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Recognizing direct & inverse variation

Sal gives many examples of two-variable equations where the variables vary directly, inversely, or neither. Created by Sal Khan.

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  • piceratops ultimate style avatar for user spongebob
    I really dont understand the difference between direct and inverse variation, whats the difference?
    (6 votes)
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    • aqualine ultimate style avatar for user Dominik.Ehlert
      Direct variation is mainly about the more A the more B, inverse variation is abotu the more A the less B. Think of yourself painting your home: If it'd take you one day to paint one room, than you will need more days, if you have more rooms. The number of days and the time taken are proportional (direct variation). Otherwise, if you're not alone but with one other person, you could paint to rooms at once or three at once with three persons etc. The more persons you have the less time you need for your whole home, so this is inverse variation.
      When you look at an equation with two variables and want to know, wether it's direct or indirect variation, just raise one of the variables als ask yourself, what has to happen to the other one, so the equation is still true.
      E.g. of you raise a in a * b = 1, you have to lower b, so the product is still the same. So this is indirect variation (the more the less).
      Otherwise a/b = 1 is direct variation: If you scale a up, you'd have to scale b by the same factor for the equation to be true.
      (23 votes)
  • marcimus pink style avatar for user Saleh Fayaz
    Shouldnt the constant always be a whole number?
    (6 votes)
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  • purple pi purple style avatar for user Alex
    If I'm correct, this is not direct or inverse variation:
    So is the general rule that no equation with addition or subtraction is direct or inverse variation?
    (4 votes)
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    • leaf green style avatar for user pandamonium123
      I believe so. Say you have y=2x. That is direct variation and will always pass through the origin when graphed. When x is multiplied by a number, the y value will also be changed by that number. If you add a y-intercept and change y=2x to, for example, y=2x+2, the x and the y will no longer be related by a constant ratio. If x is 1, y will be 4. If x is 2, y will be 6. Though x has increased by a factor of 2, y has not. Therefore, y=2x+2 is not direct variation. The same goes for inverse variation. I hope this is helpful!
      (15 votes)
  • hopper cool style avatar for user Camilla Emsbo
    what does he mean at when he says constant?
    (4 votes)
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    • male robot hal style avatar for user LBrenDrummer
      Well, what he means by constant is that when one side of a function goes up or down, multiplying or dividing, the other side will go up by the same amount, since it is a direct variation. If the function was a inverse variation, it wouldn't be a constant, because when one side of the function goes up, the other side of the function goes down, so therefore the variable isn't constant, because it doesn't do the same thing for both sides, it does the opposite.

      (6 votes)
  • male robot hal style avatar for user Connor
    At , does it matter which variable you divide both sides by?
    (3 votes)
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    • piceratops ultimate style avatar for user Wall of Words
      No. The point is that you need to recognize the formula. If you have a=1/b or b=1/a, it doesn't matter. They both have the y=k/x structure, which tells you that its an inverse formula. it's not until you start solving for those problems, that you need to be more specific on the variable you're doing things with...
      (5 votes)
  • leaf green style avatar for user caroline.hpr
    Two questions..Can "inverse variation" also be called "indirect variation"? Are the only forms of inverse variation y=kx and y=k*1/x? Thanks!
    (3 votes)
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    • leaf green style avatar for user Christopher
      Yes, you can say indirect variation. I personaly like "inverse variation" better since it explicitly says how one variable varies compared to the other.

      Be careful, y=kx is not inverse variation. This is direct variation because one variable, y, varies directly with the other variable, x, which is scaled by a constant, k.

      y=k*1/x is the only form of inverse variation, although it can look quite different when you apply some algebraic manipulation. For instance, y=k*1/x is the exact same thing as y=k/x, or xy=k. In fact, this last formula is what some people use as the basic definition of inverse variation, namely: when the product of two variables (x and y) is ALWAYS equal to a constant (k), then you have inverse variation.
      (4 votes)
  • leafers seedling style avatar for user Anshul
    I am not understanding the difference between a direct variation and an indirect variation. What would be a direct variation and wouldn't be a direct variation. Please give examples.
    (3 votes)
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    • aqualine sapling style avatar for user Kevin Yang
      A direct variation is when x and y (or f(x) and x) are directly proportional to each other... For example, if you have a chart that says x and y, and in the x column is 1, 2 and 3, and the y column says 2, 4 and 6... then you know it's proportional because for each x, y increases by 2... You can also tell a direct variation in a graph if is linear and it HAS to pass through the origin (0,0).
      Indirect variation is basically the opposite of direct variation; it isn't proportional. Also, in a graph, indirect variation will NEVER pass through the origin (0,0). It will end up as a curve.
      Hope it helps!
      (3 votes)
  • leaf green style avatar for user Rajeswari Murukan
    Can someone tell me..whether my conclusion about variation is right or not.
    "Direct/Inverse variation graphs a line passing through the origin..'.Please .Correct me if i'm wrong..
    (2 votes)
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  • male robot hal style avatar for user King Henclucky
    So if there is adding or subtracting in the situation, it is neither inverse nor direct variation?
    (3 votes)
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  • duskpin ultimate style avatar for user Code Master
    was anyone else told to watch this by connections academy?
    (3 votes)
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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.