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Manipulating expressions using structure (example 2)

Given that a+b=2a, Sal finds an expression equivalent to b-a.

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  • piceratops ultimate style avatar for user Soldier of Christ
    Why do we manipulate expressions in math and what is it used for in reality?
    (19 votes)
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    • leaf green style avatar for user Mr Mills
      We manipulate expressions for several reasons. Here are three of them.

      One reason is to make the expression easier to work with. Sometimes an expression might be written in a more complicated way than is necessary. We can manipulate the expression to change it into a simpler form.

      A second reason is to highlight an important aspect of the expression. For example, I might have an equation for a line written as "3x - 4y = 5". If I write the equation this way, I can quickly find the x-intercept and the y-intercept. But it's not so easy to see what the slope is. I can manipulate that equation to rewrite t as "y = 3/4(x) - 5/4". If I do, now I can easily see that the slope is 3/4.

      A third reason is to get the expression to fit a formula. For example, the formula for difference of squares is "a^2 minus b^2 equals (a+b) times (a-b)". That's a great tool for getting rid of some pesky exponents. But the tool only works if you have two quantities that are both squares of something.

      Could I use the difference of squares on an expression like "x^2 + 12x + 32"? At first, it doesn't look like a difference of squares. But maybe I can manipulate the expression to make it fit the formula for difference of squares. Let's try.

      x^2 + 12x + 32
      x^2 + 12x + 36 - 4 [replacing "32" with "36 - 4"]
      (x^2 + 12x + 36) - 4 [associative property]
      (x+6)^2 - 4 [factoring the perfect square trinomial]
      (x+6)^2 - 2^2 [replacing "4" with "2^2"]

      Now it's in the right format for difference of squares. "(x+6)" is the "a", and "2" is the "b".

      (x+6 + 2)(x+6 - 2) [difference of squares]
      (x+8)(x+4) [subtraction]

      By manipulating the expression, I was able to turn "x^2 + 12x + 32" into "(x+8)(x+4)". That's good, because now I can easily find the two places where a graph of the expression will cross the x-axis.
      (19 votes)
  • duskpin tree style avatar for user Wayne Riggle
    Couldn't this also be solved in this way:
    a+b=2a
    a+b-a=2a-a (subtract a from both sides)
    Answer b=a

    Next, if b=a we can either subtract b from both sides or a from both sides
    Example of subtracting a from both sides : b-a=a-a thus b-a =0, or
    subtracting b from both sides of the equation :
    b-b=a-b thus a-b=0
    a-b is the answer becuase it is equal to b-a
    No negatives involved, am I wrong about this? Please comment if I am so I can correct my mistake. Ty
    (9 votes)
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    • leaf blue style avatar for user Stefen
      Your method is fine!
      Good thinking - Nicely done.

      Now, I am not too sure what you mean by "no negatives involved", or why you think that is even a concern, but be that as it may, remember there are many different ways to arrive a a solution. Some solutions may appeal more to you than others; there is always something to be gained by looking at a problem from as many perspectives as possible.
      (3 votes)
  • aqualine seed style avatar for user cheesyseasalts
    i'm still quite confused by this 'manipulating expressions using structure' module.Can somebody explain to me how to solve this question again?
    (4 votes)
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  • blobby green style avatar for user Arbaaz Ibrahim
    Why did Sal multiply both sides by -1 to get the answer?
    (2 votes)
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    • starky ultimate style avatar for user Gabriel Barberini
      Remember that an equation is a balance. It means literally 'equivalent' (an association).

      So when you say: a = 2 you can also say that 2 = a , because they are equivalent.

      Now imagine that: If you put a rock in one plate of a balance, you need to put a rock on the other plate as well to keep the balance plates stabilized at the same height.

      That said, if I have an equation like: "a = 2", and multiply 'a' by '-1', I need to make sure that '2' gets multiplied by '-1' as well in order to maintain the balance, so "-1*a = -1*2" is the same thing as "a = 2".

