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## Algebra (all content)

### Course: Algebra (all content) > Unit 13

Lesson 7: Direct and inverse variation- Intro to direct & inverse variation
- Recognizing direct & inverse variation
- Recognize direct & inverse variation
- Recognizing direct & inverse variation: table
- Direct variation word problem: filling gas
- Direct variation word problem: space travel
- Inverse variation word problem: string vibration
- Proportionality constant for direct variation

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# Intro to direct & inverse variation

Sal explains what it means for quantities to vary directly or inversely, and gives many examples of both types of variation. Created by Sal Khan.

## Want to join the conversation?

- This might be a stupid question, but why do we use "k" as the constant?(40 votes)
- I would imnagine it is from the German "konstant."(9 votes)

- Does an inverse variation represent a line? Also, are these directly connected with functions and inverse functions?(12 votes)
- How can π*x be direct variation? Pi is irrational, and keeps going on and on, so there would be no exact scale for both x and y. Thank you for the help!(7 votes)
- The number pi is not going anywhere. It is fixed somewhere between 3 and 4. The y-scale could be indexed by pi itself.(10 votes)

- Why is 4x + 3y = 24 an equation that does not represent direct variation?(4 votes)
- The reason is that y doesn't vary by the same proportion that x does (because of the constant, 24).

In your equation, "y = -4x/3 + 6", for x = 1, 2, and 3, you get y = 4 2/3, 3 1/3, and 2. For x = -1, -2, and -3, y is 7 1/3, 8 2/3, and 10. Notice that as x doubles and triples, y does not do the same, because of the constant 6.

To quote zblakley from his answer here 5 years ago:

"The difference between the values of x and y is not what dictates whether the variation is direct or inverse. What is important is the factor by which they vary. While y becomes more negative as x becomes more positive, they will still vary by the same factor (i.e. if you increase x from 1 to 4 that's a factor of 4, the value of y [in y = -2x] will go from -2 (when x=1) to -8 (when x=4) which is also a factor of 4).(8 votes)

- I don't get what varies means? Can someone tell me.(2 votes)
- "Varies" refers to how one variable changes in relationship to the other variable.(12 votes)

- I see comments about problems in a practice section. Besides the 3 questions about recognizing direct and inverse variations, are there practice problems anywhere?(4 votes)
- https://www.khanacademy.org/math/algebra2/rational-expressions-equations-and-functions/direct-and-inverse-variation/e/direct_and_inverse_variation

Here is an exercise for recognizing direct and inverse variation. Hope it helps.(8 votes)

- At about5:20, (when talking about direct variation) Sal says that "
**in general**... if y varies directly with x... x varies directly with y." Are there any cases where this is not true? In other words, are there any cases when x**does not**vary directly with y, even when y varies directly with x?

Thanks!(7 votes) - i know this is a wierd question but what do you do when in a direct variation when your trying to find K what do you do when X wont go into Y evenly? do you just use decimal form or fraction form?(3 votes)
- You can use the form that you prefer; the two are equivalent. In the Khan A. exercises, accepted answers are simplified fractions and decimal answers (except in some exercises specifically about fractions and decimals). If you're not sure of the format to use, click on the "Accepted formats" button at the top right corner of the answer box.(5 votes)

- At6:09, where you give the formula for inverse variation, I am confused. The formula that my teacher gave us was ( y = k/x ) Please help and thanks so much!!(2 votes)
- Both your teacher's equation ( y = k / x) and Sal's equation ( y = k * 1/x) mean the same thing, like they will equal the same number. To show this, let's plug in some numbers.

How about x = 2 and k = 4?

Here's your teacher's equation:`y = k / x`

y = 4 / 2

y = 2

and now Sal's:`y = k * 1/x`

y = 4 * 1/2

(Since we know 1/2 equals .5, let's use that instead, usually people understand decimals better for multiplying, but it means the exact same as 1/2)

y = 4 * .5

(We are essentially taking half of 4)

y = 2

Would you like me to explain why? It takes a bit of explaining on fractions and how they work :)

------------------------------

If you want to see how we would multiply 4 * 1/2, here's a picture I drew to explain it =

http://i.imgur.com/EkViZOw.png(5 votes)

- Why does a graph expressing direct proportionality always go through the origin?(2 votes)
- Because in order for linear equation to not go through the origin, it has to be shifted i.e. have the form
`y=mx+c`

. But that will mean that x and y no longer vary directly (or inversely for that matter). For example, if`f(x)=x+2`

, then:`f(1) = 3`

;`f(2) = 4`

, so while x increased by a factor of 2, f(x) increased by a factor of 4/3, which means they don't vary directly.(5 votes)

## Video transcript

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.