l
j
What You've Learned
California Content Standards
1.0 Identify and use the arithmetic properties of subsets of integers and
rational, irrational, and real numbers, including closure properties for the four
basic arithmetic operations.
1.1 Use properties of numbers to demonstrate whether assertions are true
or false.
25.1 Use properties of numbers to construct simple, valid arguments for, or
formulate counterexamples to, claimed assertions.
I•
Writing an Equation
for Help to the Lesson in green.
(Lesson 1-1)
Write an equation to model each situation.
1. The total cost of n cartons of milk is $3.60. Each carton costs $.45.
2. The perimeter of an equilateral triangle is 3 times the length of a sides. The
perimeter is 124 in.
Adding and Subtracting Real Numbers
1
(Lessons 1-4 and 1-5)
Simplify each expression.
3. 6 + ( -3)
4. -4- 6
5. -5 - ( -13)
7. -4.51 + 11.65
8. 8.5 - ( -7.9)
3
3
9• 10
- 4
Multiplying and Dividing Real Numbers
6. -7 + ( -1)
10.
~ + (-~)
(Lesson 1-6)
Simplify each expression.
11. -85
7
(
-5)
12. 7(-t4)
Combining Like Terms
13. 42 ( -6) 2
14. 22
17. 6t + 2 - 4t
18. 9x- 4 + 3x
7
(
-8)
(Lesson 1-7)
Simplify each expression.
15. 14k2 - ( -2k2)
Identifying Properties
16. 4xy + 9xy
(Lesson 1-8)
Name the property that each equation illustrates.
19. (6 . 8) . 3 = 6 . (8 . 3)
66
Chapter 2
20. 7(2 - 4) = 7 . 2 - 7 . 4
21. 21 + 15 = 15 + 21
A Math models and equations
are used to meet a variety of
industrial challenges. Some
relate to vehicle design and
production.
What You'll Learn Next
California Content Standards
5.0 Solve multi -step problems, including word problems, involving linear
equations in one variable and provide justification for each step.
15.0 Apply algebraic techniques to solve rate problems and percent
mixture problems.
25.0 Use properties of the number system to judge the validity of results, to
justify each step of a procedure, and to prove or disprove statements.
nish Audio Online
• Addition Property of Equality
(p. 68)
• consecutive integers (p. 101)
• cross products (p. 94)
• Division Property of Equality (p. 68)
• equivalent equations (p. 68)
• identity (p. 86)
• inverse operations (p. 68)
• Multiplication Property of Equality
(p. 68)
• proportion (p. 94)
• rate (p. 92)
•
•
•
•
set (p. 75)
solution of an equation (p. 68)
subset (p. 75)
Subtraction Property of Equality
(p. 68)
• uniform motion (p. 101)
Solving One-Step
Equations
foR USE WITH LESSON
2-1
Prepares for 5.0 Solve
A solution of an equation is the value (or values) of the variable that
makes the equation true. To find a solution, you can use properties of equality
to form equivalent equations. Equivalent equations are equations that
have the same solution (or solutions).
A ddition Property of Equality
Subtraction Property of Equality
For all real numbers a, b, and c,
if a = b, then a + c = b + c.
For all real numbers a, b, and c,
if a = b then a - c = b - c.
Example
8 = 5+3
8 +4 =5+3 +4
8=5+3
8 -2 =5+3 -2
Example
Multiplication Property of Equality
Division Property of Equality
For all real numbers a, b, and c,
if a= b, then a · c = b · c.
For all real numbers a, b, and c,
with c =I= 0, if a = b then ~ =
~
Example
=
multi-step problems
involving linear equations in
one variab le.
£.
Example 3
3
+1=4
3 + 1 - 4
2
- 2
~ -2 =3 ·2
One way to find the solution of an equation is to get the variable alone on one side of
the equal sign. You can do this using inverse operations, which are operations that
undo one another. Addition and subtraction are inverse operations. Multiplication
and division are also inverse operations.
EXAMPLE
Solving Using Inverse Operations
a. Solve x - 3 = - 8.
X-
3 + 3= -8 + 3
X = -5
c. Solve
g + 7 - 7 = 11 - 7
g = 4
Simplify.
ix = 9.
1(ix) = 1(9)
Subtract 7 from
each side of the equation.
Simplify.
d. Solve -96 = 4c.
Multiply each side by~,
the reciprocal of
x = 12
+ 7 = 11.
b. Solve g
Add 3 to each side
of the equation.
4-
i·
96 - 4c
- 4
- 24
=
c
o·lVI'd e eac h s1'd e by 4 .
Simplify.
Simplify.
Solve each equation.
1. x -8=0
5. b
+ 5 = -13
9. 67 = w- 65
68
Review
2. c - 4 = 9
3. - 4 = ~a
4. -8n = -64
6. 6 =X + 2
7. - 7y = 28
8. -101 = -~
10. 5b = 145
Solving One-Step Equations
11.
9=
12
12. -4 = k
+ 19
Solving Two-Step
Equations
California Content Standards
5.0 Solve multi-step problems, including word problems, involving linear equations in one
variable and provide justification for each step. Introduce
25.0 Use properties of the number system to justify each step of a procedure. Develop
I
What You'll Learn
• To solve two-step equations
(f! Check Skills You'll Need
Review page 68
Solve each equation and tell which property of equality you used.
• To use deductive reasoning
1.
.. . And Why
4. x - 13 = 20
To solve a problem involving
ordering from a catalog, as in
Example 2
for Help
X -
5 = 14
2.
5.
X
+ 3.8 = 9
3. -7 +
= 8
6. 9 = 3x
*
X
= 7
Solve each equation.
7. lOx = 2
8. ~X= -6
3- 612
9• X+ 24-
Solving Two-Step Eqliations
A two-step equation involves two operations. Models can help you understand
how to solve equations. In the model below, since • represents -1 and _j
represents 1, • 0 is a zero pair representing zero. A green rectangle represents
a variable.
3x- 2
= 4
•I
•
'-J :_j
f_j _j
Model the equation.
I
• :_J _j ._j ,_J
_j :_j !_j f_j
3x- 2 + 2 = 4 + 2
_j i_j -..:J
_j l~ ._j
Simplify by removing
zero pairs.
• 18W
Divide each side into
three equal groups.
3x = 6
3x _ 6
3 - 3
x=2
Add 2 to each side.
I
l_ j
_j
Each green rectangle
equals two yellow squares,
SO X=
2.
Using the addition, subtraction, multiplication, and division properties of equality,
you can form equivalent equations. The equations 3x - 2 = 4, 3x = 6, and x = 2 are
all equivalent equations.
To solve equations, you use the properties of equality repeatedly to get the term
with a variable alone on one side of the equation and with a coefficient of 1.
