6th grade (Illustrative Mathematics)
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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- Does algebra always have varibles, or is there always an unknown?(284 votes)
- Great question, and yes it does, has, and always will include variables as that is the very definition of algebra. See Sal's video on "Why all the letters in algebra", https://www.khanacademy.org/math/algebra/introduction-to-algebra/overview_hist_alg/v/why-all-the-letters-in-algebra or this Wiki article http://en.wikipedia.org/wiki/Algebra Which says "Algebra can essentially be considered as doing computations similar to that of arithmetic with non-numerical mathematical objects." Thanks T.S.(168 votes)
- When we use "both sides" that means both sides of what?(41 votes)
- He means both sides of equation. Left of = sign, and right of = sign. Hope it helps :)(132 votes)
- Why exactly do we do the same thing to both sides?(51 votes)
- It keeps it equal. So, for example, if I had 7=7, and I wanted to add 2 to the left side, I'd have to add 2 to the right side to make both sides equal again: 7+2=7+2 or 9=9.
You can kind of think of it as "both sides are essentially saying the exact same thing." if we say x+3=9, we're saying x+3 is the exact same thing as 9. "x+3" literally means "9." So, if we were to subtract 3 from our "x+3," we'll have to do the exact same thing to our "9" (subtract 3) to make them stay the same thing.(4 votes)
- what is slope?(31 votes)
- 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.(5 votes)
- why is x the most commonly used variable(28 votes)
- 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.(9 votes)
- 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?(7 votes)
- yes, it will always be a linear equation.
when you get that type of problem, you should try to simplify the equation, here,
therefore the line is x=3(9 votes)
- At1:50why is Sal removing 6 instead of 3? The third box (lower right) that is removed has a 4 in it?(0 votes)
- Ahh, yes, I guess I'm just a little dyslexic. Thought some of those were 4s. Watching it in full screen it's a little more obvious. Thanks!(6 votes)
- why do we use the dot instead of the traditional multiplication symbol in algebra?(2 votes)
- It would be confusing if you used x as a variable and a multiplication sign, so there's a dot to replace the traditional multiplication symbol.(6 votes)
- here is a proof that this law does not work.... i think....
a x b = b x b
subtract a^2 from either side:
ab - a^2 = b^2 -a^2
or, a(b - a) = (b + a)(b - a)
now divide both sides by b - a
a = b + a
but this contradicts the fact that a = b
this was told by my friend and this racked my brains and he does not tell me the reason. anyone who explains this will be greatly appreciated.(2 votes)
- You are wrong. If a = b then a and b can’t be different numbers. This is a wrong statement(8 votes)
- What if there are variblies on both sides?(4 votes)
- 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...
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!(5 votes)
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.