In mathematics, you have four primary 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 are indirectly derived from addition. In this sense, there is truly one binary operation in mathematics—addition. Binary operation refers to the use of a mathematical operator, such as addition, on two numbers or variables, as in x + y. Since we see how important addition now is, we should understand thoroughly one of the most important tasks in all of mathematics—that of combining like terms.
Like terms are expressions that involve the same combination of variables and their respective exponents but different numerical coefficients. Coefficients, if you remember, are the numbers in front of the variable. To put this in layman’s terms, like terms are like apples and apples, oranges and oranges. Examples of like terms are 4x and 2x, or 3y and 9y. To take the abstraction out of all this business, the student needs to keep in mind that as long as the expressions are similar without regard to the coefficients, then the terms can be added or subtracted. Thus 3xy and 4xy are like terms and can be combined to give 7xy. Take away the coefficients 3 and 4, and what is left? xy.
Many times a student will not be able to get to the final answer of an algebra problem because at some point like terms were not combined correctly. In more complicated math problems, the expressions can get a bit more involved. However, if you keep in mind that like terms are similar “animals,” so to speak, then, like animals, they can mate safely. If the terms are not like, then you can never combine them. The results are always disastrous. What generally helps students is to take them away from the abstraction and put them face-on with the concrete facts: if two algebraic expressions, after taking away the numbers in front, look the same, then they are like terms and can be added and subtracted. Notice that we are only talking about the two operations of addition and subtraction as these are the two operations that require that terms be like before combining. Multiplication and division do not have this requirement.
Let us look at some examples to make this perfectly clear and to see where some possible problems might arise. Let us do the examples below.
1) 3x + 18x
2) 8xyw – 3xyw + xyw
3) 3x^2 – x^2 + 6x
The first example can be thought of as 3 x’s and 18 x’s. Think of the actual letter in plastic form in a child’s playset. Obviously, you have 21 x’s or 21x as the answer.
The second example gives an indication of when students might start having trouble. The minute more than one letter or variable is introduced, students quickly become intimidated. Don’t be. If you take away the coefficients in each of the terms, you see that they are all xyw terms. 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, just tells us how many times to use the number as a factor when multiplying by itself. Thus x^2 tells us to multiply x by itself, that is x^2 = x*x. If you take away the coefficients in this example, you see that you have 2 x^2 terms and one x term. Thus you can only combine the x^2 terms. The answer becomes 2x^2 + 6x. Notice that terms that cannot be combined just stay as is.
The information here should make you a master of combining like terms as, in reality, this is a very easy—yet extremely important—task. If you follow the precepts laid out here, you should not have any more difficulty with simplifying basic algebraic expressions.