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Basic Math Facts – Combining Like Terms

In mathematics, 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 are indirectly derived from addition. In this sense, there is really only 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 is now, we should thoroughly understand one of the most important tasks in all of mathematics—that of combining like terms.

Similar terms are expressions that involve the same combination of variables and their respective exponents but different numerical coefficients. The 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 2xor 3 years and 9 years. To eliminate the abstraction of this whole business, the student must keep in mind that as long as the expressions are similar without regard to the coefficients, 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’s left? xy.

Often a student will not be able to get the final answer to an algebra problem because at some point similar terms were combined incorrectly. In more complicated math problems, expressions can get a bit more complicated. However, if you keep in mind that like terms are like “animals”, so to speak, then, like animals, they can safely mate. If the terms don’t look alike, you can never combine them. The results are always disastrous. What generally helps students is to take them out of abstraction and put them face to face with concrete facts: if two algebraic expressions, after removing the numbers in front, resemble each other, then they are like terms and can be add and subtract. 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 the same before they combine. Multiplication and division do not have this requirement.

Let’s take a look at some examples to make this crystal clear and to see where some possible issues might arise. Let’s 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 and 18 x. Think of the actual letter in plastic form in a child’s play. Obviously you have 21x or 21x as an answer.

The second example gives an indication of when students might start having problems. The minute more than one letter or variable is introduced, students quickly become intimidated. Do not be. If you remove the coefficients of 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 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 take away the coefficients in this example, you see you have 2 x^2 terms and one x term. So you can only combine x^2 terms. The answer becomes 2x^2 + 6x. Note that terms that cannot be combined remain as they are.

The information here should make you a master of combining similar terms because, in reality, it is a very easy—but extremely important—task. If you follow the precepts laid out here, you should have no more trouble simplifying basic algebraic expressions.

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