Video Transcripts: Combining Like Terms

Intro to 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.

Combining like terms with negative coeffiicients & distribution 

I've gotten feedback that all the Chuck Norris imagery in the last video might have been a little  bit too overwhelming. So for this video, I've included something a little bit more soothing. Let's try to simplify some more expressions. And we'll see we're just applying ideas that we already  knew about. Let's say I want to simplify the expression 2 times 3x plus 5. Well, this literally  means two 3x plus 5's. So this is the exact same thing. This is one 3x plus 5, and then to that, I'm going to add another 3x plus 5. This is literally what 2 times 3x plus 5 means. Well, this is the  same thing as, if we just look at it right over here, we have now two 3x's. So we could write it as  2 times 3x. Plus, we have two 5's, so plus 2 times 5. You might say, hey, Sal, isn't this just the  distributive property that I know from arithmetic? I've essentially just distributed the 2? 2 times 3x plus 2 times 5? And I would tell you, yes, it is. And the whole reason why I'm doing this is just  to show you that it is exactly what you already know. But with that out of the way, let's  continue to simplify it. When you multiply the 2 times the 3x, you get 6x. When you multiply the 2 times the 5, you get 10. So this simplified to 6x plus 10. Now let's try something that's a little  bit more involved. Once again, really just things that you already know. Let's say I had 7 times  3y minus 5 minus 2 times 10 plus 4y. Let's see if we can simplify this. Well, let's work on the left hand side of the expression, the 7 times 3y minus 5. We just have to distribute the 7. This is  going to be 7 times 3y, which is going to give us 21y. Or if I had 3 y's 7 times, that's going to be  21 y's, either way you want to think about it. And then I have 7 times-- we've got to be careful  with the sign-- negative 5. 7 times negative 5 is negative 35. So we've simplified this part of it.  Let's simplify the right-hand side. You might be tempted to say, oh, 2 times 10 and 2 times 4y  and then subtract them. And if you do that right and you distribute the subtraction, it would  work out. But I like think of this as negative 2, and we're going to distribute the negative 2  times 10 and the negative 2 times 4y. So negative 2 times 10 is negative 20, so it's minus 20  right over here. And then negative 2 times 4, negative 2 times 4 is negative 8, so it's going to be negative 8y. Let's write a minus 8y right over here. And are we done simplifying? Well, no,  there's a little bit more that we can do. We can't add the 21y to the negative 35 or the negative  20 because these are adding different things or subtracting different things. But we do have  two things that are multiplying y. Let me do all in this green color. You have 21 y's right over  here. And then we can view it as from that we are subtracting 8 y's. So if I have 21 of something  and I take 8 of them away, I'm left with 13 of that something. So those are going to simplify to  13 y's. I'll do this in a new color. And then I have negative 35 minus 20. That's just going to  simplify to negative 55. So this whole thing simplified, using a little bit of the distributive  property and combining similar or like terms, we got to 13y minus 55. 

Combining like terms with negative coefficients 

Now we have a very, very, very hairy expression. And once again, I'm going to see if you can  simplify this. And I'll give you little time to do it. So this one is even crazier than the last few  we've looked at. We've got y's and xy's, and x squared and x's, well more just xy's and y squared  and on and on and on. And there will be a temptation, because you see a y here and a y here to  say, oh, maybe I can add this negative 3y plus this 4xy somehow since I see a y and a y. But the  important thing to realize here is that a y is different than an xy. Think about it they were  numbers. If y was 3 and an x was a 2, then a y would be a 3 while an xy would have been a 6. 

