Main content

### Course: Algebra (all content) > Unit 2

Lesson 2: Why we do the same thing to both sides of an equation# One-step equations intuition

In this math lesson, we learn how to solve a balance scale problem with unknown mass. We represent the problem using an algebraic equation, 3x = 9, where x is the unknown mass. By dividing both sides by 3, we find that x = 3, meaning each object has a mass of 3 kilograms. Created by Sal Khan.

## Want to join the conversation?

- To simplify, could sal just do the square root of 9, which would be 3?(252 votes)
- In this case, the answer is no. If you do something to one side, you have to do it to the other, and there is no way to take the square root of 3x. This is what I mean:

sqrt 3x = sqrt 9.

And taking the sqrt of 3x does not help you in solving for x.

Hope that helped!(373 votes)

- When I see 3x=9 I think of the lesson before solving equations/testing solutions. I naturally did 3(3)=9 because we were taught how to make sure one side is equal to the other. Can you explain why doing 3(3)=9 is wrong and why we would do (1/3)3x = 9(1/3) =3. or should I always do the opposite, if 3(x)=9i would do 3/3=0 and 9/3=3(15 votes)
- Well, the first method I could understand since you probably memorised that 3 times 3 is 9. But when we get to some huge numbers, we need more concrete calculations. This is where the 2nd method comes in. We isolate the variable.

Does this answer your question? Feel free to comment.(16 votes)

- are the blue boxes supposed too be variables ?(15 votes)
- Yes. There are three blue boxes with x's on them which basically means 3x.(11 votes)

- how did he draw that?! its really good!(13 votes)
- is X cubed same as 3x or is it the third power?(8 votes)
- x cubed is x to the third power, or x * x * x. Often written x³

It is NOT the same as 3x(9 votes)

- why write the divide 1/3rd instead of just putting a fractional dividing line under 3x?(7 votes)
- They achieve the same results. Multiplying by 1/3 and dividing by 3 are the same. So, pick which one you are comfortable with and use it.(6 votes)

- Wait, can x be anything else? Such as y or ! @ # etc etc ...(7 votes)
- yes x can stand for anything or any number good luck(6 votes)

- Whilst I understand where it came from, Sal should have explained where the third came from. It coming from factors is not obvious to all.(4 votes)
- This should go into the tips and thanks!(5 votes)

- kahn acemdy too hard me cry(5 votes)
- that's a very detailed scale!

i just now realised this! :)(4 votes)- i wish i could agree, but my artist brain saw it and went "AAAAHHHH, THE PERSPECTIVE IS OFF, AAHHHHHHH". so, in the moment i am VERY jealous of all of you who dont draw.(5 votes)

## Video transcript

So once again, we
have three equal, or we say three
identical objects. They all have the
same mass, but we don't know what the
mass is of each of them. But what we do know is that
if you total up their mass, it's the same exact mass
as these nine objects right over here. And each of these nine objects
have a mass of 1 kilograms. So in total, you have 9
kilograms on this side. And over here, you
have three objects. They all have the same mass. And we don't know what it is. We're just calling that mass x. And what I want
to do here is try to tackle this a little
bit more symbolically. In the last video, we
said, hey, why don't we just multiply 1/3 of this
and multiply 1/3 of this? And then, essentially,
we're going to keep things
balanced, because we're taking 1/3 of the same mass. This total is the
same as this total. That's why the
scale is balanced. Now, let's think about
how we can represent this symbolically. So the first thing I
want you to think about is, can we set up
an equation that expresses that we have these
three things of mass x, and that in total, their mass
is equal to the total mass over here? Can we express that
as an equation? And I'll give you a
few seconds to do it. Well, let's think about it. Over here, we have three
things with mass x. So their total
mass, we could write as-- we could write their
total mass as x plus x plus x. And over here, we have nine
things with mass of 1 kilogram. I guess we could
write 1 plus 1 plus 1. That's 3. Plus 1 plus 1 plus 1 plus 1. How many is that? 1, 2, 3, 4, 5, 6, 7, 8, 9. And actually, this is a
mathematical representation. If we set it up as
an equation, it's an algebraic representation. It's not the simplest
possible way we can do it, but it is a reasonable
way to do it. If we want, we can say, well, if
I have an x plus another x plus another x, I have three x's. So I could rewrite this as 3x. And 3x will be equal to? Well, if I sum up all of
these 1's right over here-- 1 plus 1 plus 1. We're doing that. We have 9 of them, so
we get 3x is equal to 9. And let me make sure I did that. 1, 2, 3, 4, 5, 6, 7, 8, 9. So that's how we
would set it up. And so the next question
is, what would we do? What can we do mathematically? Actually, to either
one of these equations, but we'll focus on
this one right now. What can we do
mathematically in order to essentially solve for the x? In order to figure out what
that mystery mass actually is? And I'll give you another
second or two to think about it. Well, when we did
it the last time with just the scales
we said, OK, we've got three of these x's here. We want to have just one x here. So we can say,
whatever this x is, if the scale stays
balanced, it's going to be the same as
whatever we have there. There might be a temptation
to subtract two of the x's maybe from this side,
but that won't help us. And we can even see it
mathematically over here. If we subtract two
x's from both sides, on the left-hand side you're
going to have 3x minus 2x. And on the right-hand
side, you're going to have 9 minus 2x. And you're just going to
be left with 3 of something minus 2 something is
just 1 of something. So you will just have an x there
if you get rid of two of them. But on the right-hand side,
you're going to get 9 minus 2 x's. So the x's still
didn't help you out. You still have a mystery
mass on the right-hand side. So that doesn't help. So instead, what we say is--
and we did this the last time. We said, well, what if we
took 1/3 of these things? If we take 1/3 of these things
and take 1/3 of these things, we should still get the
same mass on both sides because the original
things had the same mass. And the equivalent of
doing that mathematically is to say, why don't we
multiply both sides by 1/3? Or another way to say it is we
could divide both sides by 3. Multiplying by 1/3 is the
same thing as dividing by 3. So we're going to multiply
both sides by 1/3. When you multiply both sides
by 1/3-- visually over here, if you had three x's,
you multiply it by 1/3, you're only going
to have one x left. If you have nine of
these one-kilogram boxes, you multiply it by 1/3, you're
only going to have three left. And over here, you
can even visually-- if you divide by 3, which is
the same thing as multiplying by 1/3, you divide by 3. So you divide by 3. You have an x is equal
to a 1 plus 1 plus 1. An x is equal to 3. Or you see here,
an x is equal to 3. Over here you do the math. 1/3 times 3 is 1. You're left with 1x. So you're left with x
is equal to 9 times 1/3. Or you could even view
it as 9 divided by 3, which is equal to 3.