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Multiplication

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Solving an equation that contains multiplication or division is similar to solving an equation that contains addition or subtraction. Use inverse operations to undo the operations on the variable.

Multiplication

Division

Division

Multiplication

j

–8 =

3

–24

3

j

–8 =

Check

3

–8

Example 1A: Solving Equations by Using Multiplication

Solve the equation.

Since j is divided by 3, multiply both sides by 3 to undo the division.

–24 = j

To check your solution, substitute –24 for j in the original equation.

–8 –8

p

= 10

5

Check It Out! Example 1a

Solve the equation. Check your answer.

Since p is divided by 5, multiply both sides by 5 to undo the division.

p = 50

Example 2A: Solving Equations by Using Division

Solve the equation. Check your answer.

9y = 108

Since y is multiplied by 9, divide both sides by 9 to undo the multiplication.

y = 12

Example 2B: Solving Equations by Using Division

Solve the equation. Check your answer.

–4.8 = –6v

Since v is multiplied by –6, divide both sides by –6 to undo the multiplication.

0.8 = v

Check It Out! Example 2b

Solve the equation. Check your answer.

0.5y = –10

Since y is multiplied by 0.5, divide both sides by 0.5 to undo the multiplication.

y = –20

Check It Out! Example 2c

Solve the equation. Check your answer.

15k = 75

Since k is multiplied by 15, divide both sides by 15 to undo the multiplication.

k = 5

Remember that dividing is the same as multiplying by the reciprocal. When solving equations, you will sometimes find it easier to multiply by a reciprocal instead of dividing. This is often true when an equation contains fractions.

6

5

6

5

6

5

6

5

The reciprocal of is . Since w is multiplied by , multiply both sides by .

5

w = 20

Check

6

20

Example 3A: Solving Equations That Contain Fractions

Solve the equation.

5

w= 20

6

w = 24

To check your solution, substitute 24 for w in the original equation.

2020

3

2

1

8

1

1

8

8

The reciprocal of is 8. Since z is multiplied by , multiply both sides by 8.

= z

Example 3B: Solving Equations That Contain Fractions

Solve the equation.

3

= z

16

1

5

4

4

1

5

1

1

5

5

The reciprocal of is 5. Since b is multiplied by , multiply both sides by 5.

–

= b

Check It Out! Example 3a

Solve the equation. Check your answer.

–= b

4j

6

Solve the equation.

2

4j

is the same as j.

=

3

6

6

4

6

4

4

6

4

6

4

6

The reciprocal of is . Since j is multiplied by , multiply both sides by .

Check It Out! Example 3b

j = 1

4j

=

Check

6

2

3

Check It Out! Example 3b Continued

To check your solution, substitute 1 for j in the original equation.

1

Ciro puts of the money he earns from mowing lawns into a college education fund. This year Ciro added $285 to his college education fund. Write and solve an equation to find how much money Ciro earned mowing lawns this year.

4

Example 4: Application

one-fourth times earnings equals college fund

Write an equation to represent the relationship.

Substitute 285 for c. Since m is divided by 4, multiply both sides by 4 to undo the division.

Ciro earned $1140 mowing lawns.

m = $1140

Check it Out! Example 4

The distance in miles from the airport that a plane should begin descending, divided by 3, equals the plane's height above the ground in thousands of feet. A plane began descending 45 miles from the airport. Use the equation to find how high the plane was flying when the descent began.

Distance divided by 3 equals height in thousands of feet

Write an equation to represent the relationship.

Substitute 45 for d.

15 = h

The plane was flying at 15,000 ft when the descent began.

Properties of Equality

8

8

=

a

a

4

4

=

c

c

WORDS

Division Property of Equality

You can divide both sides of an equation by the same nonzero number, and the statement will still be true.

NUMBERS

8 = 8

2 = 2

ALGEBRA

a = b

(c ≠ 0)

Properties of Equality

Homework Assignment

Pg. 27-29 (22-36 even, 37-47, 57-60, 65, 77, & 78)