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Same thing to both sides of equations

The example of a scale where we try to achieve balance helps to explain why we do the same thing to both sides of an equation. Created by Sal Khan.

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  • blobby green style avatar for user Julie Zeger
    what is slope?
    (31 votes)
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    • orange juice squid orange style avatar for user Legenderz
      Slope, commonly represented as "m", is the how much the line tilts on a graph. It is commonly deciphered with the form "rise over run", basically saying that you must find two points on a line, find the y difference and the x difference, and then divide the y difference by the x difference.
      (3 votes)
  • piceratops ultimate style avatar for user VanossGaming
    Why exactly do we do the same thing to both sides?
    (2 votes)
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  • hopper cool style avatar for user tution2012
    why is x the most commonly used variable
    (2 votes)
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    • leafers tree style avatar for user Cassidy Doan
      The answer that seems more likely is that many other letters are often used for particular things (t for time, D for diameter, L for length, V for volume, E for energy, etc.), but there aren't many words that start with x. So, it makes for a good general-use variable.
      (7 votes)
  • aqualine tree style avatar for user Phoebe Dillon
    Does algebra always have varibles, or is there always an unknown?
    (0 votes)
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  • leaf blue style avatar for user Havana
    if there is a problem like x+3(squared)=12
    my question is, is if the square is on the number and not the variable, does it qualify as a linear expression?
    (6 votes)
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  • piceratops ultimate style avatar for user Blaine Moore
    At why is Sal removing 6 instead of 3? The third box (lower right) that is removed has a 4 in it?
    (0 votes)
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  • aqualine ultimate style avatar for user TheEpicOne
    why do we use the dot instead of the traditional multiplication symbol in algebra?
    (2 votes)
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  • piceratops sapling style avatar for user AGAM MAHATO
    What if there are variblies on both sides?
    (3 votes)
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    • marcimus pink style avatar for user SHADOWHUNTER
      If the variables are the same, then you cancel them out. For example...

      3x + 6= 8x -7

      You would subtract 3x. But what you do to one side you have to do to the other, leaving you with...
      6= 5x -7

      From there, you would have to add 7 -to cancel it out- and do the same to the other side. You SHOULD end up with...
      13= 5x

      It should be obvious by now -if it wasn't already- that you aren't going to get a "nice" number. But to simply it, you would have to divide 5. So your final answer should be...

      13/5 = x
      OR (if you wanted to simply it further...)
      2.6 = x

      But most math teachers will accept either answer. Although, some may say that they prefer one or the other so double check!
      (3 votes)
  • piceratops seedling style avatar for user Jordanschmidt223
    So my equation is -2(x+5)=-6x+(x-2). I solved it to -2x-10=-2x-8. I realized that if i add 2x to both sides. my x will be 0x. Would my equation be unsolvable or do i just go with 0x?
    (2 votes)
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    • mr pink green style avatar for user David Severin
      I am not sure how you got the right side of the equation as -2x - 8. If you have -6x + (x-2), the parentheses is not needed, so you have -6x + x - 2 = -5x - 2. Perhaps you left out a 4 in front of (x-2). If this is true, then you do drop the x, and end up with -10 = -8. It is not unsolvable, it is just that there is NO solution which is not quite the same, but the direction that you are aiming toward.
      (4 votes)
  • aqualine ultimate style avatar for user Satyam Rajput
    What do we have to do if one side has more weight than the other one?
    (4 votes)
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    • mr pink green style avatar for user David Severin
      When you are given a problem, the two sides are equal, so they are equally weighted. As long as you do the same thing on both sides, they remain the same weight. The only way that one side can become "more weight" on a side is if you make a math mistake, and thus the answer would not satisfy the original equation. A few common mistakes include -3x = 18, either only dividing by 3 (not -3) and dropping the negative to get x = 6 or dividing by -3 on one side and 3 on the other to get x = 6 when the answer is x = -6. A second is if you start with x + 3 = 6, and subtract 3 on one side and 6 on the other to get x = 0 instead of x = 3.
      (1 vote)

Video transcript

We've got a scale here, and as you see, the scale is balanced. And we have a question to answer. We have this mystery mass over here. It's a big question mark on this blue mass right over here. And we also have a bunch of 1-- I guess we could call them 1-kilogram masses. So these are all each a 1-kilogram mass. And my question to you is, what could we do to either side of this scale in order to figure out what the mystery mass is? Or maybe we can't figure it out at all. Is there something that we can do, either removing or adding these things, so that we can figure out what this mystery mass is? And I'll give you a couple of seconds to think about that. Well, to figure out what this mystery mass is, we essentially just want this on one side of this scale. But that by itself isn't enough. We could just remove these 3, but that won't do the job, because if we just remove these 3, then the left-hand side of the scale is clearly going to have less mass, and it's going to go up, and the right side is going to go down. And that's not going to give as much information. It's just going to tell us that this blue thing has a lower mass than what's over here. So just removing this won't help us much. It won't let us know that this is equal to that. Well, what we've got to do if we want to keep the scale balanced is we've got to remove the same amount of mass from both sides of the scale. So if we want to remove 3 things here-- so let me try my best to remove 3 things here. If we want to remove 3-- let me do it like this. I'll just color on it. I'll just erase it. So if we want to remove 3 things there, if we did this by itself, just removed these 3 things, then the two sides would not have an equal mass anymore. This side over here would have a lower mass. So we've got to remove 3 from both sides. So if we really want to make sure that our scale is balanced, we've got to remove 3 from both sides. And so if we started off with the scales balanced and then we removed 3 from both sides, the scale will still be balanced. And then when we do that, we have a clearer idea of what the mass of this object actually is. Now, when we remove 3 from both sides, the scale will still be balanced. And we know that this mass is equal to whatever's left over here. It's equal to 1, 2, 3, 4, 5, 6, 7. And if we're assuming they're kilograms, we'll know that the question mark mass is equal to 7 kilograms, that this right over here is a 7-kilogram mass.