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## Direct and inverse variation

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# Recognizing direct & inverse variation

## Video transcript

I've written some
example relationships between two variables-- in
this case between m and n, between a and b,
between x and y. And what I want to
do in this video is see if we can identify
whether the relationships are a direct relationship,
whether they vary directly, or maybe they vary inversely,
or maybe it is neither. So let's explore
it a little bit. So over here, we have
m/n is equal to 1/7. So let's see how we
can manipulate this. If we multiply both sides by n. What are we going to get? And in general, you
want to separate them so that the
two variables are on different sides
of the equation, so you can see is
it going to meet, is it going to be the pattern--
let me write it this way. m is equal to k times n. This would be direct variation. Or is it going to be the pattern
m is equal to k times 1/n? This is inverse variation. And you see in
either one of these, they're on different
sides of the equal sign. So let's take this first
relationship right now. Let's multiply both sides by n. And you get m, because
these cancel out, is equal to 1/7 times n. So this actually meets the
direct variation pattern. It's some constant times n. m is
equal to some constant times n. So this right over
here is direct. They vary directly. This is direct variation. Let's see, ab is
equal to negative 3. So if we want to separate
them-- and we could do it with either variable, we
could divide both sides. I don't know, let's
divide both sides by a. We could have done it by b. If we divide both sides by a, we
get b is equal to negative 3/a. Or we could also
write this as b is equal to negative 3 times 1/a. And once again, this is
this pattern right here. One variable is equal
to a constant times 1 over the other variable. In this case, our
constant is negative 3. So over here, they
vary inversely. This is an inverse relationship. Let's try this one over here. I'll try to do it in that same
color. xy is equal to 1/10. Once again, let's try to
separate the variables, isolate them on either
side of the equation. Let's divide both sides by x. You could divide by y, because
you're really just trying to find an inverse or
direct relationship. So divide both sides by x. You get y is equal
to 1/10 over x, which is the same
thing as 1/10x, which is the same thing
as 1/10 times 1/x. So y is equal to some
constant times 1/x. Once again, this is an inverse,
y and x vary inversely. Let's do this one over here. 9 times m-- I'll go to that same
orange color-- 9 times 1/m is equal to n. So this one's actually
already done for us. And it might be a
little bit clearer if we just flip this around. If we just flip the left
and the right hand side, we get n is equal
to 9 times 1/m. n is equal to some
constant times 1/m. So n varies inversely
with m, inverse. And remember, if I say
that n varies inversely with m, that also means that
m varies inversely with n. Those two things
imply each other. Now let's try it with
this expression over here. And this one's a little
bit of a trickier one, because we've already separated
the variables on both hands. And then we have this
kind of-- if this was b is equal to
1/3 times a, then we would have direct
variation, then b would vary directly with a. But in this case,
we have 1/3 minus a. And you say, hey, maybe
they're opposites, or whatever. And it actually turns
out that this is neither. This is neither. And to make that
point 100% clear, let's look at two
of these examples. In direct variation,
if you scale up one variable in
one direction, you would scale up the other
variable by the same amount. So if we have x going--
if x doubles from 1 to 2, when x is 1-- actually, I
should this with m and n. So m and n. So when-- and the way
I've written it here, although you could algebraically
manipulate it so that one looks more dependent than the other. But in this situation
where n is 1, m is 1/7. And when n is 7, m
is going to be 1. So you have the situation
that if n is scaled up by 7, then m is also scaled
up by 7, or vice versus. So it's more of a relationship. I could have expressed
n in terms of m, but when you scale
one variable up by 7, you also have to scale up
the other variable by 7. When you scale it
up by some amount, you have to scale the other
variable by the same amount. So this is direct variation. Let's take the inverse, or when
two variables vary inversely, this situation right over here. Let's take a and b. When a is equal to 1, b
is equal to negative 3. And we could do it
explicitly right over here. We can even go to
the original one. When a is equal to 1, we have
1b is equal to negative 3. b is equal to negative 3. Now when a-- if I were to take
a, and if I were to, let's say, I were to triple it. So far we're going
to multiply it by 3. So now a is 3. We have 1/3 times negative 3. So now b is negative. Notice we didn't
multiply b times 3 here. Now we divided b times 3, or we
divided b by 3, I should say. Or another way is we
multiplied by 1/3. So if you scale up a by 3,
you're scaling down b by 3. So they're varying inversely. What you're going to
see in this neither is that neither of these
are going to be the case. So let's try it out. I'll do it in that same
green color, that same green. So we have a and b. So when a is-- I don't know,
what a is 1, what is b? 1/3 minus 1, that's 1/3 minus
3/3, that's negative 2/3. And then let's
divide just for fun. Let's divide a by 3. So if a goes to 1/3, so over
here we're dividing by 3, or you could say we're
multiplying by 1/3. So if a is 1/3, then b
is equal to 0, right? If a is 1/3, b is equal to 0. So notice, if this
was direct variation, we would be multiplying this by
1/3 as well-- which, clearly, we didn't. And if this was
inverse variation, if they varied inversely, we
would be multiplying by 3, which clearly we didn't. We just got some other number,
which actually ended up the scaling actually
didn't matter. What happens is
that things really just got shifted by some amount. They got shifted by 2/3. These neither vary
directly nor inversely, this last one right over here.