chapter3.3

Evaluating Algebraic Expressions

3-3 Solving Multi-Step Equations

Warm UpWarm Up

California StandardsCalifornia Standards

Lesson PresentationLesson Presentation

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Warm UpSolve.

1. 3x = 102

2. = 15

3. z – 100 = 21

4. 1.1 + 5w = 98.6

x = 34

y = 225

z = 121

w = 19.5

y15



Extension of AF4.1 Solve two-step linear equations and inequalities in one variable over the rational numbers, interpret the solution or solutions in the context from which they arose, and verify the reasonableness of the results.

California Standards



A multi-step equation requires more than two steps to solve. To solve a multi-step equation, you may have to simplify the equation first by combining like terms.



Solve.

8x + 6 + 3x – 2 = 37

Additional Example 1: Solving Equations That Contain Like Terms

11x + 4 = 37 Combine like terms.

– 4 – 4 Since 4 is added to 11x, subtract 4 from both sides.

11x = 33

x = 3

Since x is multiplied by 11, divide both sides by 11.

3311

11x11

=

8x + 3x + 6 – 2 = 37Commutative Property of Addition



Solve.

9x + 5 + 4x – 2 = 42

Check It Out! Example 1

13x + 3 = 42 Combine like terms.

– 3 – 3 Since 3 is added to 13x, subtract 3 from both sides.

13x = 39

x = 3

Since x is multiplied by 13, divide both sides by 13.

3913

13x13

=

9x + 4x + 5 – 2 = 42Commutative Property of Addition



If an equation contains fractions, it may help to multiply both sides of the equation by the least common denominator (LCD) to clear the fractions before you isolate the variable.



Solve.

+ = –

Additional Example 2A: Solving Equations That Contain Fractions

34

74

5n4

Multiply both sides by 4.74

–3 4

5n4

4 + = 4 ( ) ( )

( ) ( ) ( )5n4

74

–3 44 + 4 = 4

5n + 7 = –3

Distributive Property

( ) ( ) ( )5n4

74

–3 44 + 4 = 4

Simplify.


3-3 Solving Multi-Step EquationsAdditional Example 2A Continued

5n + 7 = –3 – 7 –7 Since 7 is added to 5n, subtract

7 from both sides. 5n = –10

5n5

–10 5

= Since n is multiplied by 5, divide both sides by 5

n = –2



The least common denominator (LCD) is the smallest number that each of the denominators will divide into evenly.

Remember!



Solve.

+ – =

Additional Example 2B: Solving Equations That Contain Fractions

23

x 2

7x9

17 9

18( ) + 18( ) – 18( ) = 18( )7x9

x2

17 9

23

14x + 9x – 34 = 12

( ) ( )x2

23

7x9

17 918 + – = 18


Multiply both sides by 18, the LCD.

18( ) + 18( ) – 18( ) = 18( )7x9

x2

17 9

23 Simplify.

2

1 1

9

1

2

1

6


3-3 Solving Multi-Step EquationsAdditional Example 2B Continued

23x = 46

= 23x23

4623 Since x is multiplied by 23, divide

t both sides by 23.x = 2

+ 34 + 34 Since 34 is subtracted from 23x, add 34 to both

sides.

23x – 34 = 12 Combine like terms.



Solve.

+ = –

Check It Out! Example 2A

14

54

3n4

Multiply both sides by 4.54

–1 4

3n4

4 + = 4 ( ) ( )

( ) ( ) ( )3n4

54

–1 44 + 4 = 4

3n + 5 = –1


( ) ( ) ( )3n4

54

–1 44 + 4 = 4

Simplify.

1

1

1

1

1

1


3-3 Solving Multi-Step EquationsCheck It Out! Example 2A Continued

3n + 5 = –1 – 5 –5 Since 5 is added to 3n,

subtract 5 from both sides. 3n = –6

3n3

–6 3

= Since n is multiplied by 3, divide

both sides by 3.n = –2



Solve.

+ – =

Check It Out! Example 2B

13

x 3

5x9

13 9

9( ) + 9( ) – 9( ) = 9( )5x9

x3

13 9

13

5x + 3x – 13 = 3

x3

13

5x9

13 9 ( ) ( ) 9 + – = 9


Multiply both sides by 9, the LCD.

9( ) + 9( ) – 9( ) = 9( )5x9

x3

13 9

13 Simplify.

1

1 1

3

1

1

1

3



8x = 16

= 8x8

16 8 Since x is multiplied by 8, divide t

both sides by 8.

x = 2

+ 13 + 13 Since 13 is subtracted from 8x, add 13 to both sides.

8x – 13 = 3 Combine like terms.

Check It Out! Example 2B Continued



On Monday, David rides his bicycle m miles in 2 hours. On Tuesday, he rides three times as far in 5 hours. If his average speed for the two days is 12 mi/h, how far did he ride on Monday? Round your answer to the nearest tenth of a mile.

Additional Example 3: Travel Application

David’s average speed is his total distance for the two days divided by the total time.

average speed

=Total distance

Total time


3-3 Solving Multi-Step EquationsAdditional Example 3 Continued

Multiply both sides by 7.

Substitute m + 3m for total distance and 2 + 5 for total time.2 + 5

= 12 m + 3m

7= 12

4mSimplify.

7 = 7(12) 7

4m

4m = 84

David rode 21.0 miles.

Divide both sides by 4.

m = 21

84 4

4m 4

=


3-3 Solving Multi-Step EquationsCheck It Out! Example 3

Penelope’s average speed is her total distance for the two days divided by the total time.

average speed

=Total distance

Total time

On Saturday, Penelope rode her scooter m miles in 3 hours. On Sunday, she rides twice as far in 7 hours. If her average speed for two days is 20 mi/h, how far did she ride on Saturday? Round your answer to the nearest tenth of a mile.


3-3 Solving Multi-Step EquationsCheck It Out! Example 3 Continued

Multiply both sides by 10.

Substitute m + 2m for total distance and 3 + 7 for total time.

3 + 7= 20

m + 2m

10= 20

3mSimplify.

10 = 10(20) 10

3m

3m = 200

Penelope rode approximately 66.7 miles.

Divide both sides by 3.

m 66.67

200 3

3m 3

=



Solve.

1. 6x + 3x – x + 9 = 33

2. 29 = 5x + 21 + 3x

3. + =

5. Linda is paid double her normal hourly rate for each hour she works over 40 hours in a week. Last week she worked 52 hours and earned $544. What is her hourly rate?

Lesson Quiz

x = 1

x = 3

x = 2858

x8

33 8

6x 7

4. – =2x21

2521

$8.50

x = 1 916

chapter3.3

Technology

fractions14 x

lesson quiz x

additional example

total distance total

distributive property

penelopes average speed

multistep equation

common denominator lcd