Lesson 2-1
Solving Two-Step Equations
69
,~_- , . .-••. ,.,~··-1:~-......-~.,;-~.·"l"'-~'
~*-
., .
>••~,,. ,-~--~·-..' ·"~~-'·'-""·I-~J'o'lj".;,-;..;.._,,-e\'tf'""-:1'"~;< p;~;~t_\li,I'P~l'!!.!Y-.V'IIM......_., "'-
.P•-"'·,4.,.;.-.
:~ ,.;;.,_,,..~,._.
- ·.·:"-·:
-.-..~-
'
l
Solving a Two-Step Equation
!Jl + 2.
!fl + 2
!fl + 2 -
Solve 10 =
10 =
10 - 2 =
2
8 -- m
4
Simplify.
Multiply each side by 4.
4· 8= 4· !f/
32 = m
Check 10 =
10
:!:=
10
:!:=
Subtract 2 from each side.
Simplify.
!fl +
l
3
2
+2
Substitute 32 for m.
8 + 2
10 = 10 J'
{iffCA Standards Check
1 Solve each equation. Check your answer.
a. 7 = 2y - 3
b.~ - 15 = 12
c. -x + 15 = 12
You can write two-step equations to model real-world situations.
Application
You are ordering tulip bulbs from a flower catalog. You have $14 to spend. Use the
catalog page at the left to determine the number of bulbs that you can order.
Relate
cost per
tulip bulb
times
number of
tulip bulbs
plus
shipping
equals
amount to
spend
Define Let b = number of bulbs you can order.
Write
0.75
b
+
3
14
0.75b + 3 = 14
1
,-onps
o\ol)tor.\raor.o.\)•'\\'
""<•"-• •d"•t\~~~ \Ot 'lol)t •9t\t\ra
at\d
0.75b + 3 - 3 = 14 - 3
:~;;:~\\\ \U\\9• ~=~~~.\.
~~t\\\o\l••• \\\a\ 'fO
'\ ~u\~ !.O:t\l$3 00 sh\tlp\f\9)
(piUS
'
Subtract 3 from each side.
0.75b = 11
Simplify.
0.75b - 11
0.75 - 0.75
Divide each side by 0.75.
b = 14.6
Simplify.
You can order 14 bulbs.
Check Is the solution reasonable? You can only order whole tulip bulbs. Since
15 bulbs would cost 15 · $.75 = 11.25 plus $3 for shipping, which is more
than $14, you can only order 14 tulip bulbs.
{iffCA Standards Check
2 Suppose tulips are on sale for $.60 per bulb. What number of bulbs can you order?
In some equations, the variable may have a negative sign in front of it, such as
- x = 3. To help you solve these equations, recall that x = 1 · x and - x = -1 · x .
You can solve for x by multiplying by - 1 or dividing by -1.
70
Chapter 2 Solving Equations
Multiplying by Negative 1
Solve -b + 6 = -11.
-b+6=-11
-b + 6 - 6 = -11 - 6
-b =
-17~
-1 (-b) = -1 ( -17)
b = 17
-
@CA Standards Check
Q) Solve each equation.
a. -x + 7 = 12
Subtract 6 from each side.
Simplify.
Use the Multiplication Property of Equality.
Multiply each side by -1.
Simplify.
b. -a- 5
= -8
c. 4 = -c + 11
r -· . -______ . . . ,. -. . .. . . .. . . -- Using Deductive Reasoning
~-.,~--
~·····--····~
. ·- --- ------··---·----
~-
You can use deductive reasoning and the properties of equality to justify steps as
you solve an equation.
_
Using Deductive Reasoning
Solve 1 = 1~ + 5. Justify each step.
For help with deductive
reasoning, seep. 54.
Steps
Reasons
1 = 1~ + 5
original equation
1 - 5 = 1~ + 5 - 5
-4 = 1~
Subtraction Property of Equality
Simplify.
(12) ( -4) = (12) 1~
Multiplication Property of Equality
-48 = k
Simplify.
@CA Standards Check ~ Solve ~w + 9 = -1. Justify each step.
Remember that an expression like 8 - 3y can be rewritten as 8 + (- 3y) using the
definition of subtraction.
Using Deductive Reasoning
Solve 8 - 3y = 14. Justify each step.
Steps
8- 3y
Reasons
=
14
8+(-3y)=14
8 + ( -3y) - 8 = 14 - 8
-3y = 6
-3y
6
-3 = -3
y = -2
original equation
definition of subtraction
Subtraction Property of Equality
Simplify.
Division Property of Equality
Simplify.
@CA Standards Check ~ Solve -9- 4m = 3. Justify each step.
Lesson 2-1
Solving Two-Step Equations
71
For more exercises, see Extra Skills and Word Problem Practice.
Alg1 5.0, 25.0
- - - - -
C)
Practice by Example
Example 1
(page 70)
for
Help
Solve each equation. Check your answer.
1. 1 + ~ = -1
2. 2n - 5 = 7
4. ~
5. 3b
+ 5 = -12
7. 14 + ~
Example 2
(page 70)
=
3. -1 = 3 + 4x
8. -10 = -6 + 2c
2
~
6. 1- 9 = 0
9. i + 4 = 16
+ 7 = -2
10. ~- 21 = 7
11. 3x- 1 = 8
12. 10 = 2n + 1
13. 35 = 3 + Sx
14. 41 =~X- 7
15. -3 + }' = 12
~
Define a variable and write an equation for each situation. Then solve.
16. A library receives a large cash donation and uses the funds to double the
number of books it owns. Then a book collector gives the library 4028 books.
After this, the library has 51,514 books. How many books did the library have
before the cash donation and the gift of books?
17. Suppose you are helping to prepare a large meal. You can peel2 carrots per
minute. You need 60 peeled carrots. How long will it take you to finish if you
have already peeled 18 carrots?
---
Example 3
(page 71)
Examples 4, 5
(page 71)
Solve each equation. Check your solution.
18. - b + 5 = -16
19. - p - 24
21. -y'- 52= 33
22. -m + 2 = 1
=
20. - 7 = - c - 29
-8
23. -9 = -a + 16
Justify each step.
3+ 9 = 11
24.
3+ 9- 9 =
25.
-y - 5 + 5 = 11 + 5
11- 9
3=2
-y = 16
s(3) = 5(2)
-1( -y) = -1(16)
X=
10
27.
12 - 2h = 8
12 - 2h -
18 - n - 18 = 21 - 18
-n = 3
12 = 8 - 12
-2h = -4
-2h
-4
-1(-n) = -1(3)
-2
n = -3
Apply Your Skills
'I
= -16
y
18- n = 21
26.
