Combining like terms
Let's say that I've got 2 Chuck Norrises, or maybe it's Chuck Norri. And to that I am going to add another 3 Chuck Norrises. So I'm going to add another 3 Chuck Norrises. And this might seem a little bit obvious, but how many Chuck Norrises do I now have? Well, 2 Chuck Norrises, we can represent this as literally a Chuck Norris plus a Chuck Norris. So let me do that, a Chuck Norris plus another Chuck Norris, 2 Chuck Norrises. You could also do this 2 times Chuck Norris, and this is just another way of representing it. And 3 Chuck Norrises-- you could do that as a Chuck Norris plus a Chuck Norris plus another Chuck Norris. And so we would have a grand total-- and this might be very simple for you. But you would have a grand total of 1, 2, 3, 4, 5 Chuck Norrises. So this would be equal to 5 Chuck Norrises. Now, let's get a little bit more abstract here. Chuck Norris is a very tangible thing. So let's go to a little bit more of traditional algebraic notation. If I have 2x's and remember, you could do this as 2x's or 2 times x. And to that, I would add 3x's How many x's do I have? Well, once again, 2x's, that's 2 times x. You could do that as an x plus an x. We don't know what the value of x is. But whatever that value is, we can add it to itself. And then 3x's are they're going to be that value. Let me do that in that same green color. 3x's are going to be that value plus that value plus whatever that value is. And so how many x's do I now have? Well, I'm going to have 1, 2, 3, 4, 5 x's. So 2x plus 3x is equal to 5x. And if you think about it, all we really did-- and hopefully, you conceptually get it-- is we just added the 2 numbers that were multiplying the x. And these numbers, the 2 or the 3, they're called coefficients. Very fancy word, but it's just this constant number, this regular number that's multiplied by the variable. You just added the 2 and the 3, to get your 5x. Now, let's think about this a little bit more. Let's go back to this original expression, the 2 Chuck Norrises plus 3 Chuck Norrises. Let's say, to that, we were to add to some type of a-- let's we were to add 7 plums over here. So this is my drawing of a plum. So we have 7 plums plus 2 Chuck Norrises plus 3 Chuck Norrises. And let's say that I add another 2 plums. I add another 2 plums here. So what this whole thing be? Well, I wouldn't add the 7 to the 2 to the 3 plus the 2. We're adding different things here. You have 2 Chuck Norrises and 3 Chuck Norrises, so they're still going to simplify to 5 Chuck Norrises. . And then we would separately think about the plums. We have 7 plums, and we're adding another 2 plums. We're going to have 9 plums. Plus 9 plums, so this simplifies to five Chuck Norrises and 9 plums. Similarly, over here, instead of just 2x plus 3x, if I had 7y plus 2x plus 3x plus 2y, what do I now have? Well, I can't add the x's and the y's. They could very well represent a different number. So all I can do is really add the x's. And then I get the 5x. And then, I'd separately add the y. If I have 7y's and to that I add 2y's, I'm going to have 9y's. If I have 7 of something and I add 2 of something, I now have 9 of that something. So I'm going to have 9y's. So you add that. Do that in a different color. You add this and this. You get that. You add the x's. You get that right over there. So hopefully, that makes a little sense. Actually I'll throw out one more idea. So given this, what would happen if I were to have 2x plus 1 plus 7x plus 5? Well, once again, you might be tempted to add the 2 plus the 1, but they're adding different things. These are 2x's. This is just the number 1. So you really just have to add the x's together. So you're going to say, well, I got 2x's. And I'm going to add 7x's to that. Well, that means I now have 9x's. And then, separately, you'd say, well, I've got just the abstract number 1. And then I've got another 5. 1 plus 5 is going to be equal to 6.