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Combining like terms with negative coefficients

This example of combining like terms in an expression gets a little hairy. Pay attention. Created by Sal Khan.

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  • male robot hal style avatar for user Ben Willetts
    So x's cannot be combined with xy's or with x^2's, and y's cannot be combined with xy's or with y^2's, right? Doesn't this leave your equation containing a lot of terms still? Wouldn't it be a lot simpler if we could reduce the number of terms more?
    (297 votes)
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  • aqualine ultimate style avatar for user Wu Guanhe
    How do you recognize the difference between the 'minus' sign and the 'negative' sign.
    I'm so confused
    (81 votes)
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    • mr pink green style avatar for user David Severin
      Think of them as being the same thing, it is one or the other based on where it is in the equation. The negative sign is used at the beginning of something new, so it could be primarily in three places -3(-4 + x)/-6. A negative is beginning of an expression, the beginning of a parentheses, and the beginning of the denominator. A minus is between things such as 3(2x - 5) - 5(4x - 2) . In this case, it is between an x and a constant and between two expressions. In this case, if you try to distribute -5(4x-2) independently, the minus now is at the beginning and so it is considered a negative sign rather than the minus it was in the full expression. Does this help?
      (92 votes)
  • duskpin tree style avatar for user Jessica LeRoy
    I have been having trouble understand the -5+6=? I don't understand last video he had two numbers with - signs and he added them and kept the sign why? What am I missing maybe video that could help it seems so simple but I have no clue! Any feed back will be welcomed.
    (20 votes)
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    • cacteye blue style avatar for user Jerry Nilsson
      I think the easiest way to understand this is via the number line.
      When we add we move to the right along the number line, and when we subtract we move to the left.
      Starting at 0 (zero) we move 5 steps to the left and from there we move 6 steps to the right.

      −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7
      −−+−−+−−+−−+−−+−−+−−+−−+−−+−−+−−+−−+−−+−−+−−+−−
      <−−−−( −5 )−−−−−
      −−−−−−( +6 )−−−−−−>

      As you can see, this lands us at 1.
      So, −5 + 6 = 1

      – – –

      Adding two negative numbers, for example (−3) + (−4), is the same as (from 0) moving 3 steps to the left and then another 4 steps to the left, in total 7 steps to the left, which lands us on (−7),
      and we write −3 − 4 = −(3 + 4) = −7

      – – –

      Sal explains it more thoroughly here:
      https://www.khanacademy.org/math/arithmetic/arith-review-negative-numbers/arith-review-sub-neg-intro/v/adding-and-subtracting-negative-number-examples
      (82 votes)
  • primosaur seedling style avatar for user shemar.gardner
    at why did he leave out the y2 shouldn't that count as a y to?
    (14 votes)
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  • aqualine ultimate style avatar for user Jessiebee639
    At 0.21, what is the difference between x^2 and 2x?
    (5 votes)
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  • old spice man green style avatar for user Josiah Willis
    Im confused, please whoever explains how to do it please do so thoroughly, that i may understand.

    Please explain: Combining like terms with negative coefficient
    (4 votes)
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    • scuttlebug blue style avatar for user marshmallo
      I'm unsure what exactly you are confused about, so I will try to explain the entire concept as thoroughly as possible. If I don't cover what you are confused on, though, just tell me and I can explain that.

      First, let's set up an example problem:

      2y + 3xy + 4x - 5x^2 -7y + xy +x^2

      Remember that a term is any coefficient and any variable. So, as an example, 3x, 7y, 9z^2, and 4f are all terms. Before we do anything to combine these terms, we need to remember the rules for what terms to combine. We can only combine terms that have the same variable. That means we could combine an x with a 3x, because x is the variable in both of those terms, but we cannot combine a 4x with a 2xy. Even though both terms share an x, an xy is not the same as an x. In addition to multiplying by x, you are also multiplying by y. In short, the variables following the coefficients need to be identical in order to combine them.

      Let's recap our example

      2y + 3xy + 4x - 5x^2 -7y + xy +x^2

      Knowing what was just covered, let's rearrange the terms so that all terms with identical variables are next to each other. When doing this, remember the sign that the term carries. For example, if we wanted to rearrange x - 3xy, we would write it as -3xy + x because subtracting a term is the same as that term being negative. It helps to think of the + or - sign (and only + or - signs) in front of a term as being a part of the term itself. With that out of the way, our example becomes

      2y - 7y + 3xy +xy + 4x -5x^2 + x^2

      Now, we combine our terms. Remember that when we combine terms to think of it like combining the coefficients. No matter what numbers we combine, our variable will always stay the same. For example, the z in 3z + 9z will stay z, even after we combine the coefficients to get 12. Starting with the y variables, we have 2y - 7y. Here, we simply subtract. If you are unsure how to do this, that's okay! Don't worry. I would just go back and review negative numbers before coming back to this topic. It is important to have a good foundation. Continuing on, combining those terms gets us

      -5y

      Which added to the rest of the expression would be

      -5y + 3xy +xy + 4x -5x^2 + x^2

      Now we can combine the xy terms. 3xy and xy. Remember that there is an imaginary 1 in front of the xy. So, we can visualize the combination of these terms like 3+1. Now we have

      -5y + 4xy + 4x -5x^2 + x^2

      Our 4x is the only term with an x variable, so we leave that alone. Finally, we can combine our x^2 terms.

