1. Mr. Peter Richard,
The most profound Math teacher in all of the worlds in this universe and the rest of them will
teach you the multiple steps of stepping thru (yes it’s a word), these Multi-Step Equations
Please watch your step!

2. Warm Up
Solve.
1. 3x = 102
= 15
3. z – 100 = 21
w = 98.6
x = 34 y 15
y = 225
z = 121
w = 19.5

3. A multi-step equation requires more than two steps to solve
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.

4. Additional Example 1: Solving Equations That Contain Like Terms
Solve.
8x x – 2 = 37
Commutative Property of Addition
8x + 3x + 6 – 2 = 37
11x + 4 = 37 Combine like terms.
– 4 – 4 Since 4 is added to 11x, subtract 4 from both sides.
11x = 33 33 11
11x =
Since x is multiplied by 11, divide both sides by 11.
x = 3

5. Check It Out! Example 1
Solve.
9x x – 2 = 42
Commutative Property of Addition
9x + 4x + 5 – 2 = 42
13x + 3 = 42 Combine like terms.
– 3 – 3 Since 3 is added to 13x, subtract 3 from both sides.
13x = 39 39 13
13x =
Since x is multiplied by 13, divide both sides by 13.
x = 3

6. 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.

7. Additional Example 2A: Solving Equations That Contain Fractions
Solve.
+ = – 5n 4 7 4 3 4 7 4 –3 5n
= 4
( ) ( )
Multiply both sides by 4.
( ) ( ) ( ) 5n 4 7 –3
= 4
Distributive Property
( ) ( ) ( ) 5n 4 7 –3
= 4
Simplify.
5n + 7 = –3

8. Additional Example 2A Continued
– 7 –7 Since 7 is added to 5n, subtract from both sides.
5n = –10 5n 5
–10 =
Since n is multiplied by 5, divide both sides by 5
n = –2

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

10. Additional Example 2B: Solving Equations That Contain Fractions
Solve.
+ – = x 2 7x 9 17 2 3
( ) ( ) x 2 3 7x 9 17
– = 18
Multiply both sides by 18, the LCD.
18( ) + 18( ) – 18( ) = 18( ) 7x 9 x 2 17 3
Distributive Property
18( ) + 18( ) – 18( ) = 18( ) 7x 9 x 2 17 3 2 9 2 6
Simplify. 1 1 1 1
14x + 9x – 34 = 12

11. Additional Example 2B Continued
23x – 34 = Combine like terms.
Since 34 is subtracted from x, add 34 to both sides.
23x = 46 =
23x 23 46
Since x is multiplied by 23, divide t both sides by 23.
x = 2

12. Distributive Property
Check It Out! Example 2A
Solve.
+ = – 3n 4 5 4 1 4 5 4 –1 3n
= 4
( ) ( )
Multiply both sides by 4.
( ) ( ) ( ) 3n 4 5 –1
= 4
Distributive Property
( ) ( ) ( ) 3n 4 5 –1
= 4 1 1 1
Simplify. 1 1 1
3n + 5 = –1

13. Check It Out! Example 2A Continued
– 5 – Since 5 is added to 3n, subtract 5 from both sides.
3n = –6 3n 3 –6 =
Since n is multiplied by 3, divide
both sides by 3.
n = –2

14. Multiply both sides by 9, the LCD. x 3 1 5x 9 13 ( ) ( ) 9 + – = 9
Check It Out! Example 2B
Solve.
+ – = x 3 5x 9 13 1 3
Multiply both sides by 9, the LCD. x 3 1 5x 9 13
( ) ( )
– = 9
9( ) + 9( ) – 9( ) = 9( ) 5x 9 x 3 13 1
Distributive Property
9( ) + 9( ) – 9( ) = 9( ) 5x 9 x 3 13 1 1 3 1 3
Simplify. 1 1 1 1
5x + 3x – 13 = 3

15. Check It Out! Example 2B Continued
8x – 13 = Combine like terms.
Since 13 is subtracted from 8x, add 13 to both sides.
8x = 16 = 8x 8 16
Since x is multiplied by 8, divide t both sides by 8.
x = 2

16. Additional Example 3: Travel Application
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.
David’s average speed is his total distance for the two days divided by the total time.
Total distance
Total time =
average speed

17. Additional Example 3 Continued
2 + 5
= 12
m + 3m
Substitute m + 3m for total distance and for total time. 7
= 12 4m
Simplify.
= 7(12) 7 4m
Multiply both sides by 7.
4m = 84
84 4
4m 4 =
Divide both sides by 4.
m = 21
David rode 21.0 miles.

18. Check It Out! Example 3
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.
Penelope’s average speed is her total distance for the two days divided by the total time.
Total distance
Total time =
average speed

19. Check It Out! Example 3 Continued
3 + 7
= 20
m + 2m
Substitute m + 2m for total distance and for total time. 10
= 20 3m
Simplify.
= 10(20) 10 3m
Multiply both sides by 10.
3m = 200
3m 3 =
Divide both sides by 3.
m  66.67
Penelope rode approximately 66.7 miles.

20. Homework P-129/130 #7, 9, 21, 25, 47 Lesson Quiz Solve.
1. 6x + 3x – x + 9 = 33
2. 29 = 5x x =
5. John traveled X miles in 2 hours. He traveled four times that amount the next day in 5 hours. His average speed over the two days was 40 MPH. How far did he travel on the first day?
x = 3
x = 1 5 8 x 8 33 8
x = 28
x = 1 9 16
– = 6x 7 2x 21 25 21 56