Overview
Students can use the commutative, associative, and distributive properties of addition and multiplication to simplify algebraic expressions. Problems are more easily solved if compatible terms can be combined first. Some problems can then be solved mentally.
Compatible Terms
Real numbers, whether whole or mixed, are constants that can be added, subtracted, multiplied or divided. The only exception is dividing by zero, which is undefined. Compatible terms can also be simplified, as long as each term has the same variable or variables at the same degree. The terms 2x + 4x can be combined to equal 6x, and the terms 3y2– 8y2 equal -5y2. However, 2x + 3z cannot be added, and 3x3 + 2x -4x is in its simplest form as 3x3 –2x.
Commutative Property of Addition and Multiplication
According to the commutative property of addition or multiplication, the order of adding or multiplying numbers does not matter, as long as all of them are added or multiplied. For example, 30 + 40 = 70, and 40 +30 = 70. The law in math symbol language is a + b = b + a for addition, and ab=ba for multiplication. Similarly, 2x + 3y +3x – 2y is the same as 2x +3x +3y -2y. The commutative property is a useful tool in mental math. Suppose the column of figures is 51 + 25 + 25 +49 + 50. It can be rearranged as 51 +49 +25 + 25 +50 to equal 200.
Associative Property of Addition and Multiplication
According to the associative property of addition or multiplication, the way numbers are grouped doesn’t matter, as long as all the numbers are added or multiplied. In math symbol language, (a + b) + c = a + (b +c). The associative property is also a tool that can be used in mental math. When the mental math problem 51 +49 +25 + 25 +50 was rearranged, it was also grouped, as (51 + 49) + (25 +25 + 50). Solving within parentheses, 100 +100 =200. Both the commutative and associative problems were used together to solve the problem. An expression such as 4x + 3y – .5y +7x can be rearranged as 4x +7x +3y -.5y and then regrouped as (4x +7x) + (3y – .5y) to equal 11x + 2.5y.
Distributive Property
The long formal name of the Distributive Property is the Distributive Property of Multiplication over Addition, or the Distributive Property of Multiplication over Subtraction. In math symbol language, it means that (a +b)c = ac + bc, or (a-b)c = ac – bc. This property is very useful, because it allows monomials to be combined that use the same variables. For example, 3y + 2y = 5y because (3 +2)y equals 5y. Similarly, 17a -11a =6a because (17-11)a = 6a.
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