1. Solving Multi-Step Equations
Lesson 6.2.4
Solving Multi-Step Equations

2. Remember These? 1. −3 𝑥+4 2. −2𝑎−3+4𝑎+8 3. −4𝑎−3(𝑎+8)
Take a minute and simplify the following three expressions.
1. −3 𝑥 −2𝑎−3+4𝑎+8
3. −4𝑎−3(𝑎+8)
Give students a few minutes to simplify these, then review.

4. Prior…
In previous chapters we learned how to simplify expressions using the distributive property and by combining the “like terms”.
So far in Chapter 6, we have been concentrating mainly on equations that can be solved in one step or in two steps like the equation 2x + 5 = 17.
In this lesson we will look at solving multi-step equations. As the name suggests… there are steps you must do before you can solve the equation.

5. Steps for Solving any Equation
Distribute if you can.
Then combine the like terms.
Now you have a more simplified equation. Solve it by undoing in reverse.
Check your answer.

6. To be successful with multi-step equations… you will have to be able to analyze the equation and decide what steps must be taken to solve it…
Be on the look out for opportunities to distribute and to combine like terms when solving the equations today!
Don’t work these problems. Just review the types of problems that they will be seeing today and what to look out for.
Examples

7. Let’s Get Started…
Section 1: Combining Like Terms Section 2: Distributive Property Section 3: Multi-Steps with Fractions.

8. 𝟕𝒙 – 𝟑𝒙 – 𝟖 = 𝟐𝟒 𝟒𝐱 – 𝟖 = 𝟐𝟒 Step 1 Distribute
Step 2 Combine like terms
Step 3 Solve by undoing in reverse.
Step 4 Check
Guided Practice #1
𝟕𝒙 – 𝟑𝒙 – 𝟖 = 𝟐𝟒
Step 1 is not needed but… on the left side of the equation I notice that I have two like terms (𝟕𝐱 and −𝟑𝐱).
Since the variables are alike I can combine them to get 𝟒𝐱.
𝟒𝐱 – 𝟖 = 𝟐𝟒
Make sure student is working along with you in their notes as you review this “animated” problem.
After I combine the terms I have a 2-step equation.
To solve this equation…
Undo the subtract 1st. Then undo the multiplication.

9. 𝟒𝐱 – 𝟖 = 𝟐𝟒 𝟒𝐱 – 𝟖 = 𝟐𝟒 + 𝟖 + 𝟖 𝟒𝐱 = 𝟑𝟐 24 + 8 = 32 Step 1 Distribute
Step 2 Combine like terms
Step 3 Solve by undoing in reverse.
Step 4 Check
𝟒𝐱 – 𝟖 = 𝟐𝟒
Since this equation has – 8, I will add 8 to both sides
= 32
𝟒𝐱 – 𝟖 = 𝟐𝟒
+ 𝟖 + 𝟖
The 8’s on the left cancel out
𝟒𝐱 = 𝟑𝟐
I am left with a 1-step equation

10. The solution that makes the statement true is x = 8
𝟒𝐱 = 𝟑𝟐
In this instance 4x means 4 times x. To undo the multiplication divide both sides by 4
32  4 = 8
The 4’s on the left cancel out leaving x
𝟒𝐱 = 𝟑𝟐
𝟒 𝟒
𝐱 = 𝟖
The solution that makes the statement true is x = 8

11. Step 1 Distribute
Step 2 Combine like terms
Step 3 Solve by undoing in reverse.
Step 4 Check
𝟕𝐱 – 𝟑𝐱 – 𝟖 = 𝟐𝟒
𝟕(𝟖)−𝟑(𝟖) – 𝟖 = 𝟐𝟒
𝟓𝟔−𝟐𝟒 – 𝟖 = 𝟐𝟒
𝟐𝟒= 𝟐𝟒

12. You Try #1
𝟖𝐱−𝟑𝐱−𝟏𝟎 = 𝟐𝟎
Answer: x = 6

13. Moving On…
Section 1: Combining Like Terms Section 2: Distributive Property Section 3: Multi-Steps with Fractions.

14. 5x + 3(x +4) = 28 5x + 3(x +4) = 28 5x +3x + 12 = 28 8x + 12 = 28
Step 1 Distribute
Step 2 Combine like terms
Step 3 Solve by undoing in reverse.
Step 4 Check
Guided Practice #2
5x + 3(x +4) = 28
In this instance I begin on the left side of the equation…
I recognize the distributive property as 3(x+4). I must simplify that before I can do anything else.
5x + 3(x +4) = 28 5x
+3x
+ 12
= 28
After I do the distributive property I see that I have like terms (5x and 3x) that I have to combine them to get 8x before I can solve this equation…
8x + 12 = 28

