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Basic Math Facts – Combining Like Terms
In math, 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 derive indirectly from addition. In this sense, there really is a binary operation in mathematics: addition. 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 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. Coefficients, if you remember, are the numbers in front of the variable. To put it in simple terms, similar terms are like apples and apples, oranges and oranges. Examples of similar terms are 4x i 2xor 3 years i 9 years. To take the abstraction out of all this business, the student should note that whenever the expressions are similar regardless of the coefficients, the terms may be added or subtracted. Thus, 3xy and 4xy are like terms and can be combined to give 7xy. Take out 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 some point like terms were not combined correctly. In more complicated math problems, expressions can get a bit more involved. However, if you consider that similar terms are similar “animals”, so to speak, then, like animals, they can be safely paired. If the terms are not similar, you can never combine them. The results are always disastrous. What usually helps students is to move them away from the abstract and face the concrete facts: if two algebraic expressions, after removing the numbers in front of them, 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 the same before combining. Multiplication and division do not have this requirement.
Let’s look at some examples to make this perfectly clear and see where some potential problems could 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 18 x. Think of the actual letter in plastic form in a child’s play. Obviously, you have 21 x 21x as your answer.
The second example gives an indication of when students may start to have problems. The moment 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 will 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, only 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, meaning x^2 = x*x. If you remove the coefficients from this example, you will 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 remain as they are.
The information here should make you a master of combining similar terms, as this is actually a very easy, but very important task. If you follow the precepts outlined here, you should have no more difficulty simplifying basic algebraic expressions.
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