needed for a group of 22 diners.
Extra Examples at pre-alg.com
Don Tremain/Getty Images
Lesson 3-7 Sequences and Equations
159
Example 1
(p. 158)
Example 2
(p. 159)
Example 3
(p. 159)
Describe each sequence using words and symbols.
1. 2, 3, 4, 5, …
2. 6, 7, 8, 9, …
3. 3, 6, 9, 12, …
4. 7, 14, 21, 28, …
Write an equation that describes each sequence. Then find the
indicated term.
5. 10, 11, 12, 13, …; 10th term
6. 6, 12, 18, 24, …; 11th term
7. 2, 5, 8, 11, …; 20th term
8. 2, 6, 10, 14, …; 14th term
9. GEOMETRY Suppose each side of a square
has a length of 1 foot. Determine which
figure will have a perimeter of 60 feet.
&IGURE
HOMEWORK
HELP
For
See
Exercises Examples
10–21
1
22–29
2
30, 31
3
&IGURE
&IGURE
Describe each sequence using words and symbols.
10. 3, 4, 5, 6, …
11. 8, 9, 10, 11, …
12. 14, 15, 16, 17, …
13. 15, 16, 17, 18, …
14. 2, 4, 6, 8, …
15. 8, 16, 24, 32, …
16. 12, 24, 36, 48, …
17. 20, 40, 60, 80, …
18. 3, 5, 7, 9, …
19. 4, 6, 8, 10, …
20. 1, 4, 7, 10, …
21. 3, 7, 11, 15, …
Write an equation that describes each sequence. Then find the
indicated term.
22. 16, 17, 18, 19, …; 23rd term
23. 14, 15, 16, 17, …; 16th term
24. 4, 8, 12, 16, …; 13th term
25. 11, 22, 33, 44, …; 25th term
26. 7, 10, 13, 16, …; 20th term
27. 7, 9, 11, 13, …; 33rd term
28. 1, 5, 9, 13, …; 89th term
29. 3, 8, 13, 18, …; 70th term
30. GEOMETRY Study the pattern.
Which figure will have
40 squares?
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ˆ}ÕÀiÊÓ
ˆ}ÕÀiÊÎ
31. CONSTRUCTION A building frame consists of beams in the form of triangles.
The frame of a new office building will use 27 beams. Use the pattern
below to find the number of triangles that will be formed for the frame.
Î
EXTRA
PRACTICE
See pages 767, 796.
Self-Check Quiz at
pre-alg.com
x
Ç
THEATERS One section of a movie theater has 26 seats in the first row, 35
seats in the second row, 44 seats in the third row, and so on.
32. How many seats are in the eighth row?
33. If there are 10 rows of seats, how many seats are in the section?
160 Chapter 3 Equations
ANALYZE GRAPHS For Exercises 34 and 35, use the
graph.
34. Write an equation for the points (x, y) graphed at the
right. (Hint: Make a table of ordered pairs.)
Y
35. Find x when y is 101.
H.O.T. Problems
36. OPEN ENDED Write an arithmetic sequence whose
common difference is -8.
X
"
37. CHALLENGE Use an arithmetic sequence to find the
number of multiples of 6 between 41 and 523.
38.
Writing in Math
Explain how sequences can be used to make
predictions.
39. The expression 1 + 2n(n + 2)
describes a pattern of numbers. If n
represents a number’s position in
the sequence, which pattern does
the expression describe?
A 7, 17, 31, 49, 71, . . .
B 4, 7, 9, 17, 27, . . .
C 7, 17, 27, 31, 49, . . .
40. Use the pattern
Side Length
in the table to
1
find the
2
equation that
3
shows the
4
relationship
between the side
5
length s and
perimeter p of a pentagon.
D 7, 9, 17, 27, 31, . . .
