1. Introduction to Equations Cronnelly
Lesson 6.2.2
Teachers, be sure to look a the notes section on each slide for additional instructions and answers.

2. Prior
Earlier in this chapter you learned how to simplify expressions by combining the “like terms”.
Example: 3𝑥 − 𝑥 + 1 − 3
= 𝟐𝒙 − 𝟐
like terms
like terms

3. Prior You also learned that “equal opposites” create a zero pair.
Example: 15 – 15 = 𝟎
Example: −2𝑥 + 2𝑥 = 𝟎

4. Today…
You are going to utilize those skills to help you solve equations.

5. Let’s Get Started…
Section 1: Introduction to Equations Section 2: Review of Equations with One Operation. Section 3: Equations with Two Operations. Section 4: Combining Like Terms

6. So an equation is like a statement:
What is an Equation?
An equation says that two things are equal. It will have an equals sign "=" like this: = 10 − 1 This equation says… What is on the left (7 + 2) is equal to what is on the right (10 − 1).
So an equation is like a statement:
"this equals that"

7. What is an Equation?
Here is another example, but this one has an expression that includes the letter 𝒙.
The letter 𝒙 is called a variable. A variable is just a letter that stands for a number.
By the way…
The 4 pushed up to the 𝒙 means what?
+, −, 𝒙, 𝒐𝒓 ÷
multiply

8. We have to figure out what that number is!
Here is the dilemma …
There is some number (𝑥) that if you multiply it by 4 and then subtract 7 from that total, the result is 5.
We have to figure out what that number is!

9. Goal
To solve an equation, you need to get the variable by itself on one side of the equal sign.
On this particular equation, to get the 𝑥 by itself…
We would need to get rid of the
4 and the – 7.
However, you can’t just erase them.
That would not be a “legal” move”!

10. 𝑥 + 3 = 5 Goal Look at this very simple equation…
What would we need to get rid of
to get the 𝑥 by itself?
𝑥 + 3 = 5

11. −𝟒 + 𝒙 = 𝟐𝟓 Goal Look at this equation…
What would we need to get rid of
to get the 𝑥 by itself?
−𝟒 + 𝒙 = 𝟐𝟓

12. Before we look at exactly how to get a variable by itself, I wanted to show you a couple of pictures that I think will help you understand one of the new concepts that we will learn later on in the lesson.
I have a couple of pictures of some Weightlifters.

13. Weightlifters
I’m not sure who this athlete is, but previous students gave him a name.
He got his first name, “Harry”, from HLMS. Then, SMS students gave him his last name….
Harry Pitts
And yes… there are female Olympic weightlifters too. This is…
Zoe Smith from the UK

14. Notice in the picture that on each side of the barbell, there is the exact same number of weights.
Can anyone take a guess as to what would happen if the weightlifters took a weight off one side, but not the other side… then tried to lift the barbell? 4 4 2 2
Note: FYI…Harry has 4 on each side. Those little bitty yellow ones are weights too.
ANSWER
It might throw the weightlifter off-balance and he would not be able to lift it.
He might lift them, but then tip over and injure himself.
If he was able to lift them… then over time, one arm muscle would end up being bigger than the other, and that would be very unattractive.

15. if Harry adds weight to one side of the barbell,
Would it be safe to say…
if Harry adds weight to one side of the barbell,
then Harry must add the exact same amount of weight to the other side of the barbell…
So that it is balanced?
YES

16. if Zoe takes away weight from one side of the barbell,
Would it be safe to say…
if Zoe takes away weight from one side of the barbell,
then Zoe must take away the exact same amount of weight from the other side of the barbell…
So that it is balanced?
YES

17. Golden Rule of Algebra 𝒙 +𝟑 = 𝟓 − 𝟑 − 𝟑 𝒙 = 𝟐
What ever you do to one side of an equation, you must ALWAYS do to the other side.
𝒙 +𝟑 = 𝟓
− 𝟑 − 𝟑
𝒙 = 𝟐
The goal is to get rid of the +3 so that the x is by itself. You can subtract 3 to get rid of it, but you must also do the same to the other side.
GOAL

18. Steps to Solving any Equation
Distribute to get rid of any parentheses.
Step 2
Combine any like terms.
Step 3
Solve the simplified equation by “undoing”.
Step 4
Check your solution.
The first 2 steps are not always necessary.
Let student know that we will not have any problems today that need step 1 and step 2. That will be tomorrow.
This is where you are going to work on getting that variable
by itself.
We use inverse operations to do the “undoing”.

