1. Solving Linear Equations
Equations with Variable on Both Sides

2. Solving Equations
To find the solution to an equation we must isolate the variable.
Sometimes there is more than one variable term; they need to be combined before the variable can be isolated.
Then we isolate the variable by performing operations that will eliminate (cancel) the other numbers from the expression.

3. Variables on Same Side
If there are two or more variable terms on the SAME SIDE of the equation, combine them as you do like terms:
2x x - 7 = 6
Change Subtract (Keep-Change-Change.)
2x (-4x) + (-7) = 6
Re-arrange terms. (Commutative Property)
2x + (-4x) (-7) = 6
Add Like Terms.
-2x + (-2) = 6

4. Variables on Same Side
Finish Solving the Equation:

5. Variables on Different Sides
If there are two or more variable terms on DIFFERENT SIDES of the equation, you must move them to the same side using the Addition Property of Equality.
2x + 5 = 4x - 11
Change Subtract (Keep-Change-Change.)
2x + 5 = 4x + (-11)
Adding (-4x) to both sides, gets rid of 4x term on right side.
2x (-4x) = 4x + (-11)+ (-4x)
Re-arrange terms.
2x + (-4x) + 5 = 4x + (-4x) + (-11)
Combine like terms.
-2x + 5 = 0 + (-11)

6. Variables on Different Sides
Finish Solving the Equation:

7. Solving Equations with More Than One Variable Term
The Steps:
If more than one variable term is on the same side of the equation, add like terms.
If more than one variable term is on different sides of the equation, add the opposite of the right term to both sides. (Addition Property of Equality)
Eliminate constants from left side. (Add. Prop.)
Eliminate coefficients from left side. (Mult. Prop.)

8. Examples Check: 3x + 5 - 2x = 7 3(2) + 5 - 2(2) = 7 6 + 5 - 4 = 7
Change subtraction to addition. (Keep-Change-Change.)
Re-arrange and Add Like Terms.
Add (+4) to both sides.
Check: 3x x = 7
3(2) (2) = 7
= 7
= 7
7 = 7 √

9. Examples Check: 3x = 20 - 2x 3(4) = 20 - 2(4) 12 = 20 - 8 12 = 12 √
Change subtraction to addition. (Keep-Change-Change.)
Add 2x to both sides.
Add Like Terms.
Multiply 1/5 on both sides.
Check: 3x = x
3(4) = (4)
12 =
12 = 12 √

10. Examples Check: 4x - 8 = x - 7x + 2 4(1) - 8 = 1 - 7(1) + 2
Change subtraction to addition. (Keep-Change-Change.)
Add Like Terms.
Add 6x to both sides.
Add Like Terms.
Add +8 to both sides.
Check:
4x - 8 = x - 7x + 2
4(1) - 8 = 1 - 7(1) + 2 =
-4 =
-4 = -4 √
Multiply 1/10 on both sides.

11. Try These!

12. Solutions!

13. Solutions!