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Basic Math Facts – Combining Like Terms
In math, you have four main operations: addition, subtraction, multiplication, and division. Since subtraction is the inverse of addition, multiplication is repeated addition, and division is the inverse of multiplication, you see that the other three operations derive indirectly from addition. In this sense, there really is a binary operation in mathematics—addition. Binary operation refers to applying a mathematical operator, such as addition, to two numbers or variables, as in x + y. Since we see how important addition is now, we should thoroughly understand one of the most important tasks in all mathematics–that of combining like terms.
Like 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 it in simple terms, like terms are like apples and apples, oranges and oranges. Examples of similar terms are 4x and 2xthe 3 years old and 9 years old. To remove subtraction from all this matter, the student must bear in mind that since the expressions are similar without regard to the coefficients, then the terms may be added or subtracted. So 3xy and 4xy are like terms and can be combined to give 7xy. Subtract factors 3 and 4 and what is left? xy.
Many times a student will not be able to arrive at the final answer of an algebra problem because at some point such terms were not combined correctly. In more complex math problems, expressions can get a little 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 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 subtraction and confront them with the concrete facts: if two algebraic expressions, after subtracting 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 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 seen 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 begin to experience problems. The minute more than one letter or variable is introduced, students quickly become intimidated. Do not be. If you remove the coefficients in each of the terms, you will see that they are all xyw terms. The last term has a coefficient of 1, which is understandable. Combining, we have 6xyw.
The third example enters an expression with exponents. Remember that 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, i.e. 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. Notice that terms that cannot be combined simply remain as they are.
The information here should make you a master at combining similar terms as, in fact, this is a very easy — but extremely important — task. If you follow the principles laid out here, you should no longer have difficulty simplifying basic algebraic expressions.
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