High School Mathematics: The Distributive Law


BEDMAS Review

Recall the order of operations and BEDMAS:

\displaystyle \text{B: Brackets}

\displaystyle \text{E: Exponents}

\displaystyle \text{D: Division}

\displaystyle \text{M: Multiplication}

\displaystyle \text{A: Addition}

\displaystyle \text{S: Subtraction}

In the event where there is both multiplication and division, one can do multiplication before division. Likewise with addition and subtraction.

Sometimes BEDMAS is referred to as PEDMAS where the only difference is the letter P. The letter P stands for parantheses, a synonym for brackets.


The Distributive Law

The Distributive Law is a useful algebra technique which allows us to multiply terms through brackets such as these \displaystyle 6 (x + 3)

It is known that 7 multiplied by 10 gives 70. We can express it like this:

\displaystyle 7 * 10 = 7 * (5 + 5) = (7*5) + (7*5) = 35 + 35 = 70

We just decomposed the 10 into 5 plus 5, multiplied 7 by 5 twice and added the resulting 35s together to get 70.


Here is another motivating example:

\displaystyle 4 * (22) = 4 * (20 + 2) = (4*20) + (4*2) = 80 + 8 = 88

Here in this case, we just decomposed the 22 into 20 plus 2, had 4 multiplied by 20 and 4 multiplied by 2, added their products together to get 88.


The General Case

Now we have the general case or the theory.

Given real numbers a, b, and c, we have the general case of:

\displaystyle a * (b + c) = ab + ac

The term a multiplied by b plus the term a multiplied by c.

We can extend this further with more terms in the bracket (introducing d and e):

\displaystyle a * (b + c + d + e) = ab + ac + ad + ae

If we really want to we can use all the letters of the English alphabet (and maybe add some Greek letters too!). You get the idea of the distributive law.


Examples

Here are some examples in which we apply the distributive law.

Example 1

Expanding the expression 6(x+3) through the distributive law gives us 6x + 6(3) = 6x + 18.

Example 2

From 12(x + 2), we have 12x + 12*2 = 12x + 24.

Example 3

From x(y + 2), we have xy + 2x.

Example 4

From \dfrac{(x + 6)}{2}, we have \dfrac{x}{2}+ \dfrac{6}{2} = \dfrac{x}{2} + 3.

Example 5

Here is a more involved example. Suppose we want to solve for the value x. We are given the equation 7(x + 2) = 10. Solving for x is as follows:

 

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