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Combining like terms with negative coefficients & distribution

We've learned about order of operations and combining like terms. Let's layer the distributive property on top of this. Created by Sal Khan.

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  • mr pink red style avatar for user ani.plus.zebra
    What is the key to getting the answer to problems like these? I get lost so fast!
    (364 votes)
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    • old spice man green style avatar for user Petrie (Peter S. Asiain III)
      The key realization here is to realize that we can't combine things that are different from one another. And you must also learn or master how to distribute so you can solve these problems
      I know this is probably not such a good example, I'll improve my example as soon as I can think of a better one.
      For instance we have 3 oranges and 8 apples, in all we have 11 fruits. let o=oranges, a=apples, f=fruits
      Mathematically we can say that 3o+8a=11f
      But we know that oranges and apples are both fruits so
      (347 votes)
  • leafers tree style avatar for user emassingill1
    I'm slightly confused, in the second example it says 7(3y - 5) - 2 (10 + 4 y), but he simplifies the last parentheses as -20 -8y......What happened to the plus symbol during the simplification?
    (119 votes)
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    • male robot donald style avatar for user Jeremy

      Sal distributed the -2 over the quantity in the second parenthesis. In other words he multiplied -2 * (10 +4y), which you do by multiplying the -2 times EVERYTHING in the parentheses. So -2*10=-20 and -2*4y=-8y. Adding those together yields:
      -20+ -8y = -20-8y.
      (97 votes)
  • starky tree style avatar for user Zachary popke
    What is it called when you replace a number for a letter
    (59 votes)
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    • blobby green style avatar for user shaukatb24-25
      When you replace a number with a letter the letter is called a variable. Variables can be used to figure out questions like " If sally used 4 stars for a painting how many would she need for 79 paintings? Write an expression.
      Number of painings: x

      Hope this helped!
      (18 votes)
  • marcimus pink style avatar for user leah lewis
    At to , how come he did not do the inverse of that operational sign when it comes to subtracting 21y - 35 - 20 - 8y if the operation of the 8y is negative?
    (34 votes)
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    • hopper cool style avatar for user Chuck Towle
      If I understand your question, this answer might help.
      21y-35-20-8y is the same thing as
      21y + (-35) +(-20) +(-8y)
      and addition is commutative so we can chage the order to
      21y +(-8y) + (-35) +(-20) Now the 21y + -8y is 13y for the same reason that 21 apples miuns 8 apples is 13 apples, so
      13y + (-35) +(-20) Now the -35 and -20 can be added to be -55 so
      13y-55 is the answer.

      I hope that helps make it click for you.
      (42 votes)
  • marcimus purple style avatar for user Riley Hitchcock
    I'm kind of confused, in the second example it says 7(3y - 5) - 2 (10 + 4 y), but Sal simplifies the last parentheses as -20 -8y so I am confused What happened to the plus symbol during the simplification?
    (21 votes)
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  • ohnoes default style avatar for user GryphonClaws
    I was practicing combining like terms with negative coefficients & distribution, and in the video they distributed the 7 to -5 when I thought the "-" was for subtraction. My question is, how can you tell the difference between when a "-" is a negative and when it's for subtraction when there are no parentheses?
    (10 votes)
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  • blobby green style avatar for user Saimreen Ali
    dont 2 negetives equal a positive though?
    (11 votes)
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  • duskpin ultimate style avatar for user Shamima F Khan
    For anyone who doesn't understand why negative times negative equals to positive, I have an example to show you:

    When I say "Eat!" I am encouraging you to eat (positive) But when I say "Do not eat!" I am saying the opposite (negative). Now if I say "Do NOT not eat!", I am saying I don't want you to starve, so I am back to saying "Eat!" (positive). So, two negatives make a positive.

    Hope this helps:)
    (19 votes)
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  • male robot hal style avatar for user TheOGTristan
    Is there any time where you don't turn the equation into its simplest form
    (15 votes)
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  • blobby green style avatar for user Hayley
    I am getting really confused if the answer should be negative or not? and should I subtract or add? The signs seem like they are way different in the answer!
    (6 votes)
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    • hopper cool style avatar for user Seed Something
      Way different signs in the answer is fine if we follow the rules of math with negatives…

      In Distribution…
      negative × positive = negative
      positive × negative = negative
      Mismatch signs = Negative
      while all…
      Matching signs = Positive

      negative × negative = positive
      positive × positive = positive

      ★Combine Like Terms
      Positives and Negatives are addition and subtraction.

      Like Terms match in both variable and exponent

      to Combine Like Terms means merge matching terms, keeping the dominant ± sign

      ★an Absolute Value
      is a number's distance from zero
      and tells us which sign to keep.
      |-42| = 42
      |2| = 2
      Whichever number is the furthest from zero dominates with its sign.
      2 - 42 = -40
      -42 + 2 = -40

      Same sign Add and Keep it♪…
      -3 - 7 = -10
      -5 - 5 = -10
      Different sign Subtract, keep the sign of largest Absolute Value, then you'll be exact. ♪
      -43 + 1 = -42
      1 - 43 = -42

      Matching signs Add
      same signs stay the same

      -33 - 300 = -333

      Mismatch signs Subtract
      absolute value tells dominant sign

      2 - 22
      is like

      -|22 - 2| = -20
      - (22 - 2) = -20

      |-22| > |+2|

      Twenty-two spaces from zero
      > is greater than
      Two spaces
      from zero
      we keep the Negative sign

      3 - 36 = -33
      -36 + 3 = -33

      ★Like competing in a…
      Tug of War game over Origin,
      sign furthest from zero wins, by the answer being on its side of Origin.
      -10 + 1 = -9
      1 - 10 = -9
      yanked TEN to the Negative left
      pulled ONE to the Positive right
      Negative sign wins with nine spaces still Left of Origin!

      ★Notice that because…
      multiplication is repeat addition
      Negatives and Subtraction signs are the same thing, and Positives are Addition signs

      there's a shared pattern to which sign is correct…

      In multiplication/division:
      Matching signs = + Positive
      Mismatching signs = - Negative

      In combining terms:
      Matching signs + Add and Keep
      Mismatch signs - Subtract and Compete

      (≧▽≦) Hope this helps!
      (23 votes)

Video transcript

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.