How to Combine Like Terms
Do you know what “like terms” in algebra are? Do you know the best way to think about them? And how to combine them?
Jason Marshall, PhD
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How to Combine Like Terms
What do you think of when you think about variables in algebra? Take little x, for example. Or, if we want to let things get a bit more complex (and who doesn’t?), what do you think of when you think about some expression built from a variable like x? Perhaps 2x or x2—what do those sorts of things make you think of?
Hopefully the answer isn’t agony. But if it is, I truly am sorry for the painful experience you must have had with algebra in the past. And I want to let you know that there’s a much more pleasant way to think about these sorts of things than you were perhaps led to believe. In particular, I’d like to let you know that you can think of these things as nothing scarier than colorfully shaped objects that you can stack up and manipulate…just like the Legos or wooden blocks you probably played with as a kid.
What exactly do I mean by this? How can thinking about stacking blocks possibly help you picture these kinds of algebraic expressions and the process of combining them together known as “adding like terms”?
Let’s find out.
What Are Like Terms?
To kick things off, let’s start by talking about exactly what we mean by “like terms.” The easiest way to explain what like terms are is with an example. To begin with, we can think of a variable like x as a stack of blocks that’s x blocks high (it could be 2 blocks high, or 10, or 1 million—anything is possible—depending on the value of x). If we wanted to, we could also add x to itself to create a new expression, x + x, that now contains two terms.
If you’re like me, you think it’d be fun and helpful to visualize what this new expression might look like…so you might think of it as looking something like this:
But, if you think about it a bit, you’ll see that we don’t have to put the boxes representing the variables x and x that we’re adding together end-to-end like this. Instead, since both variables have the same length, we can nicely stack them up one on top of the other to create a lovely and orderly 2-by-x rectangle, like this:
If you think about it, you’ll see that this is the shape that’s created by “stretching” the variable x—represented by that x-block high stack—until it’s twice its original width. In other words, it’s the shape you get by multiplying 2 • x. And notice too that when we stack the boxes like this, the new expression on the right has only one term, 2x, instead of the two terms that we started with. In other words, because the two parts that we’re adding are what we call “like terms,” we can combine them together.
What Are Unlike Terms?
What if we had been adding x to y instead of x to x? Would we still have been able to combine the two terms? Well, take a look:
As you can see, the lengths of x and y are different, which means that we can’t put them into a nice and orderly perfect stack to combine them as we did with x + x. And, as a result, you can see that the expression we end up with after adding them together, x + y, still has two terms. In other words, we can’t combine x and y because they are not like terms.
What Does it Mean to Add Like Terms?
Now that we understand the basics of like terms and putting them together, let’s think about a few more complicated examples. Let’s imagine we’ve got an x2 over here, another one over there, and a final one somewhere else. How can we think of these x2s? Following on from the logic we’ve already talked about, I like to think of them as squares with widths of x blocks and heights of x blocks. The cool thing is that all of these x-by-x squares don’t have to live out their lives on their own—you can instead combine them together to get a cuboid made out of three layers of squares known in the algebra world as 3x2.
Why can you do this? Because all of the x2 terms are the same type of thing. In other words, in our view of the world, these things are all the same shape and size (they’re all x-by-x boxes)…which means that we know how to stack them up into a nice and orderly cuboid.
But what if you instead have a y2 here, another y2 there, and an x2 way over there—can you still combine them? Absolutely not! If you think about this in terms of shapes, you’ll see that you can’t add them together since they’re not the same type of object.
In other words, in our world of shapes, since x2 and y2 are not the same size, we can’t stack them up because they won’t line up to form a perfect tower with straight sides. Of course, you can stack and add up both of the y2 terms since they are the same size and shape. And that’s exactly why the final expression for a situation like this will be equal to 2y2 + x2.
What Does it Mean to Subtract Like Terms?
As you might guess, we can also subtract like terms—and subtraction is very similar to addition. As long as you’re dealing with like terms (that is, that the things you’re trying to subtract have the same size and shape when represented with boxes, rectangles, cuboids, blocks, Legos, or however else you’re picturing things in your mind), you are free to subtract the numerical coefficients of those terms. For example, since x2 and 3x2 both are some number of x2s, we can subtract x2 from 3x2 to get 2x2:
We’ve simply shortened our 3-high x-by-x block tower into a 2-high x-by-x tower.
How to Combine Like Terms
The bottom line is that for two terms to be like, they must have the exact same combination of variables and the exponents that those variables are raised to. In other words, x and x2 have the same base variable, x, but they are not both raised to the same power and are therefore not like terms. Which, of course, we can also see by noting that x and x2 have different “shapes” in the way of thinking about things that we’ve developed.
Hopefully that helps you feel a bit more comfortable with thinking about the meanings of the various bits you see floating around in algebraic expressions. And keep in mind that the system for thinking about them that I’ve talked about here—in particular the way I think about shapes—isn’t the only way of doing things. You can certainly think about them in different ways, and I encourage you to think about that a bit and come up with and use whatever works best for you.
Wrap Up
Okay, that’s all the math we have time for today. For more fun with math, please check out my book, The Math Dude’s Quick and Dirty Guide to Algebra. Also, remember to become a fan of The Math Dude on Facebook and to follow me on Twitter.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
Blocks image from Shutterstock.
About the Author
Jason Marshall is the author of The Math Dude’s Quick and Dirty Guide to Algebra. He provides clear explanations of math terms and principles, and his simple tricks for solving basic algebra problems will have even the most math-phobic person looking forward to working out whatever math problem comes their way.