-y - 5 = 11
=
-2
h=2
Solve each equation.
28.~x
+'
-!t- 2 = 4
29. ~ + 15 = 30
30.
31. -6 + 6z = 0
32. 3.5 + 10m = 7.32
33. 7 = -2x + 7
34.! = ~c - 3
35. 10.7 = -d + 4.3
36. 0.4x + 9.2 = 10
37. 4x + 92 = 100
38. -t - 0.4 = -3
39. -lOt- 4 = -30
= 3
Solve each equation. Justify each step.
40. 8 + ~4 = -6
72
Chapter 2
Solving Equations
41. 7 - 3k = -14
42.
i
+ 14
=
-1
Homework Video Tutor
43. Beneath Earth's surface, the temperature increases 10°C every kilometer.
Suppose that the surface temperature is 22°C, and the temperature at the
bottom of a coal mine is 45°C. Which equation could be used to find the depth
d of the coal mine?
co 45d - 10 = 22
(8) iod + 22 = 45
CQ) 22 = 10d + 45
® d = 22 + 10
Visit: PHSchool.com
Web Code: bae-020 1
Define a variable and write an equation for each situation. Then solve.
44. One health insurance policy pays people for claims by multiplying the
claim amount by 0.8 and then subtracting $500. Find the claim amount for
an insurance payment of $4650.
45. The Library of Congress in Washington, D.C., is the largest library in the world.
It contains nearly 128 million items. The library adds about 10,000 items to its
collection daily. Find the number of days it will take the library to reach about
150 million items.
Solve each equation.
46.
For Exercises 46-54, multiply
each side by the denominator
of the fraction as your first
step.
X
t
2 = 5
1
47. y ;
=
2
48. a
=lo = 2
7 3 = 12
49. b
27 = 6
50.
X
2 S = 10
51.
52.
! 4 = -8
53.
X
46 = -7
54. c -=-6 4 = 9
X
In each triangle, the measure of LA
Find the value of x.
55.
~A
56.
A
X
= the measure of LB.
58 o
(3x
+ It B
c (4~B
57.C ~~)'
\;t
c
B
58. Writing Miles has saved $40. He wants to buy a CD player for $129
in about four months. To find how much he should save each week,
he writes 40 + 16x = 129. Explain his equation.
Error Analysis What is the error in the work? Solve each equation correctly.
60.
59.
12~~
~1
l-
61. Write a problem that you can model with a two-step equation. Write an
equation and solve the problem.
62. Critical Thinking If you multiply each side of0.24r + 5.25 = -7.23 by 100, the
result is an equivalent equation. Explain why it might be helpful to do this.
nline Lesson Quiz Visit: PHSchool.com, Web Code: baa-0201
/
73
Use the table at the right for Exercises 63 and 64.
.63. A formula for converting
a temperature from
Celsius C to Fahrenheit F
is F = 1.8C + 32. Copy
and complete the table.
Round to the nearest
degree.
Description of
Temperature
Fahrenheit Celsius
212°
boiling point of water
I
r,
37°
68°
r
oo
human body temperature
room temperature
c
freezing point of water
64. a. Estimation Use the
r·
._j
-40°
formula F = 2C + 30
to estimate the
Fahrenheit temperatures not shown in the table.
b. Critical Thinking Compare your estimated values with the actual values.
How good is your formula at estimating the actual temperatures? Explain.
--
Challenge
Solve the first equation for x. Then substitute your result into the second equation
and solve for y.
65.x + y = 8
2x + 3y = 21
66.x + 2y = 1
6x- y = -20
67. X+ y
=
12
2x + y = 17
- .... ~~lliplg·, c;J;;I!PJ~~ -p~ilc~ir;e
For California Standards Tutorials, visit PHSchool.com. Web Code: baq-9045
Alg1 5.0
68. Last season, Everett scored 48 points. This is 6less than twice the number of
points Max scored. How many points did Max score?
®
Alg1 5.0
®
CD 21
16
69. A cable television company charges $24.95 a month for basic cable service and
$6.95 a month for each additional premium channel. If Sami's monthly bill is
$45.80, how many premium channels is he receiving?
®
Alg1 5.0, 25.0
CID 27
30
1
(]) 2
CD 3
®
4
70. To solve 9x + (4 - 3x) = 34, you first rewrite the equation as
9x + ( -3x + 4) = 34. What property justifies this step?
® Associative Property of Addition
CID Distributive Property
CD Inverse Property of Multiplication
® Commutative Property of Addition
'· Mixeit Review.
Lesson 1-6
for
Help
Simplify each expression.
71. -6( -4)
Lesson 1-4
Chapter 2
73. ~~
74.
76. 9 + ( -3)
77. -7 + ( -4)
78. 16 + ( -4)
Simplify.
75. -2 + 6
74
-5-9 31
72. 2( -5.5)
Solving Equations
The word set often indicates a group of things, like a set of dishes or set of
luggage. In mathematics, the word set indicates a well-defined collection of
elements, usually numbers. Below are two ways to write the set of integers.
Roster Notation
Set-Builder Notation
z = {... -3, -2, -1, 0, 1, 2, 3, ... }
Z = {x: xis an integer}
1.0 Identify and use the
arithmetic properties
of subsets of integers and
rational, irrational, and real
numbers, including closure
properties for the four basic
arithmetic operations. Master
The content within braces lists the elements of a set or tells how a set is built. You
read Z = {x: xis an integer} as "The set Z equals all the values of x such that xis an
integer." The notation 3 E Z means 3 is an element of the set Z. Set A = {1 , 2, 3} is
a subset of set Z because every element of set A is an element of set Z.
Write the set B of positive integers in roster notation and in set-builder notation.
B = {x: xis an integer and x > 0}
B={1 ,2,3,4, ... }
The intersection of two sets is the set of all elements that are common to both
sets. A n B means "A intersection B." The union of two sets is the set of all
elements that are in either of the sets. A U B means "A union B." The empty set, or
null set, has no elements, and is indicated by { } or 0.
a. Find C n D , where C = {9, 12, 15, 18, 21} and D = {x: xis a positive odd integer}.
C n D = {9, 15, 21} The numbers in set C that are positive odd integers are 9, 15, and 21.
b. Find P U Q, where P = {5, 10, 15, 20} and Q = {8, 10, 18, 20}.
List the numbers of both sets in order from least to
P U Q = {5, 8, 10, 15, 18, 20} greatest. If a number appears in both sets, list the
number only once.
EXERCISES
Write each set in roster notation and in set-builder notation.
1. set E = negative integers
2. set F = whole numbers greater than 10
3. set G = even prime numbers
4. set H = multiples of 3 less than 30
Let A = {2, 5, 8}, B = {5, 7, 9}, C = {2, 7, 8}, and D = {x: xis a positive even integer
less than 9}. Find the union and intersection of each of the following sets.
