1. Solving Multi-Step Equations
Lesson 2.3

2. California
Standards
4.0 Students simplify expressions before solving linear equations and inequalities in one variable, such as 3(2x – 5) + 4(x – 2) = 12.
5.0 Students solve multistep problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step.

3. Solving Multi-step Equation
2x + 3x – 4 =11 5x – 4 = x = x = 3
Steps –
Circle like terms.
Combine like terms.
Isolate Variable
Add/subtract
Multiply/divide
Solve

4. More Example 2x + 1 = 21 –1 –1 2x = 20 x = 10
Solve the equation. Check your answer.
1.undo the division.
2. undo the addition.
2x + 1 = 21
3. undo the multiplication.
4. The solution set is {10}.
–1 –1
2x = 20
x = 10

5. Now let check our answer
Solve the equation. Check your answer.
Check
To check your solution, substitute 10 for x in the original equation. 
7 7

6. Let Try again Solve 4 = 2x + 5 – 6x 4 = 2x + 5 – 6x 4 = 2x – 6x + 5
Combine like terms.
4 = 2x + 5 – 6x
4 = 2x – 6x + 5
undo the addition.
4 = –4x + 5
– –5
–1 = –4x
undo the multiplication.
The solution set is

7. Try some on your own. 1. 46 = 4x – 4 + 6x 2. 4x – 8 + 2x = 40
3. 36 = 10a – 12 – 7a 4.
5 = x 5.
6. –8 – 2d + 2 = 4
x = 6
x= 8
d = -5
16 = a

8. You may have to combine like terms or use the Distributive Property before you begin solving.

9. Using Distributive Property
Solve the equation.
5(p – 2) = –15
5(p – 2) = –15
Distribute 5.
5(p) + 5(–2) = –15
Simplify.
5p – 10 = –15
Since 10 is subtracted from 5p, add 10 to both sides.
5p = –5
Since p is multiplied by 5, divide both sides by 5.
p = –1
The solution set is {–1}.

10. Helpful Hint
You can think of a negative sign as a coefficient of –1.
–(x + 2) = –1(x + 2) and –x = –1x.

11. Another Example y= -2 10y – (4y + 8) = –20 10y +(–1)(4y + 8) = –20
Solve the equation.
10y – (4y + 8) = –20
Write subtraction as the addition of the opposite.
10y +(–1)(4y + 8) = –20
10y + (–1)(4y) + (–1)(8) = –20
Distribute –1.
10y – 4y – 8 = –20
Simplify.
6y – 8 = –20
Combine like terms.
Since 8 is subtracted from 6y, add 8 to both sides to undo the subtraction.
6y = –12
y= -2

12. Try some on your own. 1. –4(2 – y) = 8 4. d + 3(d – 4) = 20
2. 3(x + 1) – 4 = 5
3. x – (12 – x) = 38
y = 4
4. d + 3(d – 4) = 20
d = 8
X = 2 25

13. Reread and circle relevant information
Application
Lin sold 4 more shirts than Greg. Fran sold 3 times as many shirts as Lin. In total, the three sold 51 shirts. How many shirts did Greg sell?
Locate key words in the question. UNDERLINE WHAT YOU ARE BEING ASKED TO FIND 1 2
Reread and circle relevant information

14. Lin sold 4 more shirts than Greg
Lin sold 4 more shirts than Greg. Fran sold 3 times as many shirts as Lin. In total, the three sold 51 shirts. How many shirts did Greg sell? 3
Reread the part of the problem you underlined, and define an appropriate variable (or variables)
Since the information is given in relation to Lin, set an equation for each individual in terms of Lin.
Greg = L – 4
Lin = L
Fran = 3L

15. Lin sold 4 more shirts than Greg
Lin sold 4 more shirts than Greg. Fran sold 3 times as many shirts as Lin. In total, the three sold 51 shirts. How many shirts did Greg sell?
Write an equation (or inequality) and then check to see if it’s correct by rereading the circled and underlined information 4
(L – 4) + (L) + (3L) = 51
Substitute.
5L – 4 = 51
Greg + Lin + Fran = 51
Greg = L – 4
Lin = L
Fran = 3L
Combine like terms.
5L = 55
Since 4 is subtracted from 5L add 4 to both sides to undo the subtraction.
Since L is multiplied by 5, divide both sides by 5 to undo the multiplication.
L = 11

16. Step 3: Step 1: Step 4: Step 2: More Application
At a local gym, there is a joining fee of $59.95 and a monthly membership fee. Sara and Martin both joined this gym. Their combined cost for 12 months was $ How much is the monthly fee?
Step 3:
Step 1:
Step 4:
Step 2:
Let m represent the monthly fee paid by each.
Monthly fee for 2
12 months
initial fee for 2
plus is
total cost. 2
(12m +
119.90) =

17. Sara and Martin each paid $50 per month.
Distribute 2.
24m =
– –119.90
24m =
Since is added to 24m, subtract from both sides to undo the addition.
Since m is multiplied by 24, divide both sides by 24 to undo the multiplication.
m = 50
Sara and Martin each paid $50 per month.

18. Lesson Quiz: Part l 4 19 25 9 –28 Solve each equation.
1. 2y + 29 – 8y = 5
2. 3(x – 9) = 30
3. x – (12 – x) = 38 4.
5. If 3b – (6 – b) = –22, find the value of 7b. 4 19 25 9
–28

19. Lesson Quiz: Part ll
6. Josie bought 4 cases of sports drinks for an upcoming meet. After talking to her coach, she bought 3 more cases and spent an additional $6.95 on other items. Her receipts totaled $ Write and solve an equation to find how much each case of sports drinks cost.
4c + 3c = 74.15; $9.60