Perimeter
F p=5+s
H s = 5p + 5
G p = 5s
J s=5+p
5
10
15
20
25
ALGEBRA Translate each sentence into an equation. (Lesson 3-6)
41. Five more than three times a number is 20.
42. Thirty-six is 8 less than twice a number.
43. The quotient of a number and -10, less 3, is -63
ALGEBRA Solve each equation. Check your solution. (Lesson 3-5)
44. 6 - 3x = 21
45. 4y - 3 = 25
46. -3 + 2 z = -19
47. SOCCER A ticket to a soccer game is $12, a team pennant is $7, and a T-shirt
is $15. Write two equivalent expressions for the total cost of a group outing
for 10 people if each person buys a ticket, a pennant, and a T-shirt. Then
find the cost. (Lesson 3-1)
48. WEATHER On Saturday, the temperature fell 10 degrees in 2 hours. Find the
integer that expresses the temperature change per hour. (Lesson 2-5)
PREREQUISITE SKILL Solve each equation. Check your solution. (Lesson 3-4)
49. 2x = -8
50. 15s = 75
51. 108 = 18x
52. 25z = 175
Lesson 3-7 Sequences and Equations
161
3-8
Using Formulas
Main Ideas
• Solve problems by
using formulas.
• Solve problems
involving the
perimeters and areas
of rectangles.
New Vocabulary
formula
perimeter
area
Reading Math
Formulas A formula is a
concise way to describe a
relationship among
quantities.
The top recorded speed of a mallard duck in
level flight is 65 miles per hour. You can make
a table to record the distances that a mallard
could fly at that rate.
Speed
(mph)
Time
(h)
Distance
(mi)
65
1
65
b. What disadvantage is there in showing
the data in a table?
65
2
130
65
3
195
c. Describe an easier way to summarize
the relationship between the speed, time,
and distance.
65
t
?
a. Write an expression for the distance
traveled by a duck in t hours.
Formulas A formula is an equation that shows a relationship among
certain quantities. A formula usually contains two or more variables.
One of the most commonly used formulas is d = rt, which shows the
relationship between distance d, rate (or speed) r, and time t.
SCIENCE What is the rate in miles per hour of a dolphin that
travels 120 miles in 4 hours?
Method 1 Substitute first.
d = rt
120 = r · 4
120 = r · 4
4
4
30 = r
Method 2 Solve for r first.
Write the formula.
d = rt
Replace d with 120 and
t with 4.
t = t
Divide each side by t.
Divide each side by 4.
d =r
t
Simplify.
Simplify.
d
rt
120 = r
4
30 = r
Write the formula.
Replace d with 120 and
t with 4.
Simplify.
The dolphin travels at a rate of 30 miles per hour.
1. SCIENCE How long does it take a zebra to travel 160 miles at a speed
of 40 miles per hour?
162 Chapter 3 Equations
Getty Images
Perimeter and Area Formulas are commonly used in measurement. The
distance around a geometric figure is called the perimeter. One method of
finding the perimeter P of a rectangle is to add the measures of the four sides.
Perimeter of a Rectangle
Words
The perimeter of a rectangle is twice the sum of the length and width.
Symbols P = + + w + w
P = 2 + 2w or P = 2( + w)
EXAMPLE
Common
Misconception
Although the length of
a rectangle is usually
greater than the width,
it does not matter
which side you choose
to be the length.
ᐉ
Model
w
Find Perimeters and Lengths of Rectangles
a. Find the perimeter of the rectangle.
P = 2( + w)
11 in.
Write the formula.
5 in.
= 2(11 + 5) Replace with 11 and w with 5.
= 2(16)
Add 11 and 5.
= 32
Simplify. The perimeter is 32 inches.
b. The perimeter of a rectangle is 28 meters. Its width is 8 meters. Find
the length.
P = 2 + 2w
Write the formula.
28 = 2 + 2(8)
Replace P with 28 and w with 8.
28 = 2 + 16
Simplify.
28 - 16 = 2 + 16 - 16
12 = 2
6= Subtract 16 from each side.
Simplify.
Mentally divide each side by 2.
The length is 6 meters.
2A. Find the perimeter of a rectangle with length 15 meters and width
10 meters.
2B. The perimeter of a rectangle is 26 yards. Its length is 8 yards. Find the
width.
The measure of the surface enclosed by a figure is its area.
Area of a Rectangle
Words
The area of a rectangle is the product of the length and width.
Symbols A = w
ᐉ
Model
w
Extra Examples at pre-alg.com
Lesson 3-8 Using Formulas
163
EXAMPLE
Find Areas and Lengths of Rectangles
a. Find the area of a rectangle with length 15 meters and width
7 meters.
