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### Lesson 3.4: Solving Multi-step Equations

Drill

- Lenny’s Lawncare purchased a new truck for 30x + 42 dollars. One year later the value of the truck was 12x + 28 dollars. Write an expression to represent the amount that the truck’s value decreased.
- Brian bought a new drill for d dollars. He paid 5% sales tax. Write an expression to represent the total amount Brian paid for the drill.
- At JFK live, the student ticket price is p dollars and the non-student price is $2.75 more. There were 75 student tickets sold and 34 non-student tickets sold. Write an expression to represent the total ticket sales in dollars.

Solving problems by working backwards

Solving equations involving more than one operation

Working Backwards

- Starting at the end of the problem and undo each step
- Other strategies:

Solve the following problem by working backwards

- Danny took some rope with him on his camping trip. He used 32 feet of rope to tie his canoe to a log on the shore. He then gave ⅓ of the remaining rope to some fellow campers who also needed to tie a canoe. The next night, he used half of the remaining rope to secure the his tent during a thunderstorm. On the last day, he used 7 feet as a fish stringer to keep the fish he had caught. After the camping trip, he had 9 feet of rope left. How much did he have at the beginning?

Inverse operations

multiplication

by an integer

division by

an integer

/5

X 5

division by

an integer

multiplication

by an integer

/5

X 5

multiplication

by an fraction

multiplication

by its reciprocal

X 1/4

X 4/1

addition

subtraction

+ 6

- 6

subtraction

addition

- 6

+ 6

Use a table to organize

He used 7 feet as a fish stringer

9 feet + 7 feet = 16 feet

He used half of the remaining rope to secure the tent

16 feet X 2 = 32 feet

He gave 1/3 of the rope to fellow campers

32 feet X 3/2= 48 feet

which means he kept 2/3 of the rope

He used 32 feet of rope to tie his canoe

48 feet + 32 feet = 80 feet

Tips for success when solving multi-step equations…

- “Undo” the operations in reverse of the order of operations (P, E, M/D, A/S)
- So, we always start with A/S first, then move on…

- Whatever you do to one side of the equation, you have to do to the other side.
- Why? It’s like a see-saw; if you add more onto one side, the see-saw will be unbalanced!

Solve Using Addition and Division

- Solve 5q – 13 = 37. Then check your solution.
- 5q – 13 + 13 = 37 + 13
- 5q = 50
- 5q/5 =50/5
- q = 10
- Check 5(10) – 13 = 37; 50-13 = 37

add 13 to both sides

simplify

divide both sides by 5

simplify

Solving Using Subtraction and Multiplication

- s/12 + 6 = -1
- s/12 + 6 – 6 = -1 -6
- s/12 = -7
- 12(s/12 = -7)
- 12s/12 = 12(-7); s = -84
- Check: -84/12 + 6 = -1; -7 + 6 = -1

subtract 6 from both sides

simplify

multiply each side by 12

simplify

Solving Using Multiplication and Subtraction

Check

please!

Multiply both sides by -3

Subtract 8 from both sides

simplify

simplify

Vocabulary

- Consecutive integers: integers in counting order, ex: 1, 2, 3, 4… or n, n+1, n+2….
- Consecutive ODD integers
- 1, 3, 5…
- n, n+2, n+4….

- Consecutive EVEN integers
- 2, 4, 6….
- n, n + 2, n + 4….

Notice that you can use the same expression to represent either odd OR even; you just need to define the value of n to be even or odd at the beginning!

Find three consecutive odd integers whose sum is 57

Let n = the first odd integer

n+2 = the second odd integer

n+4 = the third odd integer

n + (n + 2) + (n + 4) = 57

3n = 51

3n + 6 -6 = 57 - 6

3n = 51

3 3

n = 17

n + 2 = 19

n + 4 = 21

3n

+

6

= 57

Exit Pass

- Turn to page 145 in your book. Please complete the following problems on a separate piece of paper to turn in: 5-11 (odd)
- Homework: page 146, 22-39. Work MUST be shown.

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