5. A andB
6. Band C
7. CandD
8. A and C
9. Band D
10. A and D
11. Is the set {-1, 0, 1} closed for addition and for multiplication? Explain.
Activity Lab
Set Notation
75
Solving Multi-Step Equations
California Content Standards
2.0 Understand and use such operations as finding the reciprocal. Develop
4.0 Simplify expressions before solving linear equations. Develop
5.0 Solve multi-step problems, including word problems, involving linear equations in one
variable and provide justification for each step. Develop
What You'll Learn
for Help
@ Check Skills You'll Need
• To use the Distributive
Property when combining
like terms
Lessons 1-2 and 1-7
Simplify each expression.
• To use the Distributive
Property when solving
equations
1. 2n- 3n
2. -4 + 3b + 2 + 5b
3. 9(w - 5)
4. -10(b - 12)
5. 3( -x + 4)
6. 5(6 - w)
Evaluate each expression.
. . . And Why
8. 8 +
7. 28 - a + 4a for a = 5
To solve a problem involving
construction, as in Example 2
X -
7X for X = -3
10.-(17 + 3y)fory = 6
9. (8n + 1)3 for n = -2
l.t)
'lilt
.....
<D
N
You can solve equations that require more than two steps. If there are like terms on
one side of an equation, first use the Distributive Property to combine them.
Combining Like Terms
Solve each equation.
a. 2c + c + 12 =
2c + c + 12 =
3c + 12 =
3c + 12 - 12 =
You use the Distributive
Property whenever you add
or subtract like terms.
x + 4x
= 1x + 4x
= (1 + 4)x
= Sx
78
78
78
Combine like terms.
78 - 12
Subtract 12 from each side.
3c = 66
Simplify.
3c _ 66
Divide each side by 3.
3 - 3
c = 22
Simplify.
b.8b + 16- 2b = 46
8b + 16 - 2b = 46
8b - 2b + 16 = 46
6b + 16 = 46
6b + 16 - 16 = 46 - 16
@CA Standards Check
76
Chapter 2
•
Use the Commutative Property of Addition.
Combine like terms.
Subtract 16 from each side.
6b = 30
Simplify.
6b - 30
Divide each side by 6.
6 - 6
b = 5
Simplify.
(1) Solve 7 = 4m - 2m + 1. Check your answer.
Solving Equations
l.t)
CD
You can model real-world situations using multi-step equations.
Application
X
X
A gardener is planning a rectangular garden area in a community garden.
His garden will be next to an existing 12-ft fence. The gardener has 44ft of
fencing to build the other three sides of his garden. How long will the garden
be if the width is 12 ft?
Cl0 32ft
® 24ft
© 22ft
® 16ft
Relate
length
of side
plus
12ft
plus
length
of side
equals amount of
fencing
Define Let x = length of a side adjacent to the fence.
Write
+
X
+
12
44
X
X+ 12 +X= 44
2x
2x
+ 12 = 44
Combine like terms on the left side
of the equation.
+ 12 - 12 = 44 - 12 Subtract 12 from each side.
2x = 32
2x _ 32
Simplify.
= 16
Simplify.
-z - -z
X
Divide each side by 2.
The garden will be 16 ft long. So D is the correct answer.
@CA Standards Check
@ A carpenter is building a rectangular fence for a playground. One side of the
playground is the wall of a building 70 ft wide. He plans to use 340 ft of fencing
material. What is the length of the playground if the width is 70 ft?
In the equation - 2(b - 4) = 12, the parentheses indicate multiplication. Use the
Distributive Property to multiply each term within the parentheses by -2. Then
use the properties of equality to solve the equation.
_
Solving an Equation With Grouping Symbols
Solve -2(b - 4) = 12.
Remember to multiply the
number outside the
parentheses by both terms
inside the parentheses.
-2(b - 4) = 12
- 2b + 8 = 12
- 2b + 8 - 8 = 12 - 8
-2b = 4
-2b
Subtract 8 from each side.
Simplify.
4
---=2 = ---=2
b = -2
@CA Standards Check
Use the Distributive Property.
® Solve each equation.
a. 3(k + 8) = 21
Divide each side by - 2.
Simplify.
b. 15 = -3(x - 1)
+9
Lesson 2-2 Solving Multi-Step Equations
77
3
Following are two ways you can solve an equation like 2 + ~ = 7.
Solving an Equation That Contains Fractions
Solve
I and ~x both
3+ ~ =
2
7.
Method 1 Adding fractions
represent x + 2.
2x
3
2x and lx both
3
3
represent 2x + 3.
+
:! _
2- 7
~x + ~x = 7
Rewrite the equation with fractions as coefficients.
~~ + ~~ =
Write the fractions with a denominator of 6.
~x
7
= 7
Combine like terms.
. ~(~ x) = ~(7)
Multiply each side
x = 6
by~. the reciprocal of~·
Simplify.
Method 2 Multiplying to clear fractions
2x + :! _ 7
3
26
(2x + :!) _
3
2
3) + 6(~)
6( 2
-
( )
6 7
= 6(7)
4x + 3x = 42
7x = 42
?]
~
@CA Standards Check
•
4j-
x = 6
Use the Distributive Property.
Multiply.
Combine like terms.
Divide each side by 7.
Simplify.
4 } Solve each equation. Explain why you chose the method you used.
m
m _ 5
a. 4 + 2 - 8
b 2
• 3X
-
85 X -_ 26
You can clear an equation of decimals by multiplying by a power of 10. In the
equation 0.5a + 8.75 = 13.25, the greatest number of digits to the right of a
decimal point is 2. To clear the equation of decimals, multiply each side of the
equation by 10 2 , or 100.
Powers of 10:
10 1 = 10
o
=
Multiply each side by 6,
a common multiple of 3 and 2.
oo
1 2= 1
10 3 = 1000
104 = 10,000
Solving an Equation That Contains Decimals
Solve 0.5a + 8.75 = 13.25.
100(0.5a + 8.75) = 100(13.25)
100(0.5a) + 100(8.75) = 100(13.25)
50a + 875 = 1325
50a + 875 - 875 = 1325 - 875
50a = 450
50a _ 450
so - so
•
a= 9
Multiply each side by 10 2, or 100.
Use the Distributive Property.
Simplify.
Subtract 875 from each side.
Simplify.
Divide each side by SO.
Simplify.
