If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains ***.kastatic.org** and ***.kasandbox.org** are unblocked.

Main content

Current time:0:00Total duration:9:54

I want to talk a
little bit about direct and inverse variations. So I'll do direct variation
on the left over here. And I'll do inverse variation,
or two variables that vary inversely, on the
right-hand side over here. So a very simple definition
for two variables that vary directly would
be something like this. y varies directly with x if y is
equal to some constant with x. So we could rewrite
this in kind of English as y varies directly with x. And if this constant
seems strange to you, just remember this could be
literally any constant number. So let me give you a bunch
of particular examples of y varying directly with x. You could have y is equal to x. Because in this situation,
the constant is 1. We didn't even write it. We could write y is
equal to 1x, then k is 1. We could write y is equal to 2x. We could write y
is equal to 1/2 x. We could write y is
equal to negative 2x. We are still varying directly. We could have y is
equal to negative 1/2 x. We could have y is
equal to pi times x. We could have y is equal
to negative pi times x. I don't want to beat
a dead horse now. I think you get the point. Any constant times x--
we are varying directly. And to understand this maybe
a little bit more tangibly, let's think about what happens. And let's pick one
of these scenarios. Well, I'll take a positive
version and a negative version, just because it might not
be completely intuitive. So let's take the version
of y is equal to 2x, and let's explore why we
say they vary directly with each other. So let's pick a
couple of values for x and see what the resulting
y value would have to be. So if x is equal to 1, then
y is 2 times 1, or is 2. If x is equal to 2,
then y is 2 times 2, which is going
to be equal to 4. So when we doubled x,
when we went from 1 to 2-- so we doubled x-- the
same thing happened to y. We doubled y. So that's what it means when
something varies directly. If we scale x up by
a certain amount, we're going to scale up
y by the same amount. If we scale down
x by some amount, we would scale down
y by the same amount. And just to show you it
works with all of these, let's try the situation with
y is equal to negative 2x. I'll do it in magenta. y is equal to
negative-- well, let me do a new example that I
haven't even written here. Let's try y is equal
to negative 3x. So once again, let
me do my x and my y. When x is equal to 1, y is
equal to negative 3 times 1, which is negative 3. When x is equal to 2,
so negative 3 times 2 is negative 6. So notice, we multiplied. So if we scaled-- let me do
that in that same green color. If we scale up x by 2-- it's
a different green color, but it serves the purpose--
we're also scaling up y by 2. To go from 1 to 2,
you multiply it by 2. To go from negative
3 to negative 6, you're also multiplying by 2. So we grew by the
same scaling factor. And if you wanted to go
the other way-- let's try, I don't know, let's
go to x is 1/3. If x is 1/3, then y is going
to be-- negative 3 times 1/3 is negative 1. So notice, to go from 1
to 1/3, we divide by 3. To go from negative
3 to negative 1, we also divide by 3. We also scale down
by a factor of 3. So whatever direction
you scale x in, you're going to have the
same scaling direction as y. That's what it means
to vary directly. Now, it's not always so clear. Sometimes it will be obfuscated. So let's take this
example right over here. y is equal to negative 3x. And I'm saving this real
estate for inverse variation in a second. You could write it like this,
or you could algebraically manipulate it. You could maybe divide both
sides of this equation by x, and then you would get y/x
is equal to negative 3. Or maybe you divide
both sides by x, and then you divide
both sides by y. So from this, so if you
divide both sides by y now, you could get 1/x is equal
to negative 3 times 1/y. These three statements,
these three equations, are all saying the same thing. So sometimes the
direct variation isn't quite in your face. But if you do this, what I did
right here with any of these, you will get the
exact same result. Or you could just try
to manipulate it back to this form over here. And there's other
ways we could do it. We could divide both sides of
this equation by negative 3. And then you would get
negative 1/3 y is equal to x. And now, this is kind
of an interesting case here because here, this is
x varies directly with y. Or we could say x is
equal to some k times y. And in general, that's true. If y varies directly
with x, then we can also say that x
varies directly with y. It's not going to be
the same constant. It's going to be essentially
the inverse of that constant, but they're still
directly varying. Now with that
said, so much said, about direct variation,
let's explore inverse variation a little bit. Inverse variation--
the general form, if we use the same variables. And it always doesn't
have to be y and x. It could be an a and a b. It could be a m and an n. If I said m varies
directly with n, we would say m is equal
to some constant times n. Now let's do inverse variation. So if I did it with
y's and x's, this would be y is equal to
some constant times 1/x. So instead of being
some constant times x, it's some constant times 1/x. So let me draw you
a bunch of examples. It could be y is equal to 1/x. It could be y is equal to 2
times 1/x, which is clearly the same thing as 2/x. It could be y is equal
to 1/3 times 1/x, which is the same thing as 1 over 3x. it could be y is equal
to negative 2 over x. And let's explore this, the
inverse variation, the same way that we explored the
direct variation. So let's pick-- I don't know/
let's pick y is equal to 2/x. And let me do that
same table over here. So I have my table. I have my x values
and my y values. If x is 1, then y is 2. If x is 2, then 2
divided by 2 is 1. So if you multiply x
by 2, if you scale it up by a factor of 2,
what happens to y? y gets scaled down
by a factor of 2. You're dividing by 2 now. Notice the difference. Here, however we scaled x, we
scaled up y by the same amount. Now, if we scale up
x by a factor, when we have inverse variation, we're
scaling down y by that same. So that's where the
inverse is coming from. And we could go the other way. If we made x is equal to 1/2. So if we were to
scale down x, we're going to see that it's
going to scale up y. Because 2 divided by 1/2 is 4. So here we are scaling up y. So they're going to do
the opposite things. They vary inversely. And you could try it with
the negative version of it, as well. So here we're multiplying by 2. And once again, it's not
always neatly written for you like this. It can be rearranged in a
bunch of different ways. But it will still
be inverse variation as long as they're
algebraically equivalent. So you can multiply both
sides of this equation right here by x. And you would get
xy is equal to 2. This is also inverse variation. You would get this exact
same table over here. You could divide both sides
of this equation by y. And you could get x is
equal to 2/y, which is also the same thing as 2 times 1/y. So notice, y varies
inversely with x. And you could just
manipulate this algebraically to show that x varies
inversely with y. So y varies inversely with x. This is the same thing as
saying-- and we just showed it over here with a
particular example-- that x varies inversely with y. And there's other things. We could take this and
divide both sides by 2. And you would get
y/2 is equal to 1/x. There's all sorts
of crazy things. And so in general, if you
see an expression that relates to variables,
and they say, do they vary inversely or
directly or maybe neither? You could either try to
do a table like this. If you scale up x
by a certain amount and y gets scaled up by
the same amount, then it's direct variation. If you scale up x
by some-- and you might want to try a
couple different times-- and you scale down y, you
do the opposite with y, then it's probably
inverse variation. A surefire way of knowing
what you're dealing with is to actually algebraically
manipulate the equation so it gets back to either this
form, which would tell you that it's inverse variation, or
this form, which would tell you that it is direct variation.