15 m
A = w
Write the formula.
= 15 · 7
Replace with 15 and w with 7.
= 105
Simplify.
7m
The area is 105 square meters.
b. The area of a rectangle is 45 square feet. Its length is 9 feet. Find its
width.
Method 1 Substitute, then solve
for the variable.
Method 2 Solve, then substitute.
A = w Write the formula.
45 = 9w
Replace A with 45 and with 9.
5=w
Mentally divide each side by 9.
A = w
Write the formula.
w
A
_
=_
Divide each side by .
A
_
=w
Simplify.
45
_
=w
Replace A with 45 and with 9.
9
5=w
Simplify.
The width is 5 feet.
3A. Find the area of a rectangle with a length of 14 inches and a width of
12 inches.
3B. The area of a rectangle is 198 square meters. Its width is 11 meters. Find
its length.
Personal Tutor at pre-alg.com
Example 1
(p. 162)
Examples 2, 3
(pp. 163–164)
1. ANIMALS How long would it take a bottlenose dolphin to swim 168 miles
at 12 miles per hour?
GEOMETRY Find the perimeter and area of each rectangle.
2.
3.
8 ft
3 ft
15 km
2 km
4. a rectangle with length 15 feet and width 6 feet
GEOMETRY Find the missing dimension in each rectangle.
5.
6.
12 in.
8m
w
Area 96 m
Perimeter 32 in.
164 Chapter 3 Equations
ᐉ
2
HOMEWORK
HELP
For
See
Exercises Examples
7–8
1
9–25
2, 3
7. TRAVEL Find the distance traveled by driving at 55 miles per hour for
3 hours.
8. BALLOONING What is the rate, in miles per hour, of a balloon that travels
60 miles in 4 hours?
GEOMETRY Find the perimeter and area of each rectangle.
9.
10.
3 mi
11.
9 cm
12 ft
5 ft
2 mi
12.
18 cm
13.
18 in.
14.
12 m
6m
50 in.
12 m
17 m
15. a rectangle that is 38 meters long and 10 meters wide
16. a square that is 5 meters on each side
GEOMETRY Find the missing dimension in each rectangle.
17.
18.
15 cm
Area 270 cm2
19.
w
Area 176 yd2
11 m
16 yd
Perimeter 70 m
ᐉ
20.
Freddy Adu became the
youngest professional
player in modern
American team sports
history when he joined
D.C. United at 14 years
of age. Soccer is played
on a rectangular field
that is usually 120 yards
long and 75 yards wide.
Source: sportsillustrated.
cnn.com
21.
7m
Real-World Link
ᐉ
w
w
Area 154 in2
22.
12 ft
Area 468 ft2 ᐉ
14 in.
Perimeter 24 m
23. SOCCER Find the perimeter and area of the soccer field described at the left.
24. COMMUNITY SERVICE Each participant in a community garden is allotted a
rectangular plot that measures 18 feet by 45 feet. How much fencing is
needed to enclose each plot?
25. LANDSCAPING Jordan is paving a rectangular patio with bricks. If the patio
contains a total of 308 bricks and there are 22 bricks running along the
length of the patio, how many bricks run along the width of the patio?
For Exercises 26 and 27, translate each sentence into a formula.
26. PROFITS The profit made during a year p is equal to sales s minus costs c.
27. GEOMETRY In a circle, the diameter d is twice the length of the radius r.
Lesson 3-8 Using Formulas
Thanassis Stavrakis/AP/Wide World Photos
165
28. RUNNING The stride rate r of a runner is the
number of strides n (or long steps) that he or
she takes divided by the amount of time
Runner
Number of
Strides
Time(s)
A
B
20
30
5
10
n
. The best runners usually have the
t, or r = _
t
greatest stride rate. Which runner has the
greater stride rate?
Using a
formula can
help you find
the cost of a vacation.
Visit pre-alg.com to
continue work on
your project.
GEOMETRY Draw and label the dimensions of each rectangle whose
perimeter and area are given.
29. P = 14 ft, A = 12 ft2
30. P = 16 m, A = 12 m2
31. P = 16 cm, A = 16 cm2
LANDSCAPING For Exercises 32 and 33,
use the figure at the right.