@CA Standards Check ~ Solve each equation.
a. 0.025x + 22.95 = 23.65
78
Chapter 2
Solving Equations
b. 1.2x - 3.6 + 0.3x = 2.4·
Keep the steps in the summary below in mind as you solve equations that have
variables on one side of the equation.
r
Steps for Solving a Multi-Step Equation
-----:-
·,.. ··
\l'•""\W'r:•\\:,·.:·'I"'Wll'-1
Step 1 Clear the equation of fractions and decimals.
I
Step 2 Use the Distributive Property to remove parentheses on
each side.
Step 3 Combine like terms on each side.
Step 4 Undo addition or subtraction.
Step 5 Undo multiplication or division.
For more exercises, see Extra Skills and Word Problem Practice.
0
Practice by Example
Example 1
for
Help
(page 76)
Solve each equation. Check your answer.
1. 4n - 2n =
2.y + y + 2 = 18
18
3. a + 6a - 9 = 30
5. 72
+ 4 - 14c
=
36
4. 5
- X - X=
6. 9
=
~
7. 7m - 3m - 6 = 6
9. -2y
Example 2
(page 77)
+ 5 + 5y
=
-1
-3 + n +_ 2n
._,.)f!
8. -13 = 2b - b - 10
10. - 3z + 8 + (-2z) = -12
14
Write an _equation to model each situation. Solve your equation.
11. Two friends are renting an apartment. They pay the landlord the first month's
rent. The landlord also requires them to pay an additional half of a month's
rent for a security deposit. The total amount they pay the landlord before
moving in is $1725. What is the monthly rent?
12. You are fencing a rectangular puppy kennel with 25 ft of fence. The side of the
kennel against your house does not need a fence. This side is 9 ft long. Find the
dimensions of the kennel.
Example 3
(page 77)
Solve each equation. Check your answer.
13. 2(8
+ p) = 22
16. 26
=/6(~a)
19. 48 = 89r
Example 4
(page 78)
'
+ 5(m - 1) = 7
17. m
+ 2)
20. 5(y - 3) = 19
22.7 - ; = ~
25 2
•3
15. 15 = -3(2q - 1)
14. 5(a - 1) = 35
23.
+ l 4k71
- 12
X -
26. 4
9y
_ 13
28· 21 + 7x
10 - 20
+
21. 5(2
+ 6) = -40
+ y) = 77
~ =~
24.
tg- - 7 = ~
i' = ~
27.
~ +! = 17
3
9
29· 14 + 7 = 14
f
18. -4(x
Lesson 2-2
30 1
•5
+ 3w __ 1
15 - 5
Solving Multi-Step Equations
79
Solve each equation. Check your answer.
Example 5
(page 78)
31. 3m
+ 4.5m = 15
33. 3.5 = 12s - 5s
35. 0.11p + 1.5
37. 1.12
+ 2 = 165.8
34. 1.06y - 3 = 0. 71
= 2.49
36. 25.24 = 5y
+ 1.25y = 8.62
39. 0.25m
Apply Your Skills
32. 7.8y
38. 1.025x
+ 0. 1m = 9.8
40. 0.36p
+ 3.89
+ 2.458
= 7.583
+ 0.26 = 3.86
Solve each equation.
41. 0.5t - 3t
+5 = 0
44. 0.5(x - 12) = 4
47.
2
+ !4 = ~
42. -(z
!Cm - 16)
51. 2(1.5c
ts + 1~ = {5
43.
45. 8y - (2y - 3) = 9
48.
50. 4x + 3.6 + x = 1.2
+ 5) = -14
=7
+ 4) = -1
46. ~
+y = ~
49. x
+ 3x - 7 = 29
52.26.54- p
= 0.5(50- p)
53. Error Analysis Explain the error in the student's work
at the right.
54. Critical Thinking Suppose you want to solve the
equation -3m + 4 + 5m = -6. What would you
do as your first step?
55. Writing To solve -!(3x - 5) = 7, you can use the
Distributive Property, or you can multiply each side of the equation by -2.
Which method do you prefer? Explain why.
The perimeter of each rectangle is 64 in. Find the value of x.
56.
57.
22 in.
t : t
2xin.
(3x
+ 2) in.
I
T
-t-
-I-
I
T
Use an equation to solve each problem.
58. John and two friends rent a canoe at a park. Each person must rent a life
jacket. If the bill for the rental of the canoe and life jackets is $41, for how
many hours did they rent the canoe?
59. The MacNeills rented a moving truck for $49.95 plus $.30 per mile. Before
returning the truck, they filled the tank with gasoline, which cost $18.32. The
total cost was $95.87. Find the number of miles the truck was driven.
60. Jane's cell phone plan is $40 per month plus $.15 per minute for each minute
over 200 minutes of call time. If Jane's cell phone bill is $58.00, for how many
extra calling minutes was she billed?
61. Write an expression with four terms that can be simplified to an expression
with two terms.
80
Chapter 2
Solving Equations
Homework Video Tutor
Find the value of x. (Hint: The sum of the measures of the angles of a
triangle is 180°.)
62.
64.
63.
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Web Code: bae-0202
(x
+
1St
44°
50°
65. The equation c = p
+ 0.07 p relates the price of an item p with its cost c after
7% tax is added. A student has $12.50. What is the cost of the most expensive
item he can buy?
66. Nadia has $70 in her bank account. Each week she deposits $3 from her
allowance and $15 from babysitting. The equation b = 70 + (15 + 3)w relates
her bank balance b to weeks w. When will she have $160 in her bank account?
Challenge
For Exercises 67-69, use an equation to solve each problem.
67. You fill your car's gas tank when it is about! empty. The next week, you
fill the tank a second time when it is abo~t ~ empty. If you buy a total of 18! gal
of gas on these two days, about how many gallons does the tank hold?
68. A work crew has two pumps, one new and one old. The new pump can fill a
tank in 5 hours. The old pump can fill the same tank in 7 hours.
a. How much of a tank can be filled in 1 hour with the new pump? With
the old pump?
b. Write an expression for the number of tanks the new pump can fill in t hours.
(Hint: Write the rate at which the new pump fills tanks as a fraction and then
multiply by t.)
c. Write an expression for the number of tanks the old pump can fill in t hours.
d. Write and solve an equation for the time it will take the pumps to fill one
tank if the pumps are used together.
69. Mr. Fairbanks invested half his money in land, a tenth in stock, and a twentieth
in bonds. He put the remaining $35,000 in a savings account. What is the total
amount of money that Mr. Fairbanks saved or invested?
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Alg1 2.0
70. Simplify the expression
®
Alg1 4.0
71. Solve 8n
®
Alg1 5.0
1o
3!
®
-i -
[ -(-~)].
-!
7
23
CD -20
®
-20
CD 6
®
i23
+ 5 - 2n = 41.
®
4!
72. If a number is increased by 3 and that number is doubled, the result is
-8. What was the original number?
