Chapter 1.7

The Commutative, Associative, and Distributive Properties (or Laws)

The Commutative Law:  An easy way to remember the commutative law is to think of what people are doing when they travel from a subway to the city for work—they are COMMUTING or changing positions. 

In math, the fact that the position can be changed is called the commutative property

Example 1.  5+2 = 2+5  Is an example of the commutative property for addition.

Example 2.  4(5)= 5(4)  Is and example of the commutative property of multiplication.

Note that   is not the same as .  Therefore, division is not commutative.

Similarly .  So, Subtraction is not commutative.

Quick Check:

1.      Write a problem using the commutative property of addition using the numbers 8 and 4.

2.    Write an example of the commutative property of multiplication using 3 and 4.

Answers to quick check:  1.  8+4=4+8    2.  3(4)=4(3)

The Associative Property.

To associate means to link or connect.  When adding 3 numbers, they can be associated in two ways.

1.      We can add the first and second, and then add that sum to the third.

2.    We can add the second and third and add that sum to the first.

Example 3.  Use 3, 4, and 5 to show an example of the associative property.

       

The same hold true for multiplication:

Example 4.  Use the numbers 5,2 and 6 to show an example of the associative property of multiplication.

       

The associative property allows 3 numbers to be multiplied (or added ) in any order and the product (or sum) will be the same.

The Distributive Property.

Let’s look at an example:

Example 5.  We know that 2(15) is 30.  Well this should be the same as 2(10+5) right?

So we distribute (multiply through by) the 2 like so:

In general the distributive property can be written as

Quick Check:

Simplify by using the distributive property: 

Answers to quick Check:     

Factoring:

We can use the distributive law  “in reverse” to help us to factor. 

Example 6:  Use the distributive law to factor the following:

Answers to quick Check:     

Combining like terms

In the expression , the 7x, 2y and 3x are called the terms.  When we say combine like terms we mean to combine terms that have the same variable(s).  So in the expression , the 7x and 3x are said to be “like” terms—they both contain the variable x.  We can combine them to get .  We cannot combine the 10x and 2y because x and y are different variables.

Combine like terms in the examples below:

Example 7. 

Solution: We may combine the 6 and 7 and the 3p and –5p to get:

       

Example 8. 

Solution:  If you need to rewrite the above expression so that the terms you are combing are beside each other, then do so (but you don’t have to).

  You may skip this second step

Quick check:

Combine like terms:

Answers to quick check:  1.