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5.6
Using Multiplicative Inverses
to Solve Equations
Goal: Use multiplicative inverses to solve equations.
Vocabulary
Multiplicative inverse:
The multiplicative inverse of a nonzero
number is the number’s reciprocal.
Multiplicative Inverse Property
Words The product of a number and its multiplicative inverse is 1.
3
5
5
3
Numbers ᎏᎏ p ᎏᎏ 1
a
b
b
a
Algebra ᎏᎏ p ᎏᎏ 1, where a 0, b 0
Example 1
Solving a One-Step Equation
3
ᎏᎏ x 15
5
5
ᎏᎏ
3
冢冣
3
ᎏᎏ x ᎏ5ᎏ (15)
5
3
Original equation
Multiply each side by multiplicative
inverse of ᎏ3ᎏ .
5
1 x ᎏ5ᎏ (15)
3
x 25
Multiplicative inverse property
Multiply.
Answer: The solution is 25 .
Checkpoint Solve the equation. Check your solution.
6
11
7
13
1. ᎏᎏ x 18
2. ᎏᎏ x 28
33
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Chapter 5
Notetaking Guide
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Example 2
Solving a Two-Step Equation
3
4
7
12
1
2
ᎏᎏ x ᎏᎏ ᎏᎏ
Original equation
3
1
7
3
3
ᎏᎏ x ᎏᎏ ᎏ4ᎏ ᎏᎏ ᎏ4ᎏ
12
4
3
Subtract ᎏ4ᎏ from each side.
2
7
3
2
ᎏᎏ x ᎏ4ᎏ ᎏ4ᎏ
12
7
Write fractions using LCD.
1
ᎏᎏ x ⫺ᎏ4ᎏ
12
冢
冣
7
12
12
⫺ᎏᎏ ᎏᎏ x ⫺ᎏ7ᎏ
12
7
Subtract.
冢 冣
1
4
⫺ᎏᎏ
x ᎏ3ᎏ
7
Example 3
Multiply each side by
7
multiplicative inverse of ⫺ᎏ12ᎏ .
Multiply.
Writing and Solving a Two-Step Equation
Tree Growth The height of a certain Norway Spruce is 10 feet.
1
2
If the tree’s height grows 2 ᎏᎏ feet per year, find how long it will
take the tree to reach a height of 25 feet.
Solution
Number of
New
Growth
Current
p
years
height
height
rate
1
2
10 2 ᎏᎏ x 25
Write equation.
1
2
Subtract 10 from each side.
10 2 ᎏᎏ x 10 25 10
Simplify. Write mixed number
as improper fraction.
5
ᎏᎏ x 15
2
2
ᎏᎏ
5
冢 冣
5
ᎏᎏ
2
2
x ᎏ5ᎏ
x 6
(
15
)
Multiply each side by
5
multiplicative inverse of ᎏ2ᎏ .
Multiply.
Answer: The tree will be 25 feet tall after 6 years.
Lesson 5.6
Using Multiplicative Inverses to Solve Equations
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