80 ft
32. What is the area of the lawn?
Lawn
33. Suppose your family wants to fertilize
the lawn that is shown. If one bag of
fertilizer covers 2500 square feet, how
many bags of fertilizer should
you buy?
House
28 ft x 50 ft
75 ft
Driveway
15 ft x 20 ft
34. TRAVEL An airplane flying at 500 miles
per hour leaves Minneapolis. One-half
hour later, a second airplane leaves
Minneapolis flying in the same direction at a rate of 600 miles per hour.
How long will it take the second airplane to overtake the first?
GEOMETRY Find the area of each rectangle.
35.
Y
!
X
"
$
36.
"
Y %
X
"
#
(
EXTRA
PRACTICE
See pages 767, 795.
Self-Check Quiz at
pre-alg.com
H.O.T. Problems
&
'
BICYCLING For Exercises 37 and 38, use the following information.
American Lance Armstrong won the 2005 Tour de France, completing the
2102-mile race in 83 hours, 36 minutes, 2 seconds.
37. Estimate Armstrong’s average rate in miles per hour for the race.
38. Armstrong also won the 2003 Tour de France. He completed the 2125-mile
race in 80 hours, 2 minutes, 8 seconds. Without calculating, determine
which race was completed with a faster average speed. Explain.
39. OPEN ENDED Draw and label a rectangle that has a perimeter of 18 inches.
40. CHALLENGE Is it sometimes, always, or never true that the perimeter of a
rectangle is numerically greater than its area? Give an example to justify
your answer.
166 Chapter 3 Equations
41. REASONING A rectangle has width w. Its length is one less than twice
its width. Write an expression in simplest form for its perimeter.
42.
Writing in Math
Explain why formulas are important in math and
science. Include an example of a formula from math or science that you
have used and an explanation of how you used the formula.
43. The formula d = rt can be rewritten
as dt = r. How is the rate affected if
the time t increases and the distance d
remains the same?
44. The area of each square
in the figure is 16
square units. Find the
perimeter.
A It increases.
F 16 units
B It decreases.
G 32 units
C It remains the same.
H 48 units
D There is not enough information.
J 64 units
Write an equation that describes each sequence. Then find the indicated
term. (Lesson 3-7)
45. 11, 12, 13, 14, …; 60th term
46. 4, 11, 18, 25, …; 100th term
47. Eight more than five times a number is 78. Find the number. (Lesson 3-6)
LIGHT BULBS The table shows the average life of an
incandescent bulb for selected years. (Lesson 1-6)
Incandescent Light Bulbs
1200
Hours
49. State the domain and the range of the relation.
1500
1500
48. Write a set of ordered pairs for the data.
1000
900
600
600
300
0
14
1870
1881 1910
Years
2000
Math and Technology
I Need a Vacation! It’s time to complete your Internet project. Use the information and data you have
gathered about the costs of lodging, transportation, and entertainment for each of the vacations. Prepare a
brochure or Web page to present your project. Be sure to include graphs and/or tables in the presentation.
Cross-Curricular Project at pre-alg.com
Lesson 3-8 Using Formulas
167
EXTEND
Spreadsheet Lab
3-8
Perimeter and Area
A spreadsheet allows you to use formulas to investigate problems. When you
change a numerical value in a cell, the spreadsheet recalculates the formula
and automatically updates the results.
EXAMPLE
Suppose a gardener wants to enclose a rectangular garden using part of a
wall as one side and 20 feet of fencing for the other three sides. What are
the dimensions of the largest garden she can enclose?
If s represents the length of each side attached to the wall, 20 - 2s represents the length of the
side opposite the wall. These values are listed in column B. The areas are listed in column C.
'ARDENDIMENSIONSXLS
!
"
,ENGTHOF3IDE
/PPOSITE7ALL
3HEET
#
,ENGTHOF&ENCE
,ENGTHOF3IDE
!TTACHEDTO7ALL
3HEET
4HESPREADSHEETEVALUATES
THEFORMULA"!
!REA
4HESPREADSHEETEVALUATES
THEFORMULA!"
