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Chapter Chapter 22Section Section 22
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
The Multiplication Property of Equality
Use the multiplication property of equality.Combine terms in equations, and then use the multiplication property of equality.
11
22
2.22.22.22.2
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 11
Use the multiplication property of equality.
Slide 2.2- 3
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Use the multiplication property of equality.
If , then and represent the same number. Multiplying and by the same number will also result in an equality. The multiplication property of equality states that we can multiply each side of an equation by the same nonzero number without changing the solution.
If A, B, and C (C ≠ 0) represent real numbers, then the equations
and are equivalent equations.
That is, we can multiply each side of an equation by the same nonzero number without changing the solution.
3 15x 3x 15
A B AC BC
3x 15
Slide 2.2- 4
Remember the balance analogy from Section 2.1. Whatever we do to one side of the equation, we have to do to the other side to maintain balance.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
This property can be used to solve . The on the left must be changed to 1x, or x, instead of . To isolate x, we multiply each side of the equation by . We use because is the reciprocal of 3 and .
Just as the addition property of equality permits subtracting the same number from each side of an equation, the multiplication property of equality permits dividing each side of an equation by the same number.
For example , which we just solved by multiplying each side by , could also be solved by dividing each side by 3.
3x
Slide 2.2- 5
Use the multiplication property of equality. (cont’d)
3 15x 3x
1 33 1
3 3
1
3
3 15x 1
3
1
3
1
3
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
We can divide each side of an equation by the same nonzero number without changing the solution. Do not however, divide each side by a variable, as that may result in losing a valid solution.
In practice, it is usually easier to multiply on each side if the coefficient of the variable is a fraction, and divide on each side if the coefficient is an integer. For example, to solve
it is easier to multiply by , the reciprocal of , than to divide by .
Slide 2.2- 6
Use the multiplication property of equality. (cont’d)
On the other hand, to solve it is easier to divide by −5 than to multiply by .
5 2 ,0x
312,
4x
4
3 3
4
3
4
1
5
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 1
Solve
Solution:8 0
8 8
2x
5
2x
Dividing Each Side of an Equation by a Nonzero Number
8 20.x
Check:
85
220
20 20
The solution set is .5
2
8 20x
Slide 2.2- 7
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 2
Solve
Solution:
0.7 5.04.x
0.7
0.7 4
0.7
5.0x
7.2x
Solving an Equation with Decimals
Check:
7.20.7 5.04
5.04 5.04
0.7 5.04x
The solution set is 7.2 .
Slide 2.2- 8
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 3
Solution:
Using the Multiplication Property of Equality
Solve 6.4
x
4 464
x
24x 624
4
64
x
6 6 The solution set is
Check:
24 .
Slide 2.2- 9
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 4
Solve
18h
(182
23
) 1
12 12
Using the Multiplication Property of Equality
212.
3h
Solution:
212
3 3
2 23h
Check:
212
3h
The solution set is 18 .
Slide 2.2- 10
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
In Section 2.1, we obtained the equation . We reasoned that since this equation says that the additive inverse (or opposite) of k is −17, then k must equal 17.
We can also use the multiplication property of equality to obtain the same result as detailed in the next example.
Using the multiplication property of equality when the coefficient of the variable is −1
Slide 2.2- 11
17k
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 5
Solution:71 p
1 7p
( 7) 7
Using the Multiplication Property of Equality when the Coefficient of the Variable is −1
Solve 7.p
71 11 p
1( 1) 7p
7p
Check:7p
7 7
The solution set is 7 .
Slide 2.2- 12
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 22
Combine terms in equations, and then use the multiplication property of equality.
Slide 2.2- 13
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Solve
EXAMPLE 6
4 9 20.r r
5 20
5 5
r
Combining Terms in an Equation before Solving
Solution:
5 20r
4r
4 9 20r r Check:
The solution set is 4 .
( 4) ( 4)4 9 20 16 ( 36) 20
20 20
Slide 2.2- 14