®-7
®
-5.5
nline Lesson Quiz Visit: PH School. com, Web Code: baa-0202
CD 1
® 6
81
Alg1 5.0
73. The gas tank in Royston's car holds 12 gal of gasoline. The car averages
29 mi/gal. Royston filled up the tank and then drove 140 mi. About how many
gallons of gasoline are left in the tank?
CD 6 gal
CD 8 gal
® 7gal
® 9gal
Alg1 5.0
74. Josie 's goal is to run 40 miles each week. This week she has already run
distances of 5.3 miles, 6.5 miles, and 6.2 miles. If she wants to spread out the
remaining miles evenly over the next 4 days, which equation can you use to
find how many miles (m) per day she must run?
CD 5.3 + 6.5 + 6.2 + 40 = m
®
40 - 5.2 - 6.5 - 6.2 = m
CD 5.3 + 6.5 + 6.2 +
5.3 + 6.5 + 6.2 + m
®
Alg1 5.0
4m = 40
=
~0
75. A cell phone company charges $.35 for the first minute but only $.10 every
minute after that. Which equation can you use to find how many minutes m
Eric talked if the bill for the call was $5.45?
CD 0.35 + 0.10(m - 1) = 5.45
® 0.35 + 0.10m = 5.45
CD 0.10 + 0.35(m - 1) = 5.45
® 0.10 + 0.35m = 5.45
· ~iJ~ .. .IXf!
~, M
-
ii ~ R
· ~---~-··::·,:
_ -:·, ·,
,.-
~1. ~ ~ g~!~,"";_;.,,, . :.....
Lesson 2-1
Solve each equation.
76. 2y
for
Help
+ 4 = -6
78. -4n
80. 3x
+ 20 = 36
+ 5 = 12
82. 8m- 4 = 8
77. 3x - 15 = 33
79. -8 - c = 11
81. -4y - 3 = 15
+3
83. -p
=
10
84. One cell phone plan costs $39.95 per month. The first 500 minutes of usage are
free. Each minute thereafter costs $.35. Find the number of minutes of usage
over 500 minutes for a bill of $69.70.
Lesson 1-8
Mental Math Simplify each expression.
85. 14. 4. 25
87. -8
+ 15 - 9 + 2
89. 2. 8 . 5
Lessons 1-4 through 1-6
+ 28 + 34 + 72
88. 3 . 3 . 10
90. 27
+ 46 - 17 - 16
Simplify each expression.
91. 2- 6
92. -9. ( -3)
93. -9 + ( - 4)
94. 16
95. - 7
+ (- 3)
97. -5. 6
82
86. 16
Chapter 2 · Solving Equations
-7- (
-4)
96. -5 - ( -3)
98. -25
-7- (
- 5)
5.0 Solve multi-step
problems involving
linear equations in one
variable. Develop
25.0 Use properties of the
number system to justify
each step of a procedure.
Develop
Models can help you understand how to solve equations that have
variables on both sides.
Model and solve 3a - 2 = a + 4.
3a- 2 =a+ 4
3a - 2 - a = a + 4 - a
2a - 2 = 4
2a-2 +2 =4 +2
2a = 6
2a _ 6
2 - 2
a= 3
EXERCISES
Write an equation for each model. Use models to solve each equation.
1.
_j_j
_j_j
••
3. 1
2•
'_ j
4.
_j_j
_j_j
Use models to solve each equation.
+2
5. 4x + 2 = 2x + 6
6. 2y - 2
7. 2a + 2 = a + 8
8. Sb - 4 = 2b + 5
9. z - 8 = 2z - 1
10. 4(p + 1) = 2p - 2
11. Sn - 3
=
2(n + 3)
= 4y
12. 2(k + 1) = S(k - 2)
Activity Lab
Modeling Equations
83
Equations With Variables on
Both Sides
California Content Standards
4.0 Simplify expressions before solving linear equations. Develop
5.0 Solve multi-step problems, including word problems, involving linear equations in one
variable and provide justification for each step. Develop
25.3 Given a linear equation, determine whether the statement is true. Introduce
What You'll Learn
{iff Check Skills You'll Need
• To solve equations
with variables on
both sides
Simplify each expression.
• To identify equations that
are identities or have no
solution
Solve each equation.
1. 6x - 2x
5. 4x + 3
=
for Help
To solve a problem involving
skate rentals, as in Example 2
..>)~
4. -5x + 5x
3. 5x - 5x
2. 2x - 6x
6. -x + 7 = 12
- 5
8. 0 = -7n + 4 - 5n
7. 2t - 8t + 1 = 43
. .. AndWhy
lessons 1-7 and 2-2
New Vocabulary •
identity
CA Standards
Investigation
Using a Table to Solve an Equation
Costs for a key chain business are $540 to get started plus $3 per key chain.
The cost of producing k key chains is (540 + 3k) dollars. Key chains sell for
$7 each. The revenue for selling k key chains is 7 k dollars. To make a profit,
revenue must be greater than costs.
1. Copy and complete the following table.
Key Chains
Cost
Revenue
k
540 + 3k
7k
100
840
700
110
120
130
140
150
•
•
•
•
•
0
•
•
•
•
2. For 110 key chains, which is greater, the cost or the revenue?
3. When will the revenue be greater than the cost?
4. Use your table to estimate the solution of 540 + 3k = 7 k.
5. Explain how solving an equation can help you decide whether a business
can make a profit or not.
84
Chapter 2
Solving Equations
To solve an equation that has variables on both sides, use the Addition or
Subtraction Properties of Equality to get the variables on one side of
the equation.
Variables on Both Sides
Solve 6x + 3 = 8x - 21.
+ 3 = 8x - 21
6x + 3 - 6x = 8x - 21 - 6x
6x
3 = 2x - 21
3
+ 21 =
Subtract Gx from each side.
Combine like terms.
+ 21
2x - 21
Add 21 to each side.
Simplify.
24 = 2x
2
24 = 2
2
x = 12
Divide each side by 2.
Simplify.
The value of x is 12.
@CA Standards Check
G) Solve each equation.
a. -6d = d + 4
c. m- 5 =3m
b. 2(c - 6) = 9c + 2
d. 7 k - 4 = 5k + 16
Application
You can buy used in-line skates from your friend for $40, or you can rent some.
Either way, you must rent safety equipment. How many hours must you skate
for the cost of renting and buying skates to be the same?
Relate
cost of
friend's skates
plus
safety
equipment rental
equals
skates plus
equipment rental
Define Let h = the number of hours you must skate.
Write
+
40
40
+
Safety equipment
$1.50/hour
@CA Standards Check
3.5h
1.5h = 3.5h
40 + 1.5h - 1.5h
In-line skates and
safety equipment
$3.50/hour
1.5h
=
3.5h - 1.5h
Subtract 1.5h from each side.
