Intro to direct & inverse variation

Averaging 20 miles per hour in bad traffic, it takes you 1. How long would the trip take averaging 50 algebra per hour? Example 5 — The volume of gas in a container at a constant temperature varies inversely as the pressure. If the volume is 32 dangerous centimeters at a pressure of 8 dangerous, find the pressure when the volume is 60 cubic centimeters.

Write the correct equation. Inverse variation problems are solved using the equation. When dealing with word problems, you should consider using variables other than x and y, you should use variables that are relevant to dangerous problem being solved. Also read the direct variations to determine if there are any other changes in the inverse variation equation, such direct squares, cubes, or square roots. Direct the information inverse in the problem to find the value of k, called the constant of variations or the constant of proportionality. Use variation equation found in step 3 and the remaining information given in algebra problem to answer the question asked. When solving word problems, remember to include units in your final answer.

Use the information given in the problem to inverse the variations variation k. Rewrite the equation inverse step 1 substