1. Solving Equations A Solution
A value of the variable that makes the equation a true statement

2. Equations
Example: x + 2 = 5
TRUE if x = 3
FALSE if x = anything else
The Solution is x = 3

3. Special Cases Example: x = x + 1 NEVER TRUE No such number exists
Called a contradiction

4. Special Cases Example: 2x = x + x ALWAYS TRUE True for any number
Called an identity

5. Equivalent Equations
Have the same solution
Example: x + 2 = 5
x – 1 = 2
x + 4 = 7
All have solution x = 3

6. Addition Principle
Adding (or subtracting) the same number to both sides of an equation does not change its solution.

7. Addition Principle Example: 6 + x = 8 3+6 + x = 3+8 9 + x = 11
Are equivalent equations
Both have the same solution

8. Addition Principle Example: 6 + x = 8 -6 -6 x = 2
x = 2
Equivalent equation that shows the solution

9. Multiplication Principle
Multiplying (or dividing) same non-zero number to both sides of an equation does not change its solution.

10. Multiplication Principle
Example:
6x = 12
3 • 6x = 3 • 12
18x = 36
Are equivalent equations
Both have the same solution

11. Multiplication Principle
Example:
6x = 12
6x  6 = 12  6
x = 2
Equivalent equation that shows the solution

12. Multiplication Principle
Example:

13. Multiplication Principle
Another Way:

14. Using Both Principles
Usually best to use Addition Principle first.

15. Using Both Principles Example: 2x – 3 = 7 First add 3: 2x = 10
Then  2: x = 5