      What Saul did was just the same (just to make the equation looks like some of the available choices, first he found that 'b' was equal to 'a', given that he found that "b-a = 0", but there were no choice like "b-a" or "0", so he multiplied both sides by -1 to get "-b+a=0" or "a-b=0".
      (2 votes)
  • female robot amelia style avatar for user Diyorbek Dadaboev
    Am I confused or Sal? when its written b=a, its clear that even a-b=0 consequently b-a will be 0 too. Do we have to multiply it -1 in Grid In questions ?
    (2 votes)
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  • male robot donald style avatar for user jonahmolina90
    I never would've guessed to multiply by negative 1. Does this make me incompetent?
    (1 vote)
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  • primosaur seedling style avatar for user Vi
    this uses the same format as linear equations but in variables?
    (2 votes)
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  • piceratops ultimate style avatar for user mandomerlie
    Wait. When he had
    b-a=0 why didn't he just swap them because a=b and b=a?
    (1 vote)
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    • duskpin ultimate style avatar for user Aria
      You can't swap them because subtraction is not commutative. Of course since they are equal, swapping them won't change the answer, but it's still breaking the math rules, which makes your proof incomplete/incorrect.
      (3 votes)
  • piceratops ultimate style avatar for user Elijah Watson
    I have looked at all the tips and all the questions. So now I'm going to ask this question, how did you come up with multiplying by negative one and is that legal? If it is legal, why is it legal?
    (1 vote)
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    • mr pink green style avatar for user David Severin
      When you have an equation, you can manipulate the parts by operations including multiplying by -1 as long as you do it on both sides, but with a 0, multiplying anything just gets zero
      Lets do it another way mathematically
      If we start from b - a = 0 , I can do opposites of subtracting b and adding a
      b - a = 0
      -b+a -b+a
      0 = -b + a since addition is commutative, I can switch paces for a and b
      0 = a - b
      Since both are equal to zero, they must be equal to each other,
      b - a = a - b
      So multiplying by a negative 1 is equivalent to moving all the terms to the other side by subtraction or addition, and multiplying by -1 on both sides is much easier than to move everything

      Hope this answers your question
      (2 votes)
  • leaf grey style avatar for user 𝕎𝕙i̶τε 𝕎øℓƒ
    The entire video is about finding that equation but in the end it still =0 which does not make any mathematical sense. Can someone help make sense of it for me? Thanks.
    (1 vote)
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Video transcript

- [Voiceover] We're told, suppose a+b=2a. Which of these expressions equals b-a? Alright, I encourage you to pause the video and see if you can figure that out. Which of these expressions would be equal to b-a and it's going to just involve some algebraic manipulation. Alright, let's work through this together. So we are told, we are told that a+b=2a so the first thing I would want to do is get all my a's in one place and one way I could do that is I could subract a from both sides. So if I subtract a from both sides I'm going to be left with just a b on the left-hand side and on the right-hand side I'm going to be left with 2a-a, well that's just going to be a. If I have two of something and I subtract one of them, take away one of them I'm going to have just one of those somethings equal to 1a. So we want to figure out what b-a is. Well luckily I can figure that out if I subtract a from both sides. So if I subtract a from both sides well then I'm going to get on the left-hand side b-a, which is what we want to figure out, is equal to a-a=0. So b-a=0 which is not one of the choices. Alright, so let's see if we can figure out some other things over here. So b-a=0 but that is not one of the choices. Alright, so let's see is there any other way to manipulate this? No, b minu- I could just go straight ahead and subtract 2a from both sides and I would get b-a=0. Oh, this is interesting, this is a tricky one. So b-a=0. Well, if b-a=0 then if we take the negative of both sides of this, if we take the negative of both sides, if we multiply both sides by -1. So x, I should write the times, I should write like this, alright, because we don't want to confuse it with the variable x, so if we multiply both sides by -1 what do we get? Well on the left-hand side we get a-b and on the right-hand side we still get 0. If b-a=0 then the negative of it, which is a-b is also going to be equal to 0. And that's this choice. That is, let me do that in a little darker color, that is this choice right over there. That was a good one.