Combining like terms example

We have a hairy-looking expression here. And your goal is to try to simplify it as much as you can. And I'll give you a little bit of time to do it. Let's just think about it, step by step. And it might help if we were to actually reorder the terms in this expression. So let me put all the x terms first. So I have 5x-- that's that term-- minus 2x. Then I have plus 7y plus 3y. Then i have plus 8z, and then I have minus z. And then the last term that I haven't included yet is that plus 5. Now we'll just think it through. If I have 5 x's and I were to take away 2 x's, is how many x's am I going to be left with? Well, I'm going to be left with 3 x's. That's true of anything. There's not some fancy algebraic magic going on here. 5 of anything minus 2 of that same thing, you're going to be left with 3 of that thing. In this case, that thing are x's. So this is going to simplify to 3x. Now, in a lot of algebra classes, you'll hear people say, oh, well, you know, the coefficient on 5x is 5. And the coefficient on this subtracting the 2x, the coefficient here is negative 2, and we had to add the coefficients. Let me write that word down-- coefficient. These right over here are the coefficients. They're the number that you're multiplying the variable by. So you're the 5 or the negative 2 in this case. And so you could just say, oh, I had to just add the coefficients. And that's OK, and there's nothing wrong with that. But I really want to emphasize that there's a very common sense intuition here. If you have 5 of something, you take away 2 of that something, you are left with 3 of that something. And you have to be very careful. You have to make sure that you're adding and subtracting the same things. Here, we're dealing with x's. So we can take 5 x's and take away 2 x's. We can't think about merging the x's and the y's, at least not in any simple way right now, because that, frankly, wouldn't make any intuitive sense. Now let's think about the y's. If I have 7 of something, and I were to add 3 more of that something, well, then, I'm going to have 10 of that something. So this part right over here is going to simplify to 10y. Once again, you could say the coefficient on 7y is 7. The coefficient on 3y is 3. We added the coefficients-- 7 plus 3-- to get 10y. But I really want to emphasize the intuition here. It's much more if you've got 7 of something, you add another 3 to that something, you've got 10 of that something. Now let's look at the z's. If I've got 8 of something and I take away 1 of them, I'm going to have 7 of that something. So that is 7z. And you might say, hey, wait. What was the coefficient right here on this negative z? I don't see any number out front of the z. Well, implicitly, I could have put a 1 here, and it's exactly the same thing. Subtracting a z is the exact same thing as subtracting 1z. The word "onesie" strikes a part of my brain because I have very young children, but that's a different type of onesie. And then you could see, oh, yeah, you definitely did add the two coefficients, the 8 and the negative 1. But once again, common sense tells you if you have 8 of something, and you take away 1 of them, you have 7 of that something. And then finally, you have a plus 5. So we're done. This simplified to 3x plus 10y plus 7z plus 5.