19. Inverse Operations What is the inverse of Addition (+)?
What is the inverse of Subtraction (-)?
What is the inverse of Multiplication (•)?
What is the inverse operation of Division (÷)?
Subtraction (-)
Addition (+)
Division (÷)
Multiplication (•)

20. Moving On…
Section 1: Introduction to Equations Section 2: Review of Equations with One Operation. Section 3: Equations with Two Operations. Section 4: Combining Like Terms

21. Let’s look at some simple equations first… Watch how we use the inverse operations to get the variable alone.

22. Click to Watch Demo 𝒙 + 𝟓 = 𝟐𝟖 𝟐𝟑 + 𝟓 = 𝟐𝟖 𝟐𝟖 = 𝟐𝟖 𝑥 + 5 = 28 − 𝟓 − 𝟓
Step 1
Distribute to get rid of any parentheses.
Step 2
Combine any like terms.
Step 3
Solve the simplified equation by “undoing”.
Step 4
Check your solution.
Click to Watch Demo
Check Solution
𝒙 + 𝟓 = 𝟐𝟖
𝟐𝟑 + 𝟓 = 𝟐𝟖
𝟐𝟖 = 𝟐𝟖
𝑥 + 5 = 28
− 𝟓
− 𝟓
This is an animated demonstration of how to work this problem. Click once to see the entire demo. 𝑥 = 𝟐𝟑

23. Guided Practice # 1 𝑥−6=−18 Step 1
Distribute to get rid of any parentheses.
Step 2
Combine any like terms.
Step 3
Solve the simplified equation by “undoing”.
Step 4
Check your solution.
Guided Practice # 1
Check Solution
𝑥−6=−18
Explain how to work this problem on the board. Have student write it in their notes.
Answer: x = -12

24. Guided Practice # 2 36= 𝑥 −13 Step 1
Distribute to get rid of any parentheses.
Step 2
Combine any like terms.
Step 3
Solve the simplified equation by “undoing”.
Step 4
Check your solution.
Guided Practice # 2
Check Solution
36= 𝑥 −13
Explain how to work this problem on the board. Have student write it in their notes.
Answer: x = 49

25. You Try 1) 𝑛+8=−9 2) 4=𝑝+19 3) −4+𝑎=15 Check your solutions ! n = -17
P = -15
a = 19

26. Let’s look at a couple of equations that deal with either multiplication or division… Again, your goal is to get the variable alone.

27. Click to Watch Demo 7𝑥 = −21 7𝑥 =−21 7(−3)=−21 −21=−21 7 7 = −3 𝑥
Step 1
Distribute to get rid of any parentheses.
Step 2
Combine any like terms.
Step 3
Solve the simplified equation by “undoing”.
Step 4
Check your solution.
Click to Watch Demo
Check Solution
7𝑥 = −21
7𝑥 =−21
7(−3)=−21
−21=−21 7 7
This is an animated demonstration on how to work this problem. 𝑥 = −3

28. Click to Watch Demo = 𝑥 𝟒𝟖 𝑥 −8 = −6 −𝟖 • •−𝟖 Step 1
Distribute to get rid of any parentheses.
Step 2
Combine any like terms.
Step 3
Solve the simplified equation by “undoing”.
Step 4
Check your solution.
Click to Watch Demo
Check Solution
𝒙 −𝟖 =−𝟔
𝟒𝟖 −𝟖 =−𝟔
−𝟔 =−𝟔
−𝟖 •
𝑥 −8 = −6
•−𝟖
This is an animated demonstration on how to work this problem. 𝑥 = 𝟒𝟖

29. Guided Practice #3 −3𝑥=24 Step 1
Distribute to get rid of any parentheses.
Step 2
Combine any like terms.
Step 3
Solve the simplified equation by “undoing”.
Step 4
Check your solution.
Guided Practice #3
Check Solution
−3𝑥=24
Explain how to work this problem on the board. Have student write it in their notes.
Answer: x = -8

30. Guided Practice #4 𝑦 7 =−10 Step 1
Distribute to get rid of any parentheses.
Step 2
Combine any like terms.
Step 3
Solve the simplified equation by “undoing”.
Step 4
Check your solution.
Guided Practice #4
Check Solution
𝑦 7 =−10
Answer: y = -70

31. Guided Practice #5 −𝑥 =12 Step 1
Distribute to get rid of any parentheses.
Step 2
Combine any like terms.
Step 3
Solve the simplified equation by “undoing”.
Step 4
Check your solution.
Guided Practice #5
Check Solution
−𝑥 =12
Answer: x = -12

32. You Try 4) 3𝑐=−18 5) −42=−6𝑏 6) 𝑥 4 =−8 7) −𝑦=−23
Check your solutions !
4) 3𝑐=−18 5) −42=−6𝑏 6) 𝑥 4 =−8 7) −𝑦=−23
4) c = -6
5) b = 7
6) x = -32
7) y = 23

33. Moving On…
Section 1: Introduction to Equations Section 2: Review of Equations with One Operation. Section 3: Equations with Two Operations. Section 4: Combining Like Terms

34. …of order of operations.
When an equation involves more than one operation, you “undo” in reverse.
In reverse of what?
…of order of operations.
4𝑥−7=5

35. What does that mean?
Looking at the below equation, we have multiplication (4x) and subtraction (- 7).
Order of operations requires multiplication to be done before subtraction… PEMDAS
However, when solving an equation, you undo in the reverse order. You have to undo the subtraction first, then the multiplication.
Then, get rid of this.
4𝑥−7=5
Get rid of the
subtraction first!

