solving equations by multiplying or dividing

CONFIDENTIAL 1

Solving Equations bySolving Equations byMultiplying or DividingMultiplying or Dividing

CONFIDENTIAL 2

Warm UpWarm Up

1) -5.2 + a = -8

Simplify:

2) 90 = -22 + a

3) -12 + f = 3

4) -1200 + j = 345

1) a = -2.8

2) a = 112

3) f = -9

4) j = 1545

CONFIDENTIAL 3

Introduction

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.

Inverse OperationsOperation Inverse Operation

Multiplication Division

Subtraction Addition

An equation is like a balanced scale. To keep the balance, whatever you do on one side of the equation, you must also do on the other side.

CONFIDENTIAL 4

Solve each equation. Check your answer.

Solving Equations by Using Multiplication

-4 = k -5

A)

(-5) (-4) = (-5) ( k) -5

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

20 = k

Check

-4 = k -5

-4

-4 -4

20-5

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

CONFIDENTIAL 5

m = 1.5 3

B)

(3) × m = (3) × 1.5 3

m = 4.5

Check

m = 1.5 3

1.5

1.5 1.5

4.5 3

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

To check your solution, substitute 1.5 for m in the original equation.

CONFIDENTIAL 6

Solve each equation. Check your answer.

Now you try!

1) p = 10 5

2) -13 = y 3

3) c = 7 8

1) p = 50

2) y = -39

3) c = 56

CONFIDENTIAL 7

Solving Equations by Using Division

Solve each equation. Check your answers.

A) 7x = 56

7x = 56

7x = 56 7 7

Since x is multiplied by 7, divide both sides by 7 to undo the multiplication.

Check

7x = 56

56

56 56

7(8)

x = 8

To check your solution, substitute 8 for x in the original equation.

CONFIDENTIAL 8

B) 13 = -2w

13 = -2w

13 = -2w-2 -2

Since w is multiplied by -2, divide both sides by -2 to undo the multiplication.

Check

13 = -2w

-2(6.5)

13 13

13

-6.5 = w

To check your solution, substitute 8 for x in the original equation.

CONFIDENTIAL 9

Now you try!

1) 16 = 4c

2) 0.5y = -10

Solve each equation. Check your answer.

3) 15k = 75

1) c = 4

2) y = -20

3) k = 5

CONFIDENTIAL 10

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.

CONFIDENTIAL 11

Solving Equations That Contain Fractions

Solve each equation.

A) 5v = 35 9

The reciprocal of 5/9 is 9/5 . Since v is multiplied by 5/9 , multiply both sides by 9/5.

9 × 5v = 9 ×355 9 5

v = 63

CONFIDENTIAL 12

B) 5 = 4y 2 3

The reciprocal of 4/3 is 3/4 . Since v is multiplied by 4/3 , multiply both sides by 3/4.

3 × 5 = 3 ×4y4 2 4 3

15 = y 8

5 = 4y2 3

CONFIDENTIAL 13

Now you try!

1) -1 = b 4 5

2) 4j = 2 6 3

Solve each equation. Check your answer.

3) w = 102 6

1) b = -1.25

2) j = 1

3) w = 612

CONFIDENTIAL 14

The distance in miles from the airport that a plane shouldbegin descending, divided by 3, equals the plane’s height above the ground in thousands of feet. If a plane is 10,000 feet above the ground, write and solve an equation to find the distance at which the pilot should begin descending.

Distance divided by 3 equals height in thousands of feet.

Solution:

d = h3

Write an equation to represent the relationship.

Substitute 10 for h. Since d is divided by 3, multiply both sides by 3 to undo the division.

d = 30

The pilot should begin descending 30 miles from the airport.

d = 103

(3)d = (3) × 103

CONFIDENTIAL 15

Properties of Equality

WORDS NUMBERS ALGEBRA

Multiplication Property of Equality

You can multiply both sides of an equation by the same

number, and the statement will still be true.

6 = 66 (3) = 6 (3)

18 = 18

a = bac = bc

Division Property of EqualityYou can divide both sides of anequation by the same nonzero number, and the statement will

still be true.

