evaluating algebraic expressions 3-3solving multi-step equations what are the steps for solving...

Evaluating Algebraic Expressions

3-3 Solving Multi-Step Equations

What are the steps for solving What are the steps for solving multi-step equations?multi-step equations?

How can solving equations be How can solving equations be applied to real life problems?applied to real life problems?

Multi-Step Equations Multi-Step Equations Essential QuestionsEssential Questions



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, distributing, or removing the fraction.



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



Solve 5x + 3(x + 4) = 28.

METHOD 1 Show All Steps METHOD 2 Do Some Steps Mentally

5x + 3(x + 4) = 28

5x + 3x + 12 = 28

8x + 12 = 28

8x + 12 – 12 = 28 – 12

8x = 16

x = 2

8x 168 8

=

5x + 3(x + 4) = 28

5x + 3x + 12 = 28

8x + 12 = 28

8x = 16

x = 2

METHOD 1 Show All Steps

5x + 3(x + 4) = 28

5x + 3x + 12 = 28

8x + 12 = 28

8x + 12 – 12 = 28 – 12

8x = 16

x = 2

8x 168 8

=



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

=

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