1. Unit 1 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 a solution of all real numbers?

2. you must do to the other side
Solving Equation Rule
Any change applied to one side of an equation must be applied to the other side in order to keep the balance.
What you do to one side
you must do to the other side

3. Solving Equations with variables on both sides
Step 1: Draw a line straight down from the equal sign to separate the left side from the right.
Step 2: Simplify each side separately.
Step 3: Use inverse operations to collect the variables on one side of the equation and the constants on the other side of the equation.
Step 4: Solve using the steps from Day 1 & 2.
Step 5: Check your answer by plugging it back into the original equation and simplifying.

4. Example 1: Solve the equations.
a) 7x + 19 = -2x b) x + 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 = c
+ 9y
+ 9y
- 4c
- 4c 80
= 15y 6c
= 24 15 6 15 6 80 15
= y
c = 4 16 3
= y

6. Example 3: Solve the equation.
4(1 – x) + 3x = -2(x + 1) 4
- 4x
+ 3x
= -2x
- 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
- 15n
- 15n
-13n - 36
= - 10
+ 36
+ 36
-13n
= 26
-13
-13
n = -2

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

9. Example 5: Solve the equations (True or False).
a) x - 2x + 3 = 3 - x b) x + 24 = 5(x - 5)
-x + 3 = 3 - x
5x + 24
= 5x
- 25
+ x
+ x
- 5x
- 5x 3
= 3 24
= -25
true statement
false statement
all real numbers
no solution

10. If you talk for 12 minutes, it will not matter which company you use.
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
If you talk for 12 minutes, it will not matter which company you use.
- .03x
- .03x
.36 = .03x
.03
.03
12 = x

11. 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
In 15 weeks, Justin and Tyson will weigh the same amount. x
= x
+ 1x
+ 1x
x = 195
- 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 a solution of all real numbers?
Take 1 minute to write 2 sentences answering the essential questions.