Combining Like Terms

So let's talk about combining like terms for a minute. This is a phrase that comes up a lot in algebra, as the teacher rattles off a whole list of things you are supposed to do. "Now you just combine the like terms, bring the x's over here, divide, and voile, here is your answer: x = 2!" Yeah, not only do you have no idea what he just said, you are not sure that x = 2 could ever be the answer to anything in your life!

So I guess we should first address the word "term". In an algebraic expression (click here for a short discussion on what that is) there can be one or more terms. Terms are the parts of the expression that are separated by + or - signs. Let's look at some examples:

x + 2y - 4 => There are 3 terms here: x, 2y, and -4
(Yeah, I know, you thought the minus sign was just there to separate the
2y and the 4, but it also belongs
to the 4)

3x² - 5x + 6 => There are 3 terms here as well: 3x², -5x, and 6

8y + 3 - 6y - 1 => There are 4 terms in this one: 8y, 3, -6y, and -1

Got it?

Now, like terms are terms that are enough the same that they can be combined. (I am indebted to this website for this explanation) Let's go grocery shopping again. (can you tell where the bulk of our money goes?!) Suppose you buy a bunch of gala apples, a few granny smith apples, some pears, and some cheerios. When the checker rings it up, sure they can just keep scanning individual apples, one by one -- or they can group all of the same type of apple together, as well as all of the pears. When it comes to the Cheerios, if there is more than one box, they will scan one and then hit the "mulitple" key on their register. Right?

So it's the same thing with algebra. Take the last example above. Sure, you can keep it at 8y + 3 - 6y - 1. There's nothing wrong with it! It's correct, but it is just more complicated than it needs to be. Why not combine the two terms with y's? Then the whole thing will be shorter. So we think to ourselves, 8y - 6y -- that must be 2y! (if you have trouble with this, think: 8 gala apples - 6 gala apples = 2 gala apples!)
Now instead of having to write (or say) 8y + 3 - 6y - 1,
you will write 2y + 3 - 1.

So now what do you think about the 3 and the -1? Can we combine them?? Sure!! That will make it:
2y + 2

See how easy this is? Does it matter that I kind of had to mix up the order to get this to work? Nope. It doesn't matter which order the checker rings your stuff up in, and it doesn't matter which order you write algebraic stuff in either. Just keep minus/negative signs with the numbers they "belong" to -- i.e. the number that follows them (so in our example here, the negative sign belongs to the 1)

Want to try a few? Have at it!

1) 8x - 3 + 4y - 2x
2) 2y² + 3y - y + 4
(hint: the 2y², the 3y, and the -y are all y's, but they are not all the same type -- like gala apples and granny smith. Only the y²'s will combine with each other, and the single y's (not squared) will combine together.
3) 97 - 3z + 25

Scroll down for answers:




























1) 6x - 3 + 4y
6x + 4y - 3 would also be correct.
2) 2y² + 2y + 4
3) 122 - 3z

Woo-hoo! Look how cool you are!!