8 = 88 = 8

2 = 2

a = b(c ≠ 0)a = b 4 4c c

CONFIDENTIAL 16

An orange contains about 80 milligrams of vitamin C, which is 10 times as much as an apple contains. Write

and solve an equation to find the amount ofvitamin C in an apple.

Now you try!

10x = 80

x = 10 milligrams

CONFIDENTIAL 17

Assignments

1) K = 8 4

2) z = -9 3

3) -2 = w-7

4) 6 = t-5

Solve each equation. Check your answer:

1) k = 321) k = 32

2) z = -27

3) w = 14

4) t = - 30

CONFIDENTIAL 18

5) g = 10 1.9

6) 2.4 = b 5

7) 4x = 28

8) -64 = 8c

9) -9 j = -45

Factor the trinomial. Check your answer.

5) g = 19

6) b = 12

7) x = 7

8) c = - 8

9) j = 5

CONFIDENTIAL 19

10) The Baseball Birthday Batter Package at a minor league ballpark costs $192. The package includes tickets, drinks, and cake for a group of 16 children.

Write and solve an equation to find the cost per child.

10) 16 c =$192

c = $12

CONFIDENTIAL 20

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.

Inverse OperationsOperation Inverse Operation

Multiplication Division

Subtraction Addition

An equation is like a balanced scale. To keep the balance, whatever you do on one side of the equation, you must also do on the other side.

Let’s review

CONFIDENTIAL 21

Solve each equation. Check your answer.

Solving Equations by Using Multiplication

-4 = k -5

A)

(-5) (-4) = (-5) ( k) -5

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

20 = k

Check

-4 = k -5

-4

-4 -4

20-5

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

review

CONFIDENTIAL 22

m = 1.5 3

B)

(3) × m = (3) × 1.5 3

m = 4.5

Check

m = 1.5 3

1.5

1.5 1.5

4.5 3

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

To check your solution, substitute 1.5 for m in the original equation.

review

CONFIDENTIAL 23

Solving Equations by Using Division

Solve each equation. Check your answers.

A) 7x = 56

7x = 56

7x = 56 7 7

Since x is multiplied by 7, divide both sides by 7 to undo the multiplication.

Check

7x = 56

56

56 56

7(8)

x = 8

To check your solution, substitute 8 for x in the original equation.

review

CONFIDENTIAL 24

B) 13 = -2w

13 = -2w

13 = -2w-2 -2

Since w is multiplied by -2, divide both sides by -2 to undo the multiplication.

Check

13 = -2w

-2(6.5)

13 13

13

-6.5 = w

To check your solution, substitute 8 for x in the original equation.

review

CONFIDENTIAL 25

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.

review

CONFIDENTIAL 26

Solving Equations That Contain Fractions

Solve each equation.

A) 5v = 35 9

The reciprocal of 5/9 is 9/5 . Since v is multiplied by 5/9 , multiply both sides by 9/5.

9 × 5v = 9 ×355 9 5

v = 63

review

CONFIDENTIAL 27

B) 5 = 4y 2 3

The reciprocal of 4/3 is 3/4 . Since v is multiplied by 4/3 , multiply both sides by 3/4.

3 × 5 = 3 ×4y4 2 4 3

15 = y 8

5 = 4y2 3

review

CONFIDENTIAL 28

The distance in miles from the airport that a plane shouldbegin descending, divided by 3, equals the plane’s height above the ground in thousands of feet. If a plane is 10,000 feet above the ground, write and solve an equation to find the distance at which the pilot should begin descending.

Distance divided by 3 equals height in thousands of feet.

Solution:

d = h3

Write an equation to represent the relationship.

Substitute 10 for h. Since d is divided by 3, multiply both sides by 3 to undo the division.

d = 30

The pilot should begin descending 30 miles from the airport.

d = 103

(3)d = (3) × 103

CONFIDENTIAL 29

Properties of Equality

WORDS NUMBERS ALGEBRA

Multiplication Property of Equality

You can multiply both sides of an equation by the same

number, and the statement will still be true.

6 = 66 (3) = 6 (3)

18 = 18

a = bac = bc

Division Property of EqualityYou can divide both sides of anequation by the same nonzero number, and the statement will

still be true.

8 = 88 = 8

2 = 2

a = b(c ≠ 0)a = b 4 4c c

review

CONFIDENTIAL 30

You did a great job You did a great job today!today!