3HEET
The greatest possible area is 50 square feet. It occurs when the length of each side
attached to the wall is 5 feet, and the length of the side opposite the wall is 10 feet.
ANALYZE THE RESULTS
1. What is the area if the length of the side attached to the wall is 10 feet? 11 feet?
2. Are the answers to Exercise 1 reasonable? Justify your reasoning.
3. Suppose you want to find the greatest area that you can enclose with 30 feet
of fencing. Which cell should you modify to solve this problem?
4. Use a spreadsheet to find the dimensions of the greatest area you can
enclose with 40 feet, 50 feet, and 60 feet of fencing.
5. MAKE A CONJECTURE Use any pattern you may have observed in your answers to
Exercise 4 to find the dimensions of the greatest area you can enclose with 100 feet of
fencing. Explain.
168 Chapter 3 Equations
CH
APTER
3
Study Guide
and Review
Download Vocabulary
Review from pre-alg.com
Key Vocabulary
Be sure the following
Key Concepts are noted
in your Foldable.
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Key Concepts
Distributive Property
(Lesson 3-1)
• For any numbers a, b, and c, a(b + c) =
ab + ac.
Solving Equations
formula (p. 162)
inverse operations (p. 136)
like terms (p. 129)
perimeter (p. 163)
sequence (p. 158)
simplest form (p. 130)
term (p. 129, 158)
two-step equation (p. 147)
(Lessons 3-3 through 3-6)
• When you add or subtract the same number
from each side of an equation, the two sides
remain equal.
• When you multiply or divide each side of an
equation by the same nonzero number, the two
sides remain equal.
• To solve a two-step equation, undo operations in
reverse order.
Sequences
area (p. 163)
arithmetic sequence (p. 158)
coefficient (p. 129)
common difference (p. 158)
constant (p. 129)
Distributive Property (p. 124)
equivalent equations (p. 136)
equivalent expression (p. 124)
Vocabulary Check
Complete each sentence with the correct
term. Choose from the list above.
1. Terms that contain the same variables are
called _________.
2. The ________ of a geometric figure is the
measure of the distance around it.
3. In the term 4b, 4 is the ________ of the
expression.
(Lesson 3-7)
• An arithmetic sequence is a sequence in which
the difference between any two consecutive terms
is the same.
4. The equations x + 3 = 8 and x = 5 are
________ because they have the same
solution.
• A sequence can be described by a rule or
equation that can be used to extend the pattern
or find other terms in the pattern.
5. Addition and subtraction are ________
because they “undo” each other.
Perimeter and Area Formulas
6. In the expression 10x + 6, 6 is the
________ term.
(Lesson 3-8)
• The formula for the perimeter of a rectangle is
P = 2( + w).
• The formula for the area of a rectangle is
A = w.
Vocabulary Review at pre-alg.com
7. The measure of the surface enclosed by a
geometric figure is its ________.
8. A(n) ________ is an equation that shows a
relationship among certain quantities.
Chapter 3 Study Guide and Review
169
CH
A PT ER
3
Study Guide and Review
Lesson-by-Lesson Review
3–1
The Distributive Property
(pp. 124–128)
Use the Distributive Property to write
each expression as an equivalent
algebraic expression.
9. 3(h + 6)
10. 7(x + 2)
11. -5(k + 1)
12. -2(a + 8)
13. (b - 4)(-2)
14. (y - 3)(-6)
Example 1 Use the Distributive Property
to write 2(t - 3) as an equivalent
algebraic expression.
2(t - 3) = 2[t + (-3)] Rewrite t - 3 as t + (-3).
15. BOWLING At the Bowling Palace, shoe
rental is $3.00 and each game is $2.50.
Write two equivalent expressions
for the cost of a group of 3 people to
rent shoes and play 2 games.
3–2
Simplifying Algebraic Expressions
Distributive Property
= 2t + (-6)
Simplify.
= 2t - 6
Definition of subtraction
(pp. 129–133)
Simplify each expression.
16. 4a + 5a
17. 3y + 7 + y
Example 2 Simplify 9x + 3 - 7x.
9x + 3 - 7x
18. x - 10 - 3x + 9
19. 4(m - 4) + 2
= 9x + 3 + (-7x) Definition of subtraction
20. 6w + 2(w + 9)
21. 8(n - 1) - 10n
= 9x + (-7x) + 3 Commutative Property
22. BASKETBALL Karen made 5 less than 4
times the number of free throws that
Kimi made. Write an expression in
simplest form that represents the total
number of free throws made.
3–3
= 2t + 2(-3)
Solving Equations by Adding or Subtracting
Solve each equation. Check your
solution.
23. t + 5 = 8
24. 12 = b + 4
25. z - 10 = -6
26. a + 12 = -16
27. k - 1 = 4
28. -7 = n - 6
29. REPORTS Sonia needs to add 13 more
pages to complete an assignment that is
supposed to be 37 pages long. Write
and solve an equation to find how
many pages she has already completed.
170 Chapter 3 Equations
= [9 + (-7)]x + 3 Distributive Property
= 2x + 3
Simplify.
(pp. 136–140)
Example 3 Solve x + 3 = 7.
x+3=7
Write the equation.
x + 3 - 3 = 7 - 3 Subtract 3 from each side.
x=4
Simplify.
Example 4 Solve y - 5 = -2.
y - 5 = -2
Write the equation.
y - 5 + 5 = -2 + 5 Add 5 to each side.
y=3
Simplify.
Mixed Problem Solving
For mixed problem-solving practice,
see page 796.
3–4
Solving Equations by Multiplying or Dividing
Solve each equation. Check your
solution.
30. 6n = 48
31. -3x = 30
r = -2
32. -5
d
33. 22 = -3
34. FASHION Rosa is making scarves for her
friends. Each scarf requires 48 inches of
material. Write and solve an equation to
find how many scarves Rosa can make
if she has 336 inches of material.
(pp. 141–145)
Example 5 Solve -5x = -30.
-5x = -30
Write the equation.
-5x
= -30
-5
-5
Divide each side by 5.
x=6
Simplify.
a = 3.
Example 6 Solve -8
a =3
Write the equation.
-8
a = -8(3)
-8 -8 a = -24
3–5
Solving Two-Step Equations
3
38. _c - 3 = 2
9
Writing Two-Step Equations
Example 7 Solve 6k - 4 = 14.
6k - 4 = 14
Write the equation.
6k - 4 + 4 = 14 + 4
Add 4 to each side.
6k = 18
39. BOOKS Dion’s favorite book is 35 pages
longer than Eva’s favorite book. The
number of pages in both books is 271.
Solve x + x + 35 = 271 to find the
number of pages in Dion’s book.
3–6
Simplify.
(pp. 147–151)
Solve each equation. Check your
solution.
35. 6 + 2y = 8
36. 3n - 5 = -17
37. _t + 4 = 2
Multiply each side by -8.
k=3
Simplify.
Divide each side by 6.
(pp. 154–157)
Translate each sentence into an equation.
Then find each number.
40. Three more than twice a number is 53.
41. Six less than the quotient of a number
and 4 is -3.
42. MONEY Suppose you are saving money
to buy a digital video camera that costs
$340. You have already saved $120 and
plan to save $20 each week. How many
weeks will you need to save?
Example 8 Translate the sentence into an
equation. Then find the number.
Seven less than three times a number is -22.
3n - 7 = -22
Write the equation.
3n - 7 + 7 = -22 + 7
Add 7 to each side.
3n = -15
n = -5
Simplify.
Divide each side by 3.
Chapter 3 Study Guide and Review
171
CH
A PT ER
3
3–7
Study Guide and Review
Sequences and Equations
(pp. 158–161)
Describe each sequence using words
and symbols.
43. 5, 6, 7, 8, …
44. 14, 15, 16, 17, …
45. 6, 12, 18, 24, …
46. 10, 20, 30, 40, …
Write an equation that describes each
sequence. Then find the indicated term.
47. 8, 9, 10, 11, …; 19th term
48. 6, 10, 14, 18, …; 47th term
49. 7, 14, 21, 28, …; 70th term
50. GEOMETRY Which figure in the pattern
below will have 99 squares?
3–8
ˆ}ÕÀiÊ£
ˆ}ÕÀiÊÓ
Using Formulas
(pp. 162–167)
Example 9 Find the 35th term of 9, 18,
27, 36, … .
