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Solving Equations by

Multiplying or Dividing

Solving Equations by

Warm Up

Lesson Presentation

Lesson Quiz

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Warm Up

Evaluate each expression. 1. (–7)(2.8)

2. 0.96 ÷ 6

3. (–9)(–9)

4.

5.

6.

19.6 0.16

1

81

2 3

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Solve one-step equations in one variable by

using multiplication or division.

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Inverse Operations

Operation

Inverse Operation

Multiplication Division

Division _{Multiplication}

Solving an equation that contains multiplication or division is similar to solving an equation that

contains addition or subtraction. Use inverse

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Solve the equation.

Example 1A: Solving Equations by Using Multiplication

Since j is divided by 3, multiply both sides by 3

to undo the division.

–24 = j

–8 –8 _

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

–8 = j 3

–8 –24 3

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Solving Equations by

Solve the equation.

Example 1B: Solving Equations by Using Multiplication

Since n is divided by 6, multiply both sides by 6

n = 16.8

2.8 2.8 _

To check your solution, substitute 16.8 for n in the original equation.

= 2.8

n

6

2.8

16.8

6

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Solving Equations by

Solve the equation. Check your answer. Check It Out! Example 1a

Since p is divided by 5, multiply both sides by 5

p = 50

10 10 _

To check your solution, substitute 50 for p in the original equation.

p = 10 5

10

50

5

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Solve the equation. Check your answer. Check It Out! Example 1b

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

–39 = y

–13 –13 _

To check your solution, substitute –39 for y in the original equation.

–13 = y 3

–13 –39 3 y

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Solving Equations by

Solve the equation. Check your answer. Check It Out! Example 1c

Since c is divided by 8, multiply both sides by 8

c = 56

7 7_

To check your solution, substitute 56 for c in the original equation.

c = 7 8

7

56

8

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Solve the equation. Check your answer.

Example 2A: Solving Equations by Using Division

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

y = 12

108 108 _

To check your solution, substitute 12 for y in the original equation.

9y = 108

9(12) 108

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Solving Equations by

Solve the equation. Check your answer.

Example 2B: Solving Equations by Using Division

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

0.8 = v

–4.8 –4.8 _

To check your solution, substitute 0.8 for v in the original equation.

–4.8 = –6v

–4.8 –6(0.8)

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Solving Equations by

Solve the equation. Check your answer. Check It Out! Example 2a

Since c is multiplied by 4, divide both sides by 4 to undo the multiplication.

4 = c

16 16 _

To check your solution, substitute 4 for c in the original equation.

16 = 4c

16 4(4)

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Solve the equation. Check your answer. Check It Out! Example 2b

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

y = –20

–10 –10 _

To check your solution, substitute –20 for y in the original equation.

0.5y = –10

0.5(–20) –10

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Solving Equations by

Solve the equation. Check your answer. Check It Out! Example 2c

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

k = 5

75 75 _

To check your solution, substitute 5 for k in the original equation.

15k = 75

15(5) 75

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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

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Solve the equation.

Example 3A: Solving Equations That Contain Fractions

w = 24

20 20 

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

5₆ w = 20

Check 5₆ w = 20

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

5

6 56

5 6 6

5

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Solving Equations by

Solve the equation.

Example 3B: Solving Equations That Contain Fractions



₁₆3 = z

To check your solution, substitute for z in the original equation. 3 2 = z 3 2 1 8

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

1 8 1

8

Check _{16 =}3 1₈ z

3 16

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Solving Equations by

Solve the equation. Check your answer.Check It Out! Example 3a – 1 = b

4

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

5 4

1 5

The reciprocal of is 5. Since b is multiplied by , multiply both sides by 5. 1 5 1 5 = b 5 4 –

= b 5 4

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Check It Out! Example 3b

j = 1

Solve the equation. 4j =

6

2 3

is the same as j.₆4

4j 6

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

4 6 4

6

6 4

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Solving Equations by

Check It Out! Example 3b Continued



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

Check 4j_{6 =}2₃

Solve the equation. 4j =

6

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Solving Equations by

w = 712

102 102 

To check your solution, substitute 712 for w in the original equation.

1₆ w = 102

Check 1₆ w = 102

The reciprocal of is 6. Since w is multiplied by , multiply both sides by 6 .

1 6 1

6

Solve the equation.Check It Out! Example 3c

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Example 4: Application

Write an equation to represent the relationship.

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.

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one-fourth times earnings equals college fund

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Solving Equations by

Check it Out! Example 4

Write an equation to represent the relationship.

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

15 = h

Substitute 45 for d.

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Solving Equations by

WORDS

Addition Property of Equality

You can add the same number to both sides of an equation, and the statement will still be true.

NUMBERS

3 = 3

3

+ 2

= 3

+ 2

5 = 5

ALGEBRA

a

=

b

a

+

c =

b

+

c

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WORDS

Subtraction Property of Equality

You can subtract the same number from both sides of an equation, and the

statement will still be true.

NUMBERS

7 = 7

7

– 5

= 7

– 5

2 = 2

ALGEBRA

a

=

b

a

–

c =

b

–

c

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WORDS

Multiplication Property of Equality

You can multiply both sides of an

equation by the same number, and the statement will still be true.

NUMBERS

6 = 6

6

(3)

= 6

(3)

18 = 18

ALGEBRA

a

=

b

a

c =

b

c

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Properties of Equality

Division Property of Equality

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

WORDS

a = b

(c ≠ 0)

8

=

8

2 = 2

ALGEBRA

NUMBERS

8

4

8

4

=

a

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Lesson Quiz: Part 1

Solve each equation.

1.

2.

3. 8y = 4

4. 126 = 9q

5.

6.

21

2.8

–14

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Solving Equations by

Lesson Quiz: Part 2

7. A person's weight on Venus is about his or her weight on Earth. Write and solve an equation to find how much a person weighs on Earth if he or she weighs 108 pounds on Venus.