Introduction to Equations Lesson 6.2.1 Teachers, be sure to look a the notes section on each slide for additional instructions and answers.

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

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

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: 7 + 2 = 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"

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

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!

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 and the . 4 -7 However, you can’t just erase them. That would not be a “legal” move”!

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

−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

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

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. This is where you are going to work on getting that variable by itself. We use inverse operations to do the “undoing”. Let student know that we will not have any problems today that need step 1 and step 2. That will be tomorrow.

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 (•)

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

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

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

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

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

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

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

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

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

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

…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

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!

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

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

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

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

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

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 + 4 + 2𝑥 = 16 3𝑥 + 4=16 − 𝟒 − 𝟒 3𝑥 = 12 𝟑 𝟑 𝑥=4

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

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

Practice:

Closure 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? Get the variable by it’s self distribute, combine like terms, solve simplified equation, check. 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 You undo the subtraction first because you solve equation in reverse order of order of operations. Substitute your answer back into the equation to see if it works out to be a true statement.