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Combining like terms example

We're going to simplify this expression together putting to use our new knowledge of how to combine like terms. Ok? Let's do it! Created by Sal Khan.

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  • leaf blue style avatar for user Tessah Broadhead
    why is X the most common letter used in math?
    (18 votes)
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  • mr pants teal style avatar for user Ariel
    Can we actually combine terms like that? First, it was in the right order and then Sal changed the order to gather same species. But I don't know if ...
    (7 votes)
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    • leaf blue style avatar for user Stefen
      Yes you can!
      The mathematical property which allows us to do so is the commutative property of addition, which says, essentially, that, "when adding things up, order doesn't matter," so x+y+z=x+z+y=z+y+x etc.
      So if I have 2x + 3y + 4z - x - 2y - 3z, I can rearrange that to 2x - x + 3y - 2y + 4z - 3z.
      Then I can combine the like terms, shown with parenthesis: (2x - x) + (3y - 2y) + (4z - 3z)
      2x - x = x
      3y - 2y = y
      4z - 3z = z
      Now where there is the 2x - x I can replace that with x: x + (3y - 2y) + (4z - 3z)
      Where there is the 3y - 2y I can replace that with y: x + y + (4z - 3z)
      and finally the z terms to get x + y + z which is exactly equal to the original expression, that is:
      2x + 3y + 4z - x - 2y - 3z = x + y + z.
      (8 votes)
  • old spice man green style avatar for user traynjac000
    i don t get what minus one z from 8 z and it equals 7 how? this happens around to
    (6 votes)
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  • duskpin ultimate style avatar for user angelecathie
    so i'm confused with this question:

    Combine the like terms to create an equivalent expression.
    4t−t+2

    and it said the answer is this:
    4t-t+2=(4-1)t+2
    = 3t+2

    I am confused where did the (4-1) come from? I understand where the 4 is from but where did the 1 come from? I dont see any 1 inside the question
    (4 votes)
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  • piceratops seed style avatar for user jessisaa
    Why do i hate khan
    (4 votes)
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    • mr pink green style avatar for user David Severin
      Only you can answer that, what is your attitude toward Math in general? Do you feel like you are improving or just getting by? Khan has a lot of good content that help a lot of other people, so you have to figure why it does not help you. Or maybe you are upset because someone (a teacher or parent) is making you do this, and you just protest against others telling you what to do. I cannot answer your question.
      (3 votes)
  • male robot johnny style avatar for user aden.dhies
    how would , for example 2z-7-1 = 2z + 8
    (4 votes)
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  • blobby blue style avatar for user clewis4220
    How do you Combine the like terms to create an equivalent expression? Like for example 4p +6 −3 how would you solve that?
    (3 votes)
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  • spunky sam blue style avatar for user ashley.james
    Okay now I've watched this and I'm still a little confused
    (4 votes)
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  • male robot hal style avatar for user Marvin
    when I watching this video this looks so easy but when I taking the test it's really hard!

    like this one - 4q - ( - 8q) + 10

    I thought the answer is - 12q + 10

    bcoz the the rule is "negative minus negative" the number will be just get more negative?
    (2 votes)
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  • duskpin ultimate style avatar for user Christina Gonzalez
    I understand that like terms have to go together. (ie. x goes with x, y goes with y, etc.) My question is, how do I know where the addition and subtraction symbols go? When they're all spread throughout the problem?
    (3 votes)
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    • mr pink green style avatar for user David Severin
      Maybe it will help you think about them as coefficients of the variables, so if you have 3x-2x, you add the coefficients of 3 + -2. If it is more complicated such as 4x - 3y + 2z - 4y + 3x - 4z + x -2y - 3z x has coefficients of 4, 3, and -3, y has coefficients of -3, -4, and -3, and z has coefficients of 2, -4, and -3. In each case, add the 3 numbers together.
      (1 vote)

Video transcript

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.