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Lesson 16 ! page 1
Lesson 16
Solving Equations, Part 2
The equations we’ve solved so far have had one operation on the variable. Once we’ve undone that one operation, the
equation is solved. But equations come in more complicated forms, so let’s explore the next level.
Simplify First
Algebraic equations may have expressions that can be simplified on either side of the equals sign.
Example: Solve the equation 2x + 6x = 18 – 2.
Firs simplify each side. Then solve
as usual.
2x + 6x = 18 ! 2
8x = 16
8x
=
16
8
8
x =2
Example: The cost of your crafts booth is your cost per item times the number of items, plus the rental fee. Your
booth will have 600 items, and your cost for each is $5. The total cost of the booth is $4000. What is the rental fee?
The cost of your crafts booth = your cost per item • the number of items + the rental fee.
C
=
$4000
=
Y
•
$5 / item •
N
600 items
4000 = 5 • 600 + R
4000 = 3000 + R
R + 3000 = 4000
!3000 = !3000
R = 1000
The rental fee is $1000.
More Than One Operation
Once simplified, so far only one operation has been done with
the variable in our equation. Equations get more complicated
when two or more operations are involved.
We undo one operation with the opposite operation. When you
put on your shoe, the opposite operation is to take it off. When
you put on a sock and then a shoe, there are two operations that
must be removed. To take them off, first remove the shoe (which
you put on last), then remove the sock. In equations with more
than one operation on the variable, we undo first the operation
that was done last.
© 2010 Cheryl Wilcox
Putting Them On
Taking Them Off
+
R
+
R
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Lesson 16 ! page 2
Example: Solve the equation 2x + 1 = 19.
According to the order of operations, the variable, x, has first been multiplied by 2, and then 1 has been added. When
solving, we undo the operations in the opposite order they were done. First we undo adding 1 (by subtracting 1). Then we
undo multiplying by 2 (by dividing by 2).
Write the problem:
1 Begin by answering these questions to orient yourself:
What are we trying to find? Where is the variable?
State the operations in the order they are done to the variable.
“x is multiplied by 2, then 1 is added.”
2 Undo the last operation with its opposite.
3 Continue undoing operations in the opposite order they were done,
until the variable is alone on one side of the equation.
4 Check back in the original equation to be sure the answer makes sense.
2x + 1 = 19
2x+1 = 19
2x+1 – 1
2x
2x /2
x
=
=
=
=
19 – 1
18
18 /2
9
2(9) + 1 = 19 !
Example: Solve the equation 3(x + 5) = 21.
Here the order of operations requires that we first add 5 to x, and then multiply the result by 3. To solve, we undo the last
operation first, so we first divide by 3. Then we’ll subtract the 5.
3(x + 5) = 21
3 (x + 5)
=
21
3
3
x +5=7
!5 = !5
x =2
Some people prefer to simplify first, using the distributive property, then solve.
3(x + 5) = 21
3x + 15 = 21
!15 = !15
3x = 6
3x
=
6
3
3
x =2
As you can see, either method gives the same solution.
© 2010 Cheryl Wilcox
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Lesson 16 ! page 3
Example: Solve 3x/8 = 15.
Here the two operations are multiplication and division. In this situation, either operation can be undone first.
3x
= 15
8
3x
8•
= 15 • 8
8
3x = 120
3x
=
120
3
3
x = 40
There is a shortcut, in which we do both steps at once. You can use this if you like.
3x
= 15
8
8 3x
8
•
= 15 •
3
3 8
5
x=
15 8
• = 40
1 3
Example: The cost of a flower arrangement containing N flowers is $5 per flower plus a $45 arrangement fee. If the
arrangement you ordered cost $120, how many flowers did it contain?
cost of flower arrangement = price per flower • number of flowers + arrangement fee
C
=
$120
=
P
•
$5 / flower •
N
+
F
N flowers
+
$45
120 = 5N + 45
5N + 45 = 120
!45 = !45
5N = 75
5N
=
75
5
5
N = 15
The arrangement contains 15 flowers.
© 2010 Cheryl Wilcox
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Lesson 16 ! page 4
Simplifying and Solving
First simplify when possible, then solve, undoing operations in the reverse order that they were done.
Example: Solve the equation 2(4x +3) + 5(2x + 1) = 5 • 13.
Simplify each side first:
2(4x + 3) + 5(2x + 1) = 5 • 13
8x + 6 + 10x + 5 = 65
18x + 11= 65
Then solve the equation:
18x + 11= 65
!11= !11
18x = 54
18 x
=
54
18
18
x =3
Example: The perimeter of a rectangle is 44 inches. The length is 9 inches. What is the width?
Put the numbers you know into the formula,
Solve the equation:
P = 2L + 2W
44 = 2(9) + 2W
2W + 18 = 44
2W + 18 = 44
!18 = !18
2W = 26
2W
=
26
2
2
W = 13
The width is 13 inches.
© 2010 Cheryl Wilcox
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Lesson 16 ! page 5
The Answer Might Be a Fraction
If the numbers in the equation do not divide evenly, the answer is written as a fraction (in lowest terms). In algebra problems,
leave improper fractions alone – do not convert them to mixed numbers. In word problems, the answer should be written in
the most meaningful way for the context of the situation.
Example: Solve 5x + 2 = 20.
5x + 2 = 20
!2 = !2
5x = 18
5x
=
18
5
x=
18
5
5
Example: The bus traveled 252 miles at an average rate of 56 mph. How long did the trip take?
d = rt
252 = 56t
56t 252
=
56
56
252 ÷ 28 9
1
t=
= =4
56 ÷ 28 2
2
The trip took 41/2 hours.
!
© 2010 Cheryl Wilcox
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Lesson 16: Solving Equations, Part 2
Worksheet
Name _______________________________________
Fill in the blank, then solve the equation.
1. 5x + 7x = 36
2. 5x + 6 = 36
Begin by simplifying: first __________________________.
Here x is multiplied by 5, then 6 is added. To undo in the
reverse order we must first _______________________.
3. 5(x + 1) = 35
5.
4.
5x
= 35
3
6. 5(x + 2) + 3(x + 7) = 39
x !2
= 35
3
Starting with x, we first subtract 2, then divide by 3. To undo
in the reverse order, we first undo the division by
________________________.
Fill in the formula with the numbers you know. Then solve the equation to solve the problem.
1. The Cost of a crafts booth is the Price per item times the
Number of items plus the Rental fee. Find the number of
items if the cost per item is $4, the rental fee is $1500, and
the total cost is $5000.
C=P•N+R
© 2010 Cheryl Wilcox
2. The perimeter of a rectangle is 18 feet and the width is 3
feet. What is the length?
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Lesson 16 ! page 7
Make up your own equations by following these steps. Then challenge your partner by exchanging equations and solving.
Instructions
1 Pick a variable.
2 Decide what number the variable will be equal to. This will be the solution to your equation.
3 Write some operations with the number that you can undo later.
Example
x
x=5
3•5+9
4 Figure out the answer when you do your operations with the number.
3 • 5 + 9 = 24
5 Remove the number, and write the variable in its place.
3 • x + 9 = 24
6 Your equation is ready for someone else to solve. (You already know the solution is x = 5.)
3x + 9 = 24
Let each person in your group make up two equations with solutions on scratch paper. Write your equations on a group
member’s worksheet (without the solutions) and have them write their equations on yours. Solve, then check your answer
with the person who created the equation.
Equation 1:
Solution:
Equation 2:
Solution:
© 2010 Cheryl Wilcox
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Lesson 16 ! page 8
Lesson16: Solving Equations, Part 2
Homework 16A
1. Working with Fractions.
a. Write
Name _____________________________________
2. Simplifying Algebraic Expressions.
a. Multiply (14a )(3ab )
208
in lowest terms.
312
2
b. Multiply (5x )(8x )
b. Write
45ab
in lowest terms.
48b
c. Combine like terms to simplify 4x + 2x + 7x + 1
c. Write 5 fractions equivalent to
6
.
7
2
d. Change 8 to an improper fraction.
5
d. Combine like terms to simplify 10N + 5M + 6N + M
e. Use the distributive property to simplify 9(2M + 1)
(
f. Use the distributive property to simplify 2 b + 3
e. Change
67
to a mixed number.
9
g. Simplify 8a + 3(2a + 9)
f. Multiply
22a
b
•
5 110a
1
g. Multiply 15 •
5
© 2010 Cheryl Wilcox
h. Simplify 18 + 7(5x + 11)
i. Simplify 3(3x + 3) + 5(5x + 4)
)
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Lesson 16 ! page 9
3. Solve the equations.
4. Use a formula to solve the problem.
a. 17x = 102
a. The area of a rectangle is 630 square feet, and the length
is 35 feet. What is the width?
b. x ! 19 = 16
c. 12 =
x
6
b. The perimeter of a rectangle is 234 centimeters, and the
length is 56 centimeters. What is the width?
d. a + 99 = 100
e. 4(x + 3) = 20
c. The volume of a box is 11,594 cubic inches. The length is
17 inches, and the width is 22 inches. What is the height?
f. 17 = 2 + 3n
d. The superhero traveled 201 feet in 3 seconds. What was
her speed?
g.
7x
= 14
2
e. The hero’s evil nemesis traveled 200 feet at a rate of 50
feet per second. How long did that take?
h.
x +6
=2
5
© 2010 Cheryl Wilcox
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Lesson 16 ! page 10
Lesson16: Solving Equations, Part 2
Homework 16A Answers
1. Working with Fractions.
a. Write
2. Simplifying Algebraic Expressions.
a. Multiply (14a )(3ab )
208
in lowest terms.
312
(14a )(3ab ) = 14 • 3 • a • a • b = 42a 2b
208 2 • 2 • 2 • 2 • 13 2
=
=
312 2 • 2 • 2 • 3 • 13 3
b. Write
45ab
in lowest terms.
48b
(5x 2 )(8x ) = 5 • 8 • x • x • x = 40x 3
45ab
3 •3•5•a • b
15a
=
=
48b
16
2•2•2•2• 3 • b
c. Write 5 fractions equivalent to
6
.
7
f. Multiply
(
f. Use the distributive property to simplify 2 b + 3
67
to a mixed number.
9
2b + 6
7
4
9
22a
b
•
5 110a
2 • 11 • a
b
b
•
=
5
2 • 5 • 11 • a 25
g. Multiply 15 •
16N + 6M
18M + 9
2 40 2 42
=
+ =
5 5 5 5
67 ÷ 9 = 7r 4
13x + 1
e. Use the distributive property to simplify 9(2M + 1)
2
d. Change 8 to an improper fraction.
5
e. Change
c. Combine like terms to simplify 4x + 2x + 7x + 1
d. Combine like terms to simplify 10N + 5M + 6N + M
6 12 18 24 30 36
=
=
=
=
=
7 14 21 28 35 42
8
2
b. Multiply (5x )(8x )
g. Simplify 8a + 3(2a + 9)
8a + 6a + 27 = 14a + 27
h. Simplify 18 + 7(5x + 11)
18 + 35x + 77 = 35x + 95
i. Simplify 3(3x + 3) + 5(5x + 4)
1
5
9x + 9 + 25x + 20 = 34x + 29
3
15 1 3
•
= =3
1 5 1
© 2010 Cheryl Wilcox
)
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Lesson 16 ! page 11
3. Solve the equations.
4. Use a formula to solve the problem.
a. 17x = 102
a. The area of a rectangle is 630 square feet, and the length
is 35 feet. What is the width?
17 x
17
=
102
17
A = LW
x =6
35W
b. x ! 19 = 16
x ! 19 + 19 = 16 + 19
c. 12 =
x = 35
x
6
6 • 12 =
x
6
•6
x = 72
a + 99 ! 99 = 100 ! 99
a=1
e. 4(x + 3) = 20
4(x + 3) 20
=
4
4
x +3!3= 5!3
3n + 2 ! 2 = 17 ! 2
3n
15
=
3
3
2W
=
122
2
x =2
d = rt
3
H = 31 inches
=
201= r • 3
201
3
x=4
r = 67 feet per second
d = rt
50t
50
© 2010 Cheryl Wilcox
11594
374
e. The hero’s evil nemesis traveled 200 feet at a rate of 50
feet per second. How long did that take?
x +6
=2
h.
5
=2•5
5
x + 6 ! 6 = 10 ! 6
=
11594 = (17)(22)H
d. The superhero traveled 201 feet in 3 seconds. What was
her speed?
3r
2
x +6
W = 61 cm
c. The volume of a box is 11,594 cubic inches. The length is
17 inches, and the width is 22 inches. What is the height?
374
7x
= 14
2
5•
P = 2L + 2W
234 = 2(56) + 2W
2W + 112 ! 112 = 234 ! 112
2W = 122
374 H
3n = 15
W = 18 feet
b. The perimeter of a rectangle is 234 centimeters, and the
length is 56 centimeters. What is the width?
x +3=5
n=5
2 7x
2
•
= 14 •
7 2
7
630
35
V = LWH
f. 17 = 2 + 3n
g.
35
2
d. a + 99 = 100
=
630 = 35W
x + 6 = 10
x=4
=
200 = 50t
200
50
t = 4 seconds
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Lesson 16 ! page 12
Lesson16: Solving Equations, Part 2
Homework 16B
Name _____________________________________
1. Working with Fractions.
2. Simplifying Algebraic Expressions.
a. Write
a. Multiply (8n )(13mn )
90
in lowest terms.
255
2
2
b. Multiply (15a )(10a )
b. Write
63x
54x 2
in lowest terms.
c. Combine like terms to simplify 3M + 2M + 4M + 3
c. Write 5 fractions equivalent to
2
.
13
d. Combine like terms to simplify 2x + 9 y + 7 y + 4x
d. Change 11
e. Change
3
to an improper fraction.
8
74
to a mixed number.
7
e. Use the distributive property to simplify 5(4M + 12)
f. Use the distributive property to simplify 6(x + 1)
g. Simplify 22x + 7(10x + 9)
f. Multiply
15x 8 y
•
28 y 30
h. Simplify 60 + 10(40x + 80)
g. Multiply 40 •
3
8
© 2010 Cheryl Wilcox
i. Simplify 2(3x + 4) + 9(5x + 6)
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Lesson 16 ! page 13
3. Solve the equations.
4. Use a formula to solve the problem.
a. a + 7 = 29
a. The area of a rectangle is 1344 square feet, and the
length is 32 feet. What is the width?
b. 3x = 12
c. 22 =
x
8
b. The perimeter of a rectangle is 250 centimeters, and the
length is 75 centimeters. What is the width?
d. b ! 99 = 100
e. 3(2x + 1) = 33
c. The volume of a box is 2184 cubic inches. The length is
14 inches, and the width is 12 inches. What is the height?
f. 47 = 5 + 3n
g.
5x
= 10
3
x !5
= 15
h.
2
© 2010 Cheryl Wilcox
d. The riverboat traveled 90 miles in 5 hours. What was her
speed?
e. The Coast Guard patrol boat traveled 90 miles at a rate of
45mph. How long did that take?