15. I am now left with a 2-step equation
Step 1 Distribute
Step 2 Combine like terms
Step 3 Solve by undoing in reverse.
Step 4 Check
8x + 12 = 28
I am now left with a 2-step equation
The left side has To undo the +12, I subtract 12 from both sides
8x + 12 = 28
28 – 12 = 16
The 12’s on the left cancel out leaving 8x
8x = 16
I am left with a
1-step equation

16. 8x = 16
On the left side 8x means 8 times x. To undo the multiplication I divide both sides by 8
8x = 16
The 8’s on the left cancel out leaving x
16  8 = 2
x = 2
The solution that makes the statement true is x = 2

17. Step 1 Distribute
Step 2 Combine like terms
Step 3 Solve by undoing in reverse.
Step 4 Check
𝟓𝐱 + 𝟑(𝐱 +𝟒) = 𝟐𝟖
𝟓 𝟐 +𝟑(𝟐+𝟒) = 𝟐𝟖
𝟏𝟎+𝟑(𝟔) = 𝟐𝟖
𝟏𝟎+𝟏𝟖 = 𝟐𝟖
𝟐𝟖= 𝟐𝟖

18. You Try #2
𝟐 𝐱−𝟏 +𝟑𝐱 =𝟑
Answer: x = 1

19. Distributing a Negative
Distributing a negative number is similar to using the distributive property.
However, students get this wrong because they forget to use the rules of integers
Quickly the rules are…when multiplying, if the signs are the same the answer is positive. If the signs are different the answer is negative

20. Step 1 Distribute
Step 2 Combine like terms
Step 3 Solve by undoing in reverse.
Step 4 Check
Guided Practice #3
4x – 3(x – 2) = 21
I begin by working on the left side of the equation… In this problem I have to use the distributive property. However, the 3 in front of the parenthesis is a negative 3.
When multiplying here, multiply the -3 by both terms within the parenthesis. Use the rules of integers.
4x – 3(x – 2) = 21 4x
– 3x
+ 6
= 21
After doing the distributive property, I see that I can combine the 4x and the -3x to get 1x or x :
x = 21

21. The 6’s cancel out leaving x
Step 1 Distribute
Step 2 Combine like terms
Step 3 Solve by undoing in reverse.
Step 4 Check
x = 21
After combining those like terms, you are left with a simple one step equation. To undo the +6 subtract 6 from both sides of the equation
21 – 6 = 15
x = 21
The 6’s cancel out leaving x
x = 15
The solution is x = 15

22. Step 1 Distribute
Step 2 Combine like terms
Step 3 Solve by undoing in reverse.
Step 4 Check
𝟒𝒙 – 𝟑(𝒙 – 𝟐) = 𝟐𝟏
𝟒 𝟏𝟓 −𝟑(𝟏𝟓−𝟐) = 𝟐𝟏
𝟔𝟎−𝟑(𝟏𝟑) = 𝟐𝟏
𝟔𝟎−𝟑𝟗 = 𝟐𝟏
𝟐𝟏= 𝟐𝟏

23. You Try #3
𝟔𝐱−𝟐 𝐱−𝟓 =𝟒𝟔
Answer: x = 9

24. Moving On…
Section 1: Combining Like Terms Section 2: Distributive Property Section 3: Multi-Steps with Fractions and Decimals.

25. Clearing the Fractions
Review
Step 1:
Look at all of the denominators in the equation and find the LCD.
Step 2:
Multiply both sides of the equation by the LCD.
𝐱 𝟐 + 𝟑 𝟓 =𝟑
𝟏𝟎 ( 𝐱 𝟐 + 𝟑 𝟓 )=𝟑 (𝟏𝟎)
10 ( 𝒙 𝟐 ) + 𝟏𝟎 ( 𝟑 𝟓 ) = 10(3)
The LCM is the smallest number that both 2 and 5 divide into evenly. 𝟓𝐱
+ 𝟔
= 𝟑𝟎
Fractions are gone!
LCD of 2,5 =𝟏𝟎
Clearing the Fractions