And a y is very different than a y squared. Once again, if the why it took on the value 3, then the y squared would be the value 9. So even though you see the same letter here, they aren't the  same-- I guess you cannot add these two or subtract these two terms. A y is different than a y  squared, is different than an xy. Now with that said, let's see if there is anything that we can  simplify. So first, let's think about the y terms. So you have a negative 3y there. Do we have any more y term? Yes, we do. We have this 2y right over there. So I'll just write it out-- I'll just  reorder it. So we have negative 3y plus 2y. Now, let's think about-- and I'm just going in an  arbitrary order, but since our next term is an xy term-- let's think about all of the xy terms. So  we have plus 4xy right over here. So let me just write it down-- I'm just reordering the whole  expression-- plus 4xy. And then I have minus 4xy right over here. Then let's go to the x squared  terms. I have negative 2 times x squared, or minus 2x squared. So let's look at this. So I have  minus 2x squared. Do I have any other x squared? Yes, I do. I have this 3x squared right over  there. So plus 3x squared. And then let's see, I have an x term right over here, and that actually  looks like the only x term. So that's plus 2x. And then I only have one y squared term-- I'll circle  that in orange-- so plus y squared. So all I have done is I've reordered the statement and I've  color coded it based on the type of term we have. And now it should be a little bit simpler. So  let's try it out. If I have negative 3 of something plus 2 of that something, what do I have? Or  another way to say it, if I have two of something and I subtract 3 of that, what am I left with?  Well, I'm left with negative 1 of that something. So I could write negative 1y, or I could just write negative y. And another way you could think about it, but I like to think about it intuitively  more, is what's the coefficient here? It is negative 3. What's the coefficient here? It's 2. Where  obviously both are dealing-- they're both y terms, not xy terms, not y squared terms, just y. And so negative 3 plus 2 is negative 1, or negative 1y is the same thing as negative y. So those  simplify to this right over here. Now let's look at the xy terms. If I have 4 of this, 4 xy's and I  were to take away 4 xy's, how many xy's am I left with? Well, I'm left with no xy's. Or you could  say add the coefficients, 4 plus negative 4, gives you 0 xy's. Either way, these two cancel out. If I have 4 of something and I take away those 4 of that something, I'm left with none of them. And so I'm left with no xy's. And then I have right over here-- I could have written 0xy, but that  seems unnecessary-- then right over here I have my x squared terms. Negative 2 plus 3 is 1. Or  another way of saying it, if I have 3x squared and I were to take away 2 of those x squared, so  I'm left with the 1x squared. So this right over here simplifies to 1x squared. Or I could literally  just write x squared. 1x squared is the same thing as x squared. So plus x squared, and then  these there's nothing really left to simplify. So plus 2x plus y squared. And we're done. And  obviously you might have gotten an answer in some other order, but the order in which I write  these terms don't matter. It just matters that you were able to simplify it to these four terms. 

Combining like terms with rational coefficients 

What I wanna do in this video is get some practice simplifying expressions and have some  hairier numbers involved. These numbers are kind of hairy. Like always, try to pause this video  and see if you can simplify this expression before I take a stab at it. All right, I'm assuming you  have attempted it. Now let's look at it. We have -5.55 minus 8.55c plus 4.35c. So the first thing I  wanna do is can I combine these c terms, and I definitely can. We can add -8.55c to 4.35c first,  and then that would be, let's see, that would be -8.55 plus 4.35, I'm just adding the coefficients, 

times c, and of course, we still have that -5.55 out front. -5.55. I'll just put a plus there. Now how do we calculate -8.55 plus 4.35? Well there's a couple of ways to think about it or visualize it.  One way is to say well this is the same thing as the negative of 8.55 minus 4.35, and 8.55 minus  4.35, let's see, eight minus four is going to be the negative, eight minus four is four, 55  hundredths minus 35 hundredths is 20 hundredths. So I could write 4.20, which is really just the  same thing as 4.2. So all of this can be replaced with a -4.2. So my entire expression has  simplified to -5.55, and instead of saying plus -4.2c, I can just write it as minus 4.2c, and we're  done. We can't simplify this anymore. We can't add this term that doesn't involve the variable  to this term that does involve the variable. So this is about as simple as we're gonna get. So  let's do another example. So here I have some more hairy numbers involved. These are all  expressed as fractions. And so, let's see, I have 2/5m minus 4/5 minus 3/5m. So how can I  simplify? Well I could add all the m terms together. So let me just change the order. I could  rewrite this as 2/5m minus 3/5m minus 4/5. All I did was I changed the order. We can see that I  have these two m terms. I can add those two together. So this is going to be 2/5 minus 3/5  times m, and then I have the -4/5 still on the right hand side. Now what's 2/5 minus 3/5? Well  that's gonna be -1/5. That's gonna be -1/5. So I have -1/5m minus 4/5. Minus 4/5. Now once  again, I'm done. I can't simplify it anymore. I can't add this term that involves m somehow to  this -4/5. So we are done here. Let's do one more. Let's do one more example. So here, and this  is interesting, I have a parentheses and all the rest. Like always, pause the video. See if you can  simplify this. All right, let's work through it together. Now the first thing that I want to do is let's distribute this two so that we just have three terms that are just being added and subtracted.  So if we distribute this two, we're gonna get two times 1/5m is 2/5m. Let me make sure you see  that m. M is right here. Two times -2/5 is -4/5, and then I have plus 3/5. Now how can we  simplify this more? Well I have these two terms here that don't involve the variable. Those are  just numbers. I can add them to each other. So I have -4/5 plus 3/5. So what's negative four plus  three? That's going to be negative one. So this is going to be -1/5, what we have in yellow here.  I still have the 2/5m, 2/5m minus 1/5. And we're done. We've simplified that as much as we can.