      -5x^2 and x^2

      This will give us -4x^2 (remember that the ^2 is a part of our variable, and because of that it will not change after we combine our coefficients), leaving us with

      -5y + 4xy + 4x -4x^2

      I have had teachers that were picky about having the terms being ordered in descending order, from largest exponent to smallest, so I would rearrange it like

      -4x^2 +4x -5y +4xy

      I wouldn't worry too much about this, but rearranging it this way will help you in the future.

      There we have it! We successfully combined like terms with negative coefficients. Once again, I hope my explanation was thorough enough, but feel free to reach out if it wasn't. Best of luck on your learning adventures, and remember that unlike y= mx +b, progress isn't linear. Don't let setbacks like these decrease your confidence and motivation. :)
      (19 votes)
  • primosaur seed style avatar for user sikati.mtonga
    HELP?
    On some examples it shows that we move the numbers on the other side of the equal = sign... I get that once thats done the numbers either become a negative or positive but... how do we know when to use it. There's some questions I can solve using the same method in this video but then there's this eg:
    -3n - 7 + (-6n) + 1
    If I were to use the method above would it be written as so?
    -3n - (-6n) -7 + 1 ?
    In this equation they used
    -3n - 7 + (-6n) + 1
    = (3+6)n -7 +1
    I don't get it? which method do we use? if both... how are we supposed to know which one is right? or when to use either one? Please.
    (8 votes)
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    • stelly blue style avatar for user Kim Seidel
      First, you need to recognize an equation vs an expression.
      An expression contains no equals sign to start with. All the examples in this video (and your example in your question) are expressions. An equation is made up of 2 expressions separated by an equals sign.

      We only change the signs to move terms across an equals sign. This requires that you have an equation. When you changed: -3n - 7 + (-6n) + 1 into -3n - (-6n) -7 + 1, you moved the "-(-6n)" to the other side of the -7. You did not more it across and equals sign. So you do not change the sign.

      Hope this helps.
      (10 votes)
  • starky tree style avatar for user Lord.Greenhart
    Is there any reason why this, [X+2*(X)] + [Y-1*(Y)] , wouldn't work as the answer?
    My logic here is that X^2 and Y^2 are equal to X*(X) and Y*(Y) meaning they could be combined with -1*(Y) and 2*(X).
    (8 votes)
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  • blobby green style avatar for user Chris King
    How do I type exponents? like I want to type the numbers "three times x squared", how might that be done? I was struggling doing it with Khanmigo.
    (5 votes)
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  • hopper happy style avatar for user Liberty
    how do you recognize the difference between a negative sign and a minus? it is all very confusing.
    (7 votes)
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    • hopper cool style avatar for user Seed Something
      When combining like terms with negative coefficients, negative signs and subtraction signs mean the same thing, and positive signs and addition signs are the same as well.

      ★Actually, they always have, through out all of addition and subtraction plus and minus signs, + and - have always meant positive and negative number movement.

      3 - 2 = 1
      means positive 3 combined with negative 2 equals positive 1.

      +3 combined with -2 = +1


      So we've been doing this same math all along
      Just now, at this stage, we're using a broader spread of the existing number values placed in combinations that weren't given to us before.

      -5 - 7 = -12
      Means to combine a negative 5 and a negative 7, the subtraction sign belongs to the seven, creating a negative 7.

      So…
      -5 - 7 = -12
      means…
      -5 + -7 = -12

      This is also written as:
      -5 + (-7) = -12

      • If we intend to subtract a negative value, we must show it by using parentheses.

      -5 - (-7) = 2

      The notation of 'negative parentheses negative', separates the -7 in a subtraction problem, because it's considered bad form to write the subtraction of a negative with consecutive minus signs.

      -5 - -7 = 2 ←nope, bad form

      In this -(- situation)…
      The negative signs on each side of the parentheses cancel out.
      -(-7) is just like an invisible negative one in front of parentheses that needs to be distributed, all signs within the parentheses get flipped to their opposite.

      -5 - (-7) = 2
      =
      -5 + (+7) = 2
      =
      -5 + 7 = 2

      ★Combining Like Terms means to merge the known values, (coefficients), that have the same variable and exponent.

      To merge (both positive and negative) matching terms to combine them into one term.

      ★ So for this level, we're just learning addition and subtraction of any number value combination!

      (≧▽≦) I hope this helps!
      (8 votes)

Video transcript

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.