Combining Like Terms
You probably don't keep bananas in your sock drawer or toothpaste on your bookshelf. If you do, we can recommend a good therapist. Aside from it being an awful hassle to run from the bookshelf to the bathroom sink every time you need to spit, the reason we don't do such things is because it makes sense to keep like things together. Algebraic terms are no exception.
A term in an expression has two parts: the constant part, also called the coefficient, where this word describes a term equally as efficiently as the word "constant," so it is "coefficient," and the variable part. Just as they sound, the constant part is the product of all the constants in the term, and the variable part is the product of all the variables in the term. If someone approaches you about Variable Life and Annuities, they probably want your help on an algebra problem.
1. In the term 9x, the constant part, or coefficient, is 9 and the variable part is x.
2. In the term 2y(3x2), the coefficient is 6 and the variable part is x2y.
3. In the term mc2 where c is a constant, c2 is the coefficient and m is the variable part.
In an expression, like terms are terms with the same variable part. Going back to one of our earlier examples, bananas go with bananas, and socks go with socks. Although, if bananas could go with socks, we sure could make some hilarious puppets.
1. x and 8x are like terms, since they both have x as their variable part.
2. 5x7y and 89x7y are like terms, since they both have x7y as their variable part.
3. 4 and 5 are like terms, since both are constants with no variable part. We know what we are getting them for Chanukah.
1. x and x2 are not like terms, since they have different variable parts.
2. y and x are not like terms, since they have different variable parts.
3. x2y and xy2 are not like terms, since they have different variable parts.
If we have a pile with 5 lemons and a pile with 6 lemons, we can combine them to get one pile with 11 lemons. Of course, what we actually should do is make lemonade, but that is more of a life philosophy and less of a mathematical solution. In this real-life situation—you may often be confronted with such a fruity scenario—the lemons are our variables. Therefore, we can combine like terms. Or, in this case, combine like lemons.
1. Combining 3 homework assignments with 2 homework assignments gives us 5 homework assignments. Plus 1 disgruntled student.
2. Combining 9 fingers and 1 finger gives us 10 fingers. Thank goodness. We thought we misplaced one for a second there.
3. Combining 3 copies of the letter x with 5 copies of the letter x gives us 8 of the letter x. With which we can spell "xxxxxxxx."
In order to combine terms, they need to be like terms. We can't combine x and 2x2. We could use the distributive property to rewrite x + 2x2 as (1 + 2x)x (give it a try; the math checks out), but (1 + 2x)x isn't that much prettier than x + 2x2. Not to sound superficial, but when it comes to mathematical expressions, it is all about the looks.
1. Combine like terms in the expression 7x + 3xy + 3 + 4xy + 2x2 + 6.
First, rearrange the expression to put like terms next to each other. Pull your bananas out of your sock drawer and move them over to the banana drawer of your dresser. Wait a minute...
7x + 3xy + 4xy + 2x2 + 3 + 6. Now it is easier to see what goes with what. Which is good, because you don't want your terms to clash.
Combine the terms with variable part xy to get 7x + 7xy + 2x2, and then combine the constant terms to get 7x + 7xy + 2x2 + 9.
This line is as clean and clear as it gets. Without over-the-counter acne medication, anyway.
Be careful when fractions are involved. Some of them have sharp edges. Using the distributive property to combine like terms with fractional coefficients is a little more complicated; therefore, pay extra close attention to what you are doing. Put on your thinking cap, if you have one. If you don't, a particularly snug baseball cap will do.
2. Combine like terms:
First, do some rearranging such that like terms are next to each other:
Next, use the distributive property to factor out the variables.
Finally, add the fractions as we learned to do in the section on fractions. As we are sure you recall, in order to perform the arithmetic you will need to convert these guys so that they have like denominators. If you don't remember how to do that, return to Start and draw 2 cards.
Thinking about the distributive property allows us to separate out the coefficients from the variables; therefore, we don't need to think about both at once. Remember how well that worked when you were trying to think at the same time about riding your bicycle and about that kiss you got from Heather Pelnicki? Didn't turn out well for you, did it? Didn't turn out well for Heather Pelnicki, either. To be fair, she shouldn't have been standing right in the middle of the bike lane.