1. Math Journal
− 5 8 𝑥=− 𝑥+6=15
𝑥 −(2𝑥+2) =− −3(2𝑥+8)=−18

2. Unit 3 Day 3: Solving Equations With Variables on Both Sides
Essential Questions: How do we solve equations with variables on both sides? When does an equation have no solution or many solutions?

3. Solving Equations With Variables on Both Sides
1) Simplify each side separately.
2) Use inverse operations to collect the variables on one side of the equation and the constants on the other side of the equation.
3) Continue to solve the equation.

4. Example 1: Solve the equations.
a) 7x + 19 = -2x b) 6x + 22 = 3x + 31
+ 2x
+ 2x
- 3x
- 3x 9x
+ 19 = 55 3x
+ 22 = 31
- 19
-19
- 22
-22 9x 3x
= 36
= 9 9 9 3 3
x = 4
x = 3

5. Example 2: Solve the equations. a) 80 – 9y = 6y b) 10c = 24 + 4c15 15 6 6 80 15
= y
c = 4 16 3
= y

6. Example 3: Solve the equation. 4(1 – x) + 3x = -2(x + 1)
- 2 4
- 1x
= - 2x - 2
+ 2x
+ 2x
4 + x
= - 2
- 4
- 4
x = -6

7. Example 4: Solve the equation.
9(n – 4) – 7n = 5(3n – 2) 9n
- 36
- 7n
= 15n
- 10 2n
- 36
= 15n - 10
- 2n
- 2n
-36
= 13n - 10
+ 10
+ 10
-26
= 13n 13 13
-2 = n

8. Equations With No Solution or Infinitely Many Solutions
Happens when the variable is eliminated and you are left with a true or false statement.
True Statement
Example: 5 = 5
Infinitely Many Solutions
(any number substituted for the variable will work)
False Statement
Example: 5 = 2
No Solution
(no number substituted for the variable will work)

9. infinitely many solutions
Example 5: Solve the equations.
a) x - 2x + 3 = 3 - x b) 5x + 24 = 5(x - 5)
-x + 3 = 3 - x
5x + 24
= 5x
- 25
+ x
+ x
- 5x
- 5x 3
= 3 24
= -25
true statement
false statement
no solution
infinitely many solutions

10. Example 6: Phone Company A charges an activation fee of 36 cents and then 3 cents per minute. Phone Company B charges 6 cents per minute with no activation fee. How long is a call that costs the same amount no matter which company is used?
x = .06x
- .03x
- .03x
If you talk for more than ___ minutes, Company __ has the better price.
.36 = .03x
.03
.03
12 = x

11. In 15 weeks, Justin and Tyson will weigh the same amount.
Example 7: Justin and Tyson are beginning an exercise program to train for football season. Justin weighs 150 pounds and hopes to gain 2 pounds per week. Tyson weighs 195 pounds and hopes to lose 1 pound per week. If the plan works, in how many weeks will the boys weigh the same amount?
Justin
Tyson x
= x
+ 1x
+ 1x
x = 195
In 15 weeks, Justin and Tyson will weigh the same amount.
- 150
- 150
3x = 45
x = 15

12. Summary
Essential Questions: How do we solve equations with variables on both sides? When does an equation have no solution or many solutions?
Take 1 minute to write 2 sentences answering the essential question.