Term Number (n)
1
2
3
4
Term (t)
9
18
27
36
The common difference is 9. Each term is 9
times the term number. So, t = 9n.
t = 9n
Write the equation.
t = 9(35) Replace n with 35.
t = 315
Simplify.
The 35th term of the sequence is 315.
ˆ}ÕÀiÊÎ
Find the perimeter and area of each
rectangle.
51.
Example 10 Find the perimeter and area
of a 14-meter by 6-meter rectangle.
P = 2( + w) Formula for perimeter
= 2(14 + 6) Replace with 14 and w with 6.
= 40 m
™Êˆ˜°
™Êˆ˜°
52. a rectangle with length 8 feet and
width 9 feet
53. a square with sides that are 2.5 yards
54. SWIMMING Karl’s rectangular pool
is 15 feet by 9.5 feet. What are the
perimeter and area of the pool?
55. TYPING Toya typed a 1140-word essay
in 12 minutes. At what rate is she
typing? (Hint: Typing is measured
in words per minute.)
172 Chapter 3 Equations
A=·w
= 14 · 6
= 84
m2
Simplify.
Formula for area
Replace with 14 and w with 6.
Simplify.
Example 11 The speed of light is
299,792,458 meters per second. Use the
formula d = rt to calculate how far light
travels in one minute.
d=r·t
Write the formula.
= 299,792,458 · 60
Substitute.
= 17,987,547,480
Simplify.
CH
A PT ER
3
Practice Test
ENTERTAINMENT Suppose you pay $15 per hour
to go horseback riding. You ride 2 hours today
and plan to ride 4 more hours this weekend.
1. Write two different expressions to find the
total cost of horseback riding.
2. Find the total cost.
Simplify each expression.
3. x + 3x
4. 9x + 5 - x + 3
5. 10(y + 3) - 4y
6. -7b - 5(b - 4)
22. MULTIPLE CHOICE A carpet store advertises
16 square yards of carpeting for $300, which
includes the $60 installation charge. Which
equation could be used to determine the
cost of one square yard of carpet x?
A 16x = 300
C 60x + 16 = 300
B x + 60 = 300
D 16x + 60 = 300
23. MULTIPLE CHOICE In the sequence below,
which expression can be used to find the
value of the term in the nth position?
7. MUSIC Omar and Deb each have a digital
music player. Deb has 37 more songs on
her player than Omar has on his player.
Write an expression in simplest form that
represents the total number of songs on
both players.
Position
Value
1
5
2
14
3
23
4
32
n
?
Solve each equation. Check your solution.
8. 19 = f + 5
9. -15 + z = 3
10. x - 7 = 16
11. g - 9 = -10
12. -8y = 72
n = -6
13. -30
14. 25 = 2d - 9
15. 4w - 18 = -34
16. 6v + 10 = -62
d +1
17. -7 = 18. x + 7 - 2x = 18
19. b - 7b + 6 = -30
F 5n
H 9n
G 5n + 4
J 9n - 4
24. Find the perimeter and area of the rectangle.
48 m
20 m
-5
20. TRAVEL Ms. Carter is renting a car from
an agency that charges $20 per day plus
$0.15 per mile. She has a budget of $80 per
day. Use the equation 80 = 20 + 0.15m to
find the maximum number of miles she can
drive each day.
21. GENETICS Approximately one-seventh of the
people in the world are left-handed. Write
and solve an equation to estimate how many
people in the United States are left-handed if
the population of the United States is about
300 million.
Chapter Test at pre-alg.com
25. MULTIPLE CHOICE The rectangle below has a
length of 20 centimeters and a perimeter of
P centimeters. Which equation could be
used to find the width of the rectangle?
ÓäÊV“
w
A P = 40 + 2
B P = 40 + 2w
C P = 20 + w
D P = 20 + 2w
Chapter 3 Practice Test
173
CH
A PT ER
3
Standardized Test Practice
Cumulative, Chapters 1–3
Read each question. Then fill in the correct
answer on the answer document provided by
your teacher or on a sheet of paper.
1. Which expression can be used to find the
nth term of the following sequence, where
n represents a number’s position in the
sequence?
4. Let n represent the position of a number in
the following sequence.
