# The Distributive Property

The Distributive Property states that, for all real numbers $x$ , $y$ , and $z$ ,

$x\left(y+z\right)=xy+yz$

Example:

Take $3\left(6+7\right)$ . We can add within the parentheses, and then multiply:

$3\left(6+7\right)=3\left(13\right)=39$

Or, you can multiply each addend by the $3$ , and then add.

$3\left(6+7\right)=3\left(6\right)+3\left(7\right)=18+21=39$

Either way, you get the same answer.

## Using the Distributive Property with Variables

You can use the Distributive Property to simplify algebraic expressions.

Example:

$\begin{array}{l}7p+3q-21p+8q=\left(7-21\right)p+\left(3+8\right)q\\ =-14p+11q\end{array}$

## Using the Distributive Property to do Mental Multiplication

You can sometimes use the Distributive Property to break difficult multiplication problems into two or more easy ones that you can do in your head.

Example 1:

$\begin{array}{l}7×997=7\left(1000-3\right)\\ =7\left(1000\right)-7\left(3\right)\\ =7000-21\\ =6979\end{array}$

Example 2:

$\begin{array}{l}1309×3=\left(1000+300+9\right)3\\ =1000\left(3\right)+300\left(3\right)+9\left(3\right)\\ =3000+900+27\\ =3927\end{array}$