Download Use the multiplication property of equality.

Chapter 2
Section 2
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2.2
1
2
The Multiplication Property of Equality
Use the multiplication property of equality.
Combine terms in equations, and then use
the multiplication property of equality.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 1
Use the multiplication property of
equality.
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Slide 2.2- 3
Use the multiplication property of equality.
If 3x  15, then 3x and 15 represent the same number.
Multiplying 3x and 15 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
AB
and
AC  BC
are equivalent equations.
That is, we can multiply each side of an equation by the same
nonzero number without changing the solution.
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
Slide 2.2- 4
Use the multiplication property of
equality. (cont’d)
This property can be used to solve 3x  15. The 3x on the left
must be changed to 1x, or x, instead of 3x . To isolate x, we
multiply each side of the equation by 1 . We use 1 because 1 is the
3
3
3
1
3
reciprocal of 3 and  3   1.
3
3
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 3x  15, which we just solved by multiplying each
side by 1 , could also be solved by dividing each side by 3.
3
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2.2- 5
Use the multiplication property of
equality. (cont’d)
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
4
3
 x  12,
4
3
it is easier to multiply by  , the reciprocal of  , than to divide
3
4
3
by  .
4
On the other hand, to solve 5 x  20, it is easier to divide by
1
−5 than to multiply by  .
5
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Slide 2.2- 6
EXAMPLE 1
Dividing Each Side of an
Equation by a Nonzero Number
Solve 8x  20.
Solution:
Check:
8x  20
5
8    20
2
20  20
8 x 20

8
8
5
x
2
The solution set is
5
 .
2
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Slide 2.2- 7
EXAMPLE 2
Solving an Equation with
Decimals
Solve 0.7 x  5.04.
Check:
Solution:
0.7 x  5.04
0.7 x 5.04

0.7 0.7
0.7  7.2  5.04
x  7.2
5.04  5.04
The solution set is 7.2 .
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2.2- 8
EXAMPLE 3
Using the Multiplication
Property of Equality
x
 6.
Solve
4
Solution:
x
 4   6  4 
4
x  24
The solution set is 24 .
Check:
x
 6
4
24
 6
4
6  6
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Slide 2.2- 9
EXAMPLE 4
Using the Multiplication
Property of Equality
2
Solve  h  12.
3
Solution:
Check:
 3 2
 3
    h  12   
 2 3
 2
h  18
The solution set is 18 .
2
 h  12
3
2
 (18)  12
3
12  12
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Slide 2.2- 10
Using the multiplication property of equality
when the coefficient of the variable is −1
In Section 2.1, we obtained the equation k  17. 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.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2.2- 11
EXAMPLE 5
Using the Multiplication
Property of Equality when the
Coefficient of the Variable is −1
Solve  p  7.
Solution:
1  p  7
1 1 p   7  1
1(1)  p  7
Check:
p  7
(7)  7
77
1 p  7
p  7
The solution set is 7 .
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2.2- 12
Objective 2
Combine terms in equations, and
then use the multiplication
property of equality.
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Slide 2.2- 13
EXAMPLE 6
Combining Terms in an Equation
before Solving
Solve 4r  9r  20.
Solution:
Check:
5r  20
5r 20

5 5
r  4
4r  9r  20
4(4)  9(4)  20
16  (36)  20
20  20
The solution set is 4 .
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2.2- 14