3
5 3
1 , 1 , , 1, , , 7 , 2, . . .
4 2 4
4 2 4
Which expression can be used to find any
term in the sequence?
n
F 4
Position in Sequence
1
2
3
4
n
n
G
2
Term
4
6
8
10
?
H 2n
J 3n
A 2n
B 2n + 1
5. A movie theater sells large boxes of popcorn
for $7.50, medium boxes of popcorn for
$5.75, and small boxes of popcorn for $3.75.
Suppose a group of friends orders 2 large
popcorns, 1 medium popcorn, and 2 small
popcorns. Which equation can be used to
find the total cost of the popcorn?
C 2n + 2
D 3n + 1
2. Mrs. Kelly’s deck has an area of
660 square feet.
A 2(7.5) + 5.75 + 2(3.75)
B (2 + 1 + 2)(7.5 + 5.75 + 3.75)
FT
FT
7.5 + 5.75 + 3.75
C (2 + 1 + 2) 3
D 2(7.5) + 5.75 + 3.75
What is the length of the deck if the width is
22 feet?
F 25 ft
H 32 ft
G 30 ft
J 35 ft
3. On Saturday the low temperature in Detroit,
Michigan, was -7°F, and the high temperature
was 23°F. How much warmer was the high
temperature than the low temperature?
A -30°F
B -16°F
6. GRIDDABLE The ordered pairs (-7, -2),
(-3, 5), and (-3, -2) are coordinates of
three vertices of a rectangle. What is the
y-coordinate of the ordered pair that
represents the fourth vertex?
7. Todd is 5 inches taller than his brother.
The sum of their heights is 139 inches.
Find Todd’s height.
F 67 in.
H 77 in.
G 72 in.
J 82 in.
8. Which expression represents the greatest
integer?
C 16°F
A ⎪4⎥
C ⎪-8⎥
D 30°F
B ⎪-3⎥
D -9
174 Chapter 3 Equations
Standardized Test Practice at pre-alg.com
Preparing for
Standardized Tests
For test-taking strategies and more practice,
see pages 809–826.
9. The scatter plot shows how many cups of
hot chocolate are sold on average at a ski
resort based on the outside temperature.
12. GRIDDABLE Six tables positioned in a row
will be used to display science projects. Each
table is 8 feet long. How many yards of
fabric are needed to make a banner that will
extend from one end of the row of tables to
the other?
œÌÊ
…œVœ>ÌiÊ->iÃ
Õ«ÃÊ-œ`Ê*iÀʜÕÀ
xx
xä
{x
{ä
Îx
Îä
Óx
Óä
£x
£ä
x
Y
ä
x yd
Óä Óx Îä Îx {ä {x
"ÕÌÈ`iÊ/i“«iÀ>ÌÕÀiÊ­®
13. Pedro is buying a DVD player that is on sale
for $49.89 plus tax. If he pays with a $100
bill, what other information is needed to
determine how much change he should
receive?
X
A The brand of the DVD player
Which description best represents the
relationship of the data?
F Negative trend
B The amount of money Pedro has in his
wallet
G No trend
C The amount of Pedro’s weekly income
H Positive trend
D The sales tax rate
J Cannot be determined
Pre-AP
10. Henry paid $32 to rent a table saw for a
two-day period. The maximum length of
time a customer can rent the saw is 10 days.
Which equation can be used to find c, the
cost of renting the table saw for the
maximum number of days?
A c = 32 10
C c = 32 5
B c = 2 32
D c = 32 15
Record your answers on a sheet of paper.
Show your work.
14. A basketball team can score 3-point baskets,
2-point baskets, and 1-point free throws.
Josh heard the Springdale Stars scored a
total of 63 points in their last game. Soledad
says that they made a total of two 3-point
baskets and 12 free throws in that game.
a. Write an equation to represent the total
points scored p. Use f for the number of
free throws, g for the number of 2-point
baskets, and h for the number of
3-point baskets.
b. Can both Josh and Soledad be correct?
Explain your reasoning.
11. Find - 19 - (-10).
F -29
G -9
H9
J 29
Question 11 When a question involves
integers, check to be sure that you have
chosen an answer with the correct sign.
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Chapter 3 Standardized Test Practice
175