40 = 2h
Combine like terms.
40 - 2h
-y - z-
Divide each side by 2.
20 = h
Simplify.
You must skate for 20 hours for the cost to be the same.
Check Is the solution reasonable? Buying skates and renting safety equipment
for 20 hours costs 40 + 1.5(20) = 70, or $70. The cost of renting both
skates and safety equipment for 20 hours is 3.5(20) = 70, or $70. The
answer is correct.
Q) A hairdresser is considering ordering a certain shampoo. Company A charges $4
per 8-oz bottle plus a $10 handling fee per order. Company B charges $3 per 8-oz
bottle plus a $25 ha.n dling fee per order. How many bottles must the hairdresser
buy to justify using Company B?
Lesson 2-3
Equations With Variables on Both Sides
85
-····-· ·-···" ---
r-····--'"·········~~·--·
-~----
----..-.. . . . . . . .
~
Special Cases: Identities and No Solutions
An equation has no solution if no value of the variable makes the equation true.
The equation 2x = 2x + 1 has no solution. An equation that is true for every value
of the variable is an identity. The equation 2x = 2x is an identity.
Identities and Equations With No Solutions
I
a. Solve 10 - 8a = 2(5 - 4a).
I~
10 - 8a = 2(5 - 4a)
--'·
. . . . _ _ _ _
...
- ·--
10 - 8a
= 10 -
8a
10 - 8a + 8a = 10 - 8a + 8a
10 = 10
For: Solving Equations Activity
Use: Interactive Textbook, 2-3
Use the Distributive Property.
Add Sa to each side.
Always
. true!
This equation is true for every value of a, so the equation is an identity.
b. Solve 6m - 5 = 7m + 7- m.
6m - 5 = 7m + 7 - m
6m- 5 = 6m + 7
@CA Standards Check
Combine like terms.
6m - 5 - 6m
=
6m + 7 - 6m
Subtract Gm from each side.
~
-5
=
7
Not true for any value for m!
e
This equation has no solution.
3) Determine whether 9 + 5n = 5n - 1 is an identity or whether it has no solution.
EXERCISES
For more exercises, see Extra Skills and Word Problem Practice.
Standards Practice
Q
Practice by Example
Alg1 4.0,
Solve each equation. Check your answer.
Example 1
1. 6x - 2 = x + 13
2. 5y - 3 = 2y + 12
(page 85)
3. 4k - 3 = 3k + 4
4. 5m + 3 = 3m + 9
5. 8- x = 2x- 1
6. 2n - 5 = 8n + 7
7. 3a + 4 = a + 18
8. 6b + 14 = -7 - b
for
Help
-5 + 8a
10. 3 + 4x = 3x + 6
11. 30 - 7z = 10z - 4
12. 8x - 3 = 7x + 2
13. -36 + 2w = -8w + w
14. 4p - 10 = p + 3p - 2p
9. Sa - 14
Example 2
(page 85)
s.o, 25.3
=
Write and solve an equation for each situation. Check your solution.
15. One phone company charges $16.95 per month and $.05 per minute for local
calls. Another charges $22.95 per month and $.02 per minute for local calls.
For what number of minutes of local calls per month is the cost the same?
16. One health club charges a $44 sign-up fee and $30 per month. Another health
club charges a $99 sign-up fee and $25 per month. For what number of months
is the cost the same?
86
Chapter 2
Solving Equations
Example 3
17. a. Use the equation 9 - 6x = 3(3 - 2x). Substitute four different values for x
and simplify.
b. What kind of equation is 9 - 6x = 3(3 - 2x)?
(page 86)
Determine whether each equation is an identity or whether it has no solution.
18. 14 - (2q
20. -8x
+ S) = - 2q + 9
+ 14 = -2(4x- 7)
22. a - 4a = 2a
0
Apply Your Skills
+ 1 - Sa
19. 6x
21. y 23. 9x
+ 1 = 6x - 8
s=
- (s
-
y)
+ 3x - 10 = 3(3x + x)
Solve each equation. If the equation is an identity, write identity. If it has no
solution, write no solution.
+ Sa = 2a + 9
24. 18x - 5 = 3(6x - 2)
25. 9
26. 3(x - 4) = 3x - 12
27. 6x = 4(x
28. ~ k -
lo k ! k + 1
+ S)
29. 0 = 0.98b + 0.02b - b
=
30. Sm - 2(m + 2) = -(2m + 1S)
7 - 4
6
31. 8
w - 8w + 8 w
32. A toy company spends $1SOO per day for factory expenses plus $8 to make each
toy. Each toy sells for $12. Which equation could be used to find the number of
toys t the company has to sell in one day to equal its daily cost?
®
®
1SOO + 8t = 12
12 + 8t = 1SOO
CD 1SOO + 8t = 12t
® 8t = 12t + 1SOO
33. A company manufactures tote bags. The company spends $1200 each day for
overhead expenses plus $9 per tote bag for labor and materials. The tote bags
sell for $2S each. How many tote bags must the company sell each day to
equal its daily costs for overhead, labor, and materials? Write an equation
and solve.
Find the value of x.
34.
35.
(12x
Vertical angles are congruent,
so their measures are equal.
+ 4t
(llx + 33t
(13x-
(lSx +
st
st
Error Analysis Find the mistake in the solution of each equation. Explain the
mistake and solve the equation correctly.
36.
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Web Code: bae-0203
37.
38. Writing Is an equation that has 0 for a solution the same as an equation with
no solution? Explain.
Lesson 2-3
Equations With Variables on Both Sides
87
39. Don set up a table to solve
5(x - 3) = 4 - 3(x + 1).
a. Does Don's table show a solution to
the equation?
b. Between which two values of x is the
solution to the equation? How do
you know?
c. For what values of xis 4 - 3(x + 1)
less than 5(x - 3)?
(t Challenge
6
3
-8
0
Write an equation with a variable on each side such that you get the given solution.
40.x = 0
41. x is a positive number.
42. x is a negative number.
43. All values of x are solutions.
44. No values of x are solutions.
45.x = 1
46. Use the equations below to find the length of the bat.
1-a-1
1-c--1
b
a+ b- 22 = c
b-a=4
a+ b = 32
47. The perimeters of the rectangles bel0w are equal. Find the length and width of
each rectangle.
lzx
I
4x
+2
D4x
4x
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Alg1 5.0
48. Ace Truck Rental charges $54.00 a day plus 9¢ per mile. Roni's Truck Rental
charges $38.00 a day plus 13¢ per mile. For how many miles will the cost of
renting a truck for one day at Ace equal the cost at Roni 's?
