1. 3.4 Solving Equations with Variables on Both Sides
Objective: To solve equations with variables on both sides of the equation.
Warm-up:
6x – 2 = –10
2x – 6x + 20 = –16
6x = –8
–4x + 20 = –16
x = –4/3
–4x + 20 = –16 –5
–20
–4x = –36
x = 9 +5
How would you rate yourself on solving these problems?
GREAT!! OK – getting there Need some help, but ok Need to come in for help

2. Can you check you answer? How?
1st: Variables to one side  How do you decide who to move? 2nd: Constants to the other side  Who must you move?
1) Which side has the smaller coefficient?
7x + 19 = -2x – 17
+2x x
2) Add 2x to both sides.
9x + 19 = –17
3) Simplify.
–19 –19
4) Subtract 19 from both sides.
9x = –36
5) Simplify.
6) Divide both sides by 9.
7) Simplify.
Can you check you answer? How?

3. Can you check you answer? How?
1st: Variables to one side  How do you decide who to move? 2nd: Constants to the other side  Who must you move?
1) Which side has the smaller coefficient?
6x + 22 = -3x + 31
+3x x
2) Add 3x to both sides.
9x + 22 = 31
3) Simplify.
–22 –22
4) Subtract 22 from both sides.
9x = 9
5) Simplify.
6) Divide both sides by 9.
7) Simplify.
Can you check you answer? How?

4. How is this one different?
Let’s try some!!
1) 80 – 9y = 6y
2) 64 – 12 w = 6w
+9y
+12w
3) 4(1 – x) + 3x = –2(x + 1)
How is this one different?
4 – 4x + 3x = –2x – 2
3x + 4 = 10 – 3x + 6
4 – x = –2x – 2
3x + 4 = 16 – 3x
+2x +2x
+3x x
4 + x = – 2
6x + 4 = 16 –4 –4
x = – 6
x = 2
6x = 12

5. 1st: Distribution  How do you decide? Why?
Ex.
4(1 – x) + 3x = –2(x + 1)
1) Distribution.
4 – 4x + 3x = –2x – 2
2) Simplify.
–x + 4 = –2x – 2
+2x x
3) Add 2x to both sides.
x + 4 = –2
4) Simplify.
–4 –4
5) Subtract 4 from both sides.
x = –6
6) Simplify.
Can you check you answer? How?

6. +3x +3x –4 –4 6 6 1st: Distribution  How do you decide? Why?
2) Simplify.
+3x x
3) Add 3x to both sides.
4) Simplify.
– –4
5) Subtract 4 from both sides.
6) Simplify.
7) Divide 6.
8) Simplify.

7. More Examples…You try these!!
5) 9x + 22 = –3x + 46
+3x x
4x + 6 = 18 – 4x + 12
12x + 22 = 46
4x + 6 = 30 – 4x
–22 –22
+4x x
12x = 24
8x + 6 = 30
–6 –6
8x = 24

8. 2 More of the fun type!! +3x +3x +3x +3x 6 – 4x + x = –3x – 3
–21 = –21
identity
6 = – 3
inconsistent
This kind of equation is called “identity equation”. When you get an identity equation, you just write
and conclude that
Many solutions
This kind of equation is called “inconsistent equation”. When you get an inconsistent equation, you just write
and conclude that
No solution

9. 2 More questions of the fun type!!
9) -4(1+2x) - 2x = –5(2x+1)
10) 2(3x – 1) – 6 = x
– 4 – 8x – 2x = –10x – 5
6x – 2 – 6 = –8 + 6x
6x – 8 = –8 + 6x
–4 – 10x = –10x – 5
–6x –6x
+10x x
–8 = –8
identity
–4 = – 5
inconsistent
This kind of equation is called “identity equation”. When you get an identity equation, you just write
and conclude that
Many solutions
This kind of equation is called “inconsistent equation”. When you get an inconsistent equation, you just write
and conclude that
No solution

10. Summary
When solving the equations with variables at both sides, you need apply the distributive property, combining like terms, and other skills.
Always simplify the equation first.
Always work on the variable before to work on the constant. Move the variables to one side and the constants to the other side.
Moving the variable term with less coefficient to the other side that the variable term with larger coefficient.
Be aware of two extreme situations:
No solution  you meet an inconsistent equation when solving.
Many solutions  you meet an identity equation when solving.

11. Assignment
P 157 #’s