 ### 7.2b Evaluate and Simplify Algebraic Expressions – Combine Like Terms

Now let’s talk about terms, and the roles the terms play in expressions. Terms are numbers and variables multiplied together. I often look at expressions, and I think of the terms as the clumps. You can kinda of see that if you know how to look at it, if you look at example one here, you’ll see it has one, two, three, four, five, six terms. Now the thing about simplifying, is that expressions like what we see in these two examples down here, are unnecessarily complicated; they can be made simpler. To do that we look at like terms. Like terms, are terms that have matching variables and exponents. Think of like terms as terms that are only different in their coefficients. So their variables are the exact same, their exponents are the exact same, but the numbers can be different. Now you have to be careful here because this is a very very rigid thing. So for instance if I had, say 5 x to the tenth, y to the ninth z, and I had 11 x to the tenth, y to the ninth z, those would be like terms. But if I put z squared here, they would cease to be like terms, because now they don’t have the same power. So even the most minute difference in exponents can be a deal breaker there. So look out for that. Remember that they have to be the exact same with variables and exponents. And if they are the exact same that allows us to combine them and make these problems much simpler. To do that we just add or subtract their coefficients from the like terms. An easy way to do that often is to just pretend the variables aren’t there, and see what the number would come out to if the variables weren’t there. So for instance for example one, let’s look for some matching sets of like terms. We see that we have two cubic terms here, we also have two squared terms here, and we also have just two regular x terms there. So we can combine those, 4 x cubed plus 5 x cubed, again if you have four of something and five of something, that comes out to 9 x cubed. Be careful the variable never changes. The variable and it’s power never change as a result of adding or subtracting like terms. So, you’re going to have 9 x cubed. Now let’s look at the green terms. Negative 2 x squared minus 4 x squared, again it’s easy to combine those if you just imagine what they would look like without the variables there. Negative two minus four is negative six; therefore, negative 2 x squared minus 4 x squared is negative 6 x squared. As for 2 x minus 6 x again two minus six is negative four, and therefore 2 x minus 6 x is negative 4 x. So again with combining like terms, you just identify things that match and put them together. We’ll see the same thing over here with example two. So, with example two here we have three y terms, we have two constants, and we just have a single x term that does not combine with anything. So you can just write it down immediately, 2 x like that. As for the red terms, four minus six plus seven, well, four plus seven is eleven, eleven minus six is five, so we get 5 y. Five minus nine for the grey terms is negative four, and now this is as simple as it gets. And so, we have now successfully combined both of these. They cannot go any further, so remember you can’t add or subtract terms if their variable parts do not match. These two expressions we just did actually cannot be made any simpler. That’s all order of operations is, so practice that a little bit and I think you’ll be surprised at how easy it is, just remember don’t change the variable parts and remember to only combine terms that match.