®
Alg1 4.0
Alg1 5.0
170mi
®
CD 400mi
418mi
49. Which equation is NOT equivalent to 3p - 2 = 6p + 4?
®
3p = 6p + 6
®
-6 = 3p
CD 3p = 6
® -3p- 2
= 4
50. A record store sells CDs for $12.00 each. A music club offers 5 free CDs and
charges $15.00 for each additional CD. Which equation can you use to find the
number of CDs x that would cost the same under both plans?
®
®
88
®
40mi
Chapter 2 . Solving Equations
15x - 5
=
12x
12x - 5 = 15x
CD 12x = 15(x - 5)
® 12(x - 5) = 15x
~
Alg1 5.0
51. Solve 2y = 3y - 20.
®
Alg1 4.0, 5.0
Alg1 4.0, 5.0
®
-2(y - 3) = -6y
CD y = -~
®
-2y- 6 = -6y
®
4y = -6
®
-1.6
®
CD 1.6
1.4
2.4
54. Which of the following is NOT a subset of rational numbers?
..-... IYJI~,~~~!ili i;I~~I~W... ~. ~,, .
Lesson 2-2
20
53. Solve 2(y - 3) = 1.2 - y.
®
®
for
Help
®
CD 4
-4
52. Which of the following equations is NOT equivalent to the others?
®
Alg1 4.0, 5.0
®
-20
CD {-1.3, V25,8.4}
® {-3, 0, ~}
{1 ,2,3, ... }
{-5, Vf2, 6.1}
·b.: .• 't·'v'k , "'""'·'"::
~··~ ...~.· ·..r• .~.,... ·~ '-~ ,.-;_.,
.•
.
-·
.
:,""~''''"'J
: /i.:' ...;;;.,, ' ·,
Solve each equation.
55. 9 = -4y
58.
+ 6y - 5
!x + 4 = ~
56. -2(a - 3) = 14
57. 0.5m
59. 4.8 = 1.25(y - 17)
60.
+_..2.8
=
3.64
4(i + X) = 5
Lesson 2-1
61. An art gallery owner is framing a painting. The width of the painting to be
displayed is 30 in. He wants the width of the framed painting to be 38! in.
How wide should each section of the frame be?
Lesson 1-3
Order the numbers in each group from least to greatest.
62 -~ -~ _1
. 5' 8' 5
63. 5.04, 5.009, 5.043
64. 8.1, 8.02, 8.3
65. -100,93, -87,500
66. 0.45, -1.24, 2.24, 1.23
67. 9.7, -9.8, 8.6, 0.9
Solve each equation.
1. 9
3.
+ 5d = 19
z; + i =
5. 4n
2. -x- 4 = -20
4. 6.85 - 0.25t = 10.6
2
+ 7 + 6n = 32
7. 8(h- 1) = 6h
+ 4 + 2h
6. 5(3 -d) = 2d
+1
8. 43 = 8y- 11
Write and solve an equation for each situation. Check your solution.
9. Peter was building a porch. Placing boards of equal length from end to end,
Peter found that 4 boards were 3 ft too long for the porch length, while
3 boards were 5 ft too short. How long was each board?
10. You and a pilot friend decide to rent an airplane to do some sightseeing. One
service charges $100 plus $80 per hour, while another charges $250 plus $70 per
hour for the same airplane. At what number of hours is the cost the same?
nline Lesson Quiz Visit: PHSchool.com, Web Code: baa-0203
89
Transforming Formulas
fOR USE WITH LESSON 2-4
A literal equation is an equation involving two or more variables. Formulas
are special types of literal equations. To transform a literal equation, you
solve for one variable in terms of the others. This is helpful when you need to
use a formula numerous times.
5.0 Solve mult i-step
problems, includ ing
word problems, and provide
j ustification for each step.
Develop
Transforming Geometric Formulas
Solve the formula for the area of a triangle A = !bh
for height h. Then find the height of the triangle
if the area is 48 in. 2 and the base is 4 in.
Step 1 Solve for h.
A= !bh
b
2A = 2(!)bh
Multiply each side by 2.
2A = bh
Simplify.
2A _ bh
Divide each side by b to get h alone on
one side of the equation.
b - b
t=
2
h
Step 2 Evaluate
t
2(48) - h
4
~
=
h for A = 48 and b = 4.
Substitute 48 for A and 4 for b.
-
964 - h
24 =
Simplify.
2
Simplify.
h
The height is 24 in.
Sometimes an equation will only have variables. Transforming this type of equation
is no different from transforming equations with numbers. The goal is to get the
variable you are solving for alone on one side of the equation.
Transforming Equations With Only Variables
Solve ab - d = c for b in terms of a, d, and c.
ab - d
+d
=
c+d
Add d to each side.
+d
Combine like terms.
ab = c
a/1
•
90
= c~ d
b = c ~ d
Divide each side by a, a
Simplify.
Review Transforming Formulas
=I=
0.
~
Transforming Formulas for Real-World Problems
The formula C = ~ (F - 32) gives the Celsius temperature C in terms of the
Fahrenheit temperature F. Transform the formula to find Fahrenheit temperature
in terms of Celsius temperature. Then find the Fahrenheit temperature when the
Celsius temperature is 30°.
Step 1 Solve for F.
C = ~(F- 32)
~ · C = ~ · ~(F - 32)
~c = F - 32
Multiply each side
by~. the reciprocal of i·
Simplify.
~C + 32 = F - 32 + 32
Add 32 to each side.
~c + 32 = F
Simplify.
Step 2 Find F when C
= 30.
~(30) + 32 = F
Substitute 30 for C.
Find ~{30).
54 + 32 = F
86 = F
30°C is equivalent to 86°F.
EXERCISES
1. a. Solve the formula V = Cwh for h.
b. Copy and complete the table to find the height for rectangular prisms with
the given volumes, lengths, and widths.
v
54 in. 3
64 in. 3
72 in. 3
90 in. 3
t
3 in.
4 in.
3 in.
18 in.
h
I
)-----
'
w
2 in.
4 in.
8 in.
2 in.
h
•
•
•
•
w
e
2. a. Bricklayers use the formula N = 7LH to estimate the number of bricks N
needed to build a wall of height H given a length L in feet. Transform the
formula to find the height of a wall in terms of the length and the number
of bricks.
b. What is the height of a wall that is 30 feet long and requires 2135 bricks?
3. You can use the number of chirps a cricket makes in one minute to estimate
the outside temperature Fin Fahrenheit. Transform the formula F = ~ + 37 to
find the number of chirps a cricket makes in a minute in terms of a given
temperature. How many chirps can you expect if the temperature is 60°F?
Review Transforming Formulas
91