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

Lesson 7: Direct and inverse variation

# Recognizing direct & inverse variation

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

## Want to join the conversation?

• I really dont understand the difference between direct and inverse variation, whats the difference?
• 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.
• Shouldnt the constant always be a whole number?
• A constant is a term that doesn't contain a variable. So no, it doesn't have to be a whole number.
• If I'm correct, this is not direct or inverse variation:
x=1/3-2y
So is the general rule that no equation with addition or subtraction is direct or inverse variation?
• 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!
• what does he mean at when he says constant?
• 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.

HOPE THIS HELPS!!
• At , does it matter which variable you divide both sides by?
• 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...
• 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!
• 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.
• 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.
• 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!
http://www.regentsprep.org/regents/math/algtrig/ate7/indexATE7.htm
• 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..
• An inverse portion is undefined at the origin, so it does not pass through the origin.

However, a direct proportion does pass through the origin.