36. Click to Watch Demo 4𝑥−7 = 5 4𝑥−7=5 4 3 −7=5 5 =5 + 𝟕 4𝑥 = 12 𝟒 𝟒 𝑥 =
Step 1
Distribute to get rid of any parentheses.
Step 2
Combine any like terms.
Step 3
Solve the simplified equation by “undoing”.
Step 4
Check your solution.
Click to Watch Demo
Check Solution
4𝑥−7 = 5
4𝑥−7=5
4 3 −7=5
5 =5
+ 𝟕
+ 𝟕
This is an animated demo of how to work this problem. 4𝑥 = 12 𝟒 𝟒 𝑥 = 𝟑

37. Guided Practice #6 −5𝑥 −6=14 Step 1
Distribute to get rid of any parentheses.
Step 2
Combine any like terms.
Step 3
Solve the simplified equation by “undoing”.
Step 4
Check your solution.
Guided Practice #6
Check Solution
−5𝑥 −6=14
Answer: x = -4

38. Guided Practice #7 6−3𝑥=21 Step 1
Distribute to get rid of any parentheses.
Step 2
Combine any like terms.
Step 3
Solve the simplified equation by “undoing”.
Step 4
Check your solution.
Guided Practice #7
Check Solution
6−3𝑥=21
Answer: x = -5

39. Guided Practice #8 𝑥 2 −4=−10 Step 1
Distribute to get rid of any parentheses.
Step 2
Combine any like terms.
Step 3
Solve the simplified equation by “undoing”.
Step 4
Check your solution.
Guided Practice #8
Check Solution
𝑥 2 −4=−10
Answer: x = -12

40. You Try 8) 3𝑥+2=20 9) 5+2𝑛=−1 10) −𝑥 4 +6=−5 11) n − 2 8 =4
Check your solutions !
8) 3𝑥+2=20 9) 5+2𝑛=−1 10) −𝑥 4 +6=−5 11) n − 2 8 =4
8) x = 6
9) n = -3
10) x = 44
11) n = 34

41. Last Section…
Section 1: Introduction to Equations Section 2: Review of Equations with One Operation. Section 3: Equations with Two Operations. Section 4: Combining Like Terms

42. 𝑥 + 4 + 2𝑥 = 16 3𝑥 + 4=16 − 𝟒 − 𝟒 3𝑥 = 12 𝟑 𝟑 𝑥=4 Click to Watch Demo
Step 1
Distribute to get rid of any parentheses.
Step 2
Combine any like terms.
Step 3
Solve the simplified equation by “undoing”.
Step 4
Check your solution.
Click to Watch Demo
𝑥 𝑥 = 16 3𝑥
+ 4=16
− 𝟒
− 𝟒 3𝑥 = 12 𝟑 𝟑
𝑥=4

43. Guided Practice #9 7𝑎+2−5𝑎=10 Step 1
Distribute to get rid of any parentheses.
Step 2
Combine any like terms.
Step 3
Solve the simplified equation by “undoing”.
Step 4
Check your solution.
Guided Practice #9
Check Solution
7𝑎+2−5𝑎=10
Answer: x = 4

44. You Try
Check your solutions !
12) −5𝑎+𝑎+6=26
Answer: −5

45. Closure What was the male weightlifter’s name?
What was the female weightlifter’s name?
What does weightlifting have to do with equations?
What is the Golden Rule of Algebra?
What are the 4 steps of solving any equation?
When solving an equation with multiplication and subtraction, which do you “undo” first? Why?
How do you check your solution after solving an equation?
1)Harry Pitts
2)Zoe Smith
3)When a weightlifter adds or takes away weight from one side of the barbell, then he must take it away from the other side of the barbell. Solving equation is similar.
4)Whatever you do to one side of an equation, you must always do the same to the other side.
5)distribute, combine like terms, solve simplified equation, check.
6)You undo the subtraction first because you solve equation in reverse order of order of operations.
7)Substitute your answer back into the equation to see if it works out to be a true statement.
−𝟗𝒙 − 𝟑 = 𝟐𝟒

46. Closure
DON”T FORGET…
Always balance or the result can be bad.