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## Basic Math Facts – Combining Like Terms

In math, you have four main operations: addition, subtraction, multiplication, and division. Given that subtraction is the inverse of addition, multiplication is repeated addition, and division is the inverse of multiplication, you see that the other three operations follow indirectly from addition. In this sense, there really is a binary operation in mathematics — addition. A binary operation refers to using a mathematical operator, such as addition, on two numbers or variables, as in x + y. Since we see how important addition is now, we must fully understand one of the most important tasks in all mathematics—that of combining like terms.

*Like the terms* are expressions involving the same combination of variables and their corresponding exponents, but different numerical coefficients. Coefficients, if you remember, are the numbers in front of the variable. To put this in layman’s terms, the terms are like apples and apples, oranges and oranges. Examples of similar terms are *4x *AND *2x*or *3 years *AND *9 years old*. To remove the abstraction from all this work, the student should note that as long as the expressions are similar regardless of the coefficients, then the terms can be added or subtracted. Thus, 3xy and 4xy are like terms and can be combined to give 7xy. Remove the coefficients 3 and 4 and what is left? xy.

Many times a student will not be able to arrive at the final answer to an algebra problem because at that point the terms have not been combined correctly. In more complicated math problems, expressions can be a bit more involved. However, if you keep in mind that similar terms are similar “animals”, so to speak, then, like animals, they can be safely paired. If the conditions are not like, then you can never combine them. The results are always disastrous. What generally helps students is to move them away from abstraction and confront them with concrete facts: if two algebraic expressions, after removing the numbers in front, look the same, then they are like terms and can be added and subtracted. Note that we are only talking about the two operations of addition and subtraction as these are the two operations that require the terms to be as before the combination. Multiplication and division do not have this requirement.

Let’s look at some examples to make this perfectly clear and to see where some potential problems can arise. Let’s do the following examples.

1) 3x + 18x

2) 8xyw – 3xyw + xyw

3) 3x^2 – x^2 + 6x

The first example can be considered as 3 x and 18 x. Think of the actual letter in plastic form on a child’s play set. Obviously, you have 21 x or 21x as the answer.

The second example gives an indication of when students may start to have problems. As soon as more than one letter or variable is entered, students quickly panic. Do not be. If you remove the coefficients in each of the terms, you see that they are all *xyw *the conditions. The last term has a coefficient of 1, which is understood. Combining, we have 6xyw.

The third example introduces an expression with exponents. Remember the exponent, or power, simply tells us how many times to use the number as a factor when multiplying by itself. So x^2 tells us to multiply x by itself, which is x^2 = x*x. If you remove the coefficients in this example, you see that you have 2 x^2 terms and an x term. So you can only combine the x^2 terms. The answer becomes 2x^2 + 6x. Note that terms that cannot be combined simply stand as they are.

The information here should make you a master of combining similar terms, as, in reality, this is a very easy—but extremely important— task. If you follow the principles presented here, you should have no more difficulty simplifying basic algebraic expressions.

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