2.2 The Multiplication Property of Equality

Objective 1

Use the multiplication property of equality.

Slide 2.2-3

The addition property of equality is not enough to solve some equations, such as

Since the coefficient of x is 3 rather than 1, the multiplication property of equality is needed to change the equation to the form x = a number, after the 2 is subtracted from both sides of the equation and we are left with

3 2 17.x

3 15.x

Multiplication Property of EqualityIf 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.

A B AC BC

Slide 2.2-4

Use the multiplication property of equality.

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.

DO NOT, however, divide each side by a variable, since the variable might be equal to 0.

This property can be used to solve . The on the left must be changed

to 1x, or x. To isolate x, we multiply each side of the equation by , the

reciprocal of 3, which will result in a coefficient of 1 when multiplied.

3 15x 3x

1 33 1

3 3

1

3

Slide 2.2-5

Use the multiplication property of equality. (cont’d)

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.

CLASSROOM EXAMPLE 1

Solve.

Solution:

15 1

5

5

1 75x

5x

15 75x

Check:

15(5) 75

75 75

The solution set is 5 .

Slide 2.2-6

Applying the Multiplication Property of Equality

CLASSROOM EXAMPLE 2

Solve.

Solution:

8 2

8 8

0x

5

2x

8 20x

Check:

8 05

22

20 20

The solution set is5.

2

8 20x

Slide 2.2-7

Applying the Multiplication Property of Equality

CLASSROOM EXAMPLE 3

Solve.

Solution:

0.7 5.04.x

0.7

0.7 4

0.7

5.0x

7.2x

Check:

7.20.7 5.04

5.04 5.04

0.7 5.04x

The solution set is 7.2 .

Slide 2.2-8

Solving an Equation with Decimals

Solution:

Solve.

64

x

4 464

x

24x 624

4

64

x

6 6

Check:

The solution set is 24 .

Slide 2.2-9

Applying the Multiplication Property of EqualityCLASSROOM EXAMPLE 4

CLASSROOM EXAMPLE 5

Solve.

18t (182

23

) 1

12 12

212

3t

Solution:

212

3 3

2 23t

Check:

212

3t

The solution set is 18 .

Slide 2.2-10

Applying the Multiplication Property of Equality

Using the multiplication property of equality when the coefficient of the variable is −1.In Section 2.1, we obtained the equation We reasoned that since this equation says that the additive inverse (or opposite) of x is −17, then x must equal 17. We can also use the multiplication property of equality to obtain the same result as detailed in the next example.

17.x

Slide 2.2-11

Solution:

71 p

1 7p

( 7) 7

CLASSROOM EXAMPLE 6

Solve. 7p

71 11 p

1( 1) 7p

7p

Check:

7p

7 7

The solution set is 7 .

Slide 2.2-12

Applying the Multiplication Property of Equality

Objective 2

Simplify, and then use the multiplication property of equality.

Slide 2.2-13

CLASSROOM EXAMPLE 7

Solve.

4 9 20r r

5 20

5 5

r

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

Combing Like Terms When Solving