Overview:
One of the first steps in solving an equation is to combine like terms. Terms can be combined when they have the same variables and they have the same exponents. Otherwise they are unlike and cannot be combined to solve the equation.
Real Numbers and Variables Are Unlike Terms
Suppose the expression is 3z + 9. The number 9 is a real number in this expression and doesn’t have a variable. It cannot be combined with 3z until the value of z is defined. Similarly, suppose the expression is 7mn – 12. The number 12 is a real number without a variable and cannot be combined with 7mn.
Combining the Same Variables
Suppose the expression is 3x + 2z. Those terms are in the simplest form possible, because they have two different variables, x and z. Until the values of x and z are defined, those expressions cannot be combined. However, if the expression were 4x + 2x, the variables are the same and can be added as 6x.
Combining the Same Exponents
Suppose the expression is 5x2 + x – 2z. That trinomial is also in the simplest form possible, because it also has three different terms that cannot be combined. Until the values of x and z are defined, those terms cannot be combined. In addition, the 5x2 and x cannot be combined, because the value of x is not defined. Similarly, suppose the expression is 6a2b + 2ab3. Those terms cannot be combined, since even though 6a2b and 2ab3 have the same variables, ab, the variables are not of the same degree.
Using the Commutative and Associative Properties
Suppose the expression is 7x – 2y + 3 + 2x + 3x + 10y. It could be simplified by using the Commutative Property of Addition to arrange the like terms, such as 7x + 2x + 3x – 2y + 10y + 3. Then, the Associative Property of Addition can be used to group them together, as (7x +2x +3x) + (-2y + 10y) + 3. The 7x + 2x + 3x can be simplified as 12x and the – 2y + 10y can be simplified as 8y. The best expression is then 12x + 8y + 3.
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