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### Course: 7th grade > Unit 6

Lesson 4: Two-step equations intro# Same thing to both sides of equations

In this fun math problem, we have a balanced scale with a mystery mass on one side and several 1-kilogram masses on the other. To find the mystery mass, we remove the same number of 1-kilogram masses from both sides, keeping the scale balanced. Then, we can easily determine the mystery mass's weight. Created by Sal Khan.

## Want to join the conversation?

- Does algebra always have varibles, or is there always an unknown?(301 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.(183 votes)

- When we use "both sides" that means both sides of what?(42 votes)
- He means both sides of equation. Left of = sign, and right of = sign. Hope it helps :)(135 votes)

- Why exactly do we do the same thing to both sides?(54 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.(2 votes)

- what is slope?(33 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(31 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.(13 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?(8 votes)- yes, it will always be a linear equation.

when you get that type of problem, you should try to simplify the equation, here,

x+9=12

x=12-9=3

therefore the line is x=3(12 votes)

- why do we use the dot instead of the traditional multiplication symbol in algebra?(3 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.(7 votes)

- if you look too hard at the scale it starts to look weird, like how the base is from a slight top view but the weighty things are from a direct side view, the whole perspective is just off. or maybe that's just me being a nit-picky artist, idk.(8 votes)
- lol agreed i noticeed thaat too(1 vote)

- 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!(5 votes)

- here is a proof that this law does not work.... i think....

let a=b

then

a x b = b x b

then ab=b^2

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

then,

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(9 votes)

## 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.