Multiplication Principle

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

Example:

so if 6x = 12, then 18x = 36 for the same value of x (which in this case is x = 2).

The way we use the multiplication principle to solve equations is that it allows us to isolate the variable by getting rid of a factor that is multiplying the variable.

Example:  2x = 6

To get rid of the 2 that is multiplying the x, we can divide both sides of the equation by 2, or multiply by its reciprocal (one-half).

Either divide both sides by 2:

 

or multiply both sides by a half:

• Whether you prefer to think of it as dividing by the number or multiplying by its reciprocal is not important, although when the coefficient is a fraction it is easier to multiply by the reciprocal:
•  

Example:  

Multiply both sides by the reciprocal of the coefficient, or