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

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

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

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

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

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

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

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

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

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

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

Multiplication Principle Example:

Multiplication Principle Another Way:

Using Both Principles Usually best to use Addition Principle first.

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