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Equivalent expressions: negative numbers & distribution

Learn how to identify equivalent expressions using your knowledge of the distributive property and negative numbers.

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  • mr pink green style avatar for user abdullahaliabbasi1
    This has been bothering me for ages, and a thanks to Sal Khan for answering my other 24(maybe it was 240, I'm not sure)...but I still have one question...

    1. I'm not sure if this appeals to THIS video... But it's related. Sometimes Sal changes a negative into a minus sign... How is that possible?

    Any and all help will be appreciated,

    If you answered this, Thanks!!!
    (41 votes)
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  • starky seed style avatar for user Vinay Garg
    I am confused when I use this in word problems.
    (20 votes)
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  • blobby green style avatar for user charlie.duff
    does anyone else speed the videos x2
    (22 votes)
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  • starky tree style avatar for user Rudy Sanchez
    it sort of helps but im still confused
    (9 votes)
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  • blobby green style avatar for user asantiago
    how do you do it if it is a negative minus aa number times a positive minus three
    example -c-6(2c-6)
    please answer
    (8 votes)
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    • hopper cool style avatar for user Seed Something
      Hi again!
      (answer to your question from Comments on my original reply)

      Yes! we can multiply unlike terms.

      We can multiply the coefficients, then place the variables together, behind and against our new coefficient.

      2x • 5y = 10xy

      Why? The concepts in action


      The coefficient is the number that is smooshed in front of a variable, the coefficient is the count of that variable's repeats combined, so it represents the number of times the variable occurs.)

      (x + x) • (y + y + y + y + y)
      =
      2x • 5y

      So…
      ★ That's one of the reasons we can multiply Unlike Terms is because the relationship between the coefficient and the variable is Multiplication!
      Let's take a closer look…

      2x • 5y
      =
      2 • x • 5 • y

      Another reason we can is the useful Property of Multiplication, (a truth about it), it's Commutative (interchangeable), which means we can rearrange multiplication terms, and still get the same answer!
      4 • 3 • 2 = 24 = 2 • 4 • 3 = 24

      Which is what we're doing (when we multiply unlike terms), just rearranging a multiplication expression, by completing the calculation for the values we do know (the coefficients), and keeping the unknown values (variables) set up properly to multiply!

      2x • 5y
      =
      2 • x • 5 • y
      =
      2 • 5 • x • y
      =
      10 • x • y
      =
      10xy ←yay! 🥳

      So that's why when multiplying unlike terms, we can just multiply the coefficients, and tack the variables on at the end!

      It works when there are exponents >1 as well, but take note of addition or subtraction operations zones, they act like barriers for the multiplication, only cross them after you complete the multiplication and only if you can combine like terms!

      2ab • 4w^2 + 5x • 3y - z^2 • 5u
      =
      (2ab • 4w^2) + (5x • 3y) - (z^2 • 5u)
      =
      8abw^2 + 15xy - 5uz^2 ←yay! 🥳

      We rarely need to show all the details, but knowing and understanding the 'how' and 'why' of 'what' is happening can help us take the next step up in comprehension!

      I hope this helps someone! (≧▽≦)
      (22 votes)
  • area 52 green style avatar for user HOTCOCOA
    TO ANY ONE WHO IS CONFUSED ON THE DIFFERENCE BETWEEN TERMS AND EXPRESSIONS, I CAN HELP OUT:
    Hey guys, HotCocoa here! I had trouble with this at first, too.
    Expressions are like this simple one: 2+4, or -3 + (5 x 8v) x 29v. You see, it doesn't have an equal sign, it doesn't have an answer, it's not meant to be solved. now, an EQUATION is meant to be solved, like this: 99-37=?
    Please like if this helped you, and comment if you have any more questions or want to add on or give me feedback.
    Happy to help :)
    (12 votes)
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  • leaf grey style avatar for user Dyre
    i love kahn academy and i appreciate the help but sometimes... i think sal took a knife to my brain with this math stuff.
    (12 votes)
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  • starky tree style avatar for user AnnaW
    Just so yall know I read all of yalls bios. Every one of them. On every video.
    (8 votes)
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  • blobby green style avatar for user SHARINEKIAS
    I'm confused in large equations
    (7 votes)
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  • sneak peak blue style avatar for user 𝕯𝖊𝖗𝕶𝖆𝖎𝖟𝖊𝖗
    does anyone else love it when he gives you the answer to a test question
    (6 votes)
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Video transcript

- Let's get some practice identifying equivalent expressions. So I have an expression written here in yellow and then I have two more written in this light green color. And I want you to pause this video and see if you can figure out which of these expressions, and it's possible that neither of them are, which of these are equivalent to the one in yellow. So I'm assuming you've had a go at it. So the way I like to tackle it is just to simplify all of them as much as possible. So this one up here is clearly not that simplified. So let's distribute this two. So if I distribute the two, what does it become? This is equal to two times negative six-C, is negative 12-C. Two times positive three is positive six. And then we have plus four-C, plus four-C. And then we can simplify it further cause I have both of these terms that involve C. I have negative 12-C plus four-C. So what's that going to be? Negative 12 of something plus four of something is going to be negative eight of that something. So this is going to be equal to negative eight-C. So these two blue terms when I add them, I'm going to get negative eight-C. And then finally plus six. Plus six. Now just doing that that's exactly what this first green expression is. So this one is definitely, this one is definitely going to be equivalent. Now what about this one down here? Well to figure that out let's simplify it. So let's distribute the three. Three times negative four-C is negative 12-C. Three times positive two is positive six. So plus six and then we have the plus four-C over there. It's lookin' good. And then we can add the terms that involve C. Negative 12-C and four-C, you add those together, you're gonna get negative eight-C. Negative eight-C plus six. Plus six, which is exactly what these other ones are. So all of these, all of these expressions, are actually equivalent. This one, that one, and that one. Let's do another example. And just like the last time pause the video and see which of these two expressions, it could be both of them or it could be none of them, or it could be one of them. Which of them, if any, are equivalent to this yellow expression? Alright let's do it together. And like before let's just simplify it. So the first thing my brain wants to do is let's take the terms involving N and add those together. So negative six of something, in this case N, plus four of that something, in that case N. So negative six-N plus four-N that's gonna leave you with negative two of that something. You add the coefficients. Negative six plus four is negative two Ns. So we have negative two-N, and then plus negative 12, that's the same thing as just minus 12. So minus, minus 12. So I simplified our original expression. Let's see these ones, these down here. So if I distribute the four, if I distribute the four I get four times N is four-N. And then four times negative three is minus 12. And then we are going to subtract six-N. So minus six-N. So what does this give us? We get, let me get another color here. So we have four-N, I'm adding all the terms with N, minus six-N, that's gonna give us negative two-N. And then we have the minus 12. And then we have the minus 12. So this expression when I simplify it got me the exact same place as the first expression. So these two, these two are equivalent. This is equivalent to that. Now let's check this one out. So two, let me just distribute, let me just distribute the two. Two times two-N is four-N. And then two times negative six is negative 12. So this simplified to four-N minus 12 which is clearly different than negative two-N minus 12. So this one, this one, is not the same as the other two.