26. Clearing the Decimals Review 𝟏𝟎𝟎 𝟎.𝟐𝟓𝒙+𝟎.𝟔 =𝟏𝟎𝟎 𝟎.𝟏
Step 1:
Look for the decimal with the most digits.
Step 2:
Multiply both sides of the equation by that power of 10.
Decimal with the most digits.
𝟎.𝟐𝟓𝒙+𝟎.𝟔=𝟎.𝟏
𝟏𝟎𝟎 𝟎.𝟐𝟓𝒙+𝟎.𝟔 =𝟏𝟎𝟎 𝟎.𝟏
Teachers… skip this slide if you have your students deal with decimals the normal way.
𝟏𝟎𝟎 𝟎.𝟐𝟓𝒙 +𝟏𝟎𝟎 𝟎.𝟔 =𝟏𝟎𝟎(𝟎.𝟏)
𝟐𝟓𝒙+𝟔𝟎 = 𝟏𝟎
Decimals are gone!
Clearing the Decimals

27. 𝟑 𝟏𝟎 𝐱+𝟐 = 𝟏𝟐 𝟑 𝟏𝟎 𝐱+𝟐 =𝟏𝟐 𝟑 𝟏𝟎 𝐱 + 𝟔 𝟏𝟎 =𝟏𝟐 Step 1 Distribute
Step 2 Combine like terms
Step 3 Solve by undoing in reverse.
Step 4 Check
Guided Practice #4
𝟑 𝟏𝟎 𝐱+𝟐 = 𝟏𝟐
In this example, there are actually a couple of ways that you can correctly handle the fraction… However, we are going to keep with our steps and distribute first. Then we will clear fractions.
𝟑 𝟏𝟎 𝐱+𝟐 =𝟏𝟐
𝟑 𝟏𝟎 𝐱
+ 𝟔 𝟏𝟎
=𝟏𝟐
Now we will clear these fractions.

28. 𝟑 𝟏𝟎 𝐱 + 𝟔 𝟏𝟎 = 𝟏𝟐 𝟑𝐱+𝟔=𝟏𝟐𝟎 Step 1 Distribute
Step 2 Combine like terms
Step 3 Solve by undoing in reverse.
Step 4 Check 10
𝟑 𝟏𝟎 𝐱 + 𝟔 𝟏𝟎 = 𝟏𝟐 10 10
To clear the fractions, we multiply both sides of the equation by the LCD… which is 10.
The result is an equation without fractions.
𝟑𝐱+𝟔=𝟏𝟐𝟎
There are no like terms to combine.
All we need to do now is to solve the 2-step equation by undoing in reverse.

29. 𝟑𝐱+𝟔=𝟏𝟐𝟎 − 𝟔 −𝟔 𝟑𝐱 = 𝟏𝟏𝟒 𝟑 𝟑 𝐱 =𝟑𝟖 Step 1 Distribute
Step 2 Combine like terms
Step 3 Solve by undoing in reverse.
Step 4 Check
Subtract 6 from both sides
𝟑𝐱+𝟔=𝟏𝟐𝟎
− 𝟔 −𝟔
𝟑𝐱 = 𝟏𝟏𝟒
Then divide by 3.
𝟑 𝟑
𝐱 =𝟑𝟖
The solution to the equation is x = 38
Let’s move on to another practice problem… but don’t forget that you should always check your solution.

30. Guided Practice #4
𝟑 𝟐 𝟑𝐱+𝟓 =−𝟐𝟒
Answer: x = -7

31. You Try #4
𝟏 𝟒 𝐝−𝟑 =−𝟏𝟓
Answer: x = -57

32. Guided Practice #5
𝟎.𝟕𝟓 𝟔+𝐝 =𝟏𝟐
Answer: d = 10

33. You Try #5
𝟎.𝟐 𝐜−𝟑 =−𝟏𝟎
Answer: c = -47

34. Always More Than One Way
Take this opportunity to show students that there is always more than one way to solve a problem. Problems like this may appear on state test and students need to know how to answer it.

35. Always More Than One Way

36. Summary
With multi-step equations… you have to be able to analyze the equation and decide what steps must be taken to solve it…

37. Summary Distribute if you can. Then combine the like terms.
Now you have a more simplified equation. Solve it by undoing in reverse.
Check your answer.
These steps can be used to solve any equation. You will not always have to do Step 1 and Step 2.

38. Summary There is always more than one way to solve a problem.
In this equation, you can distribute first if you want to, but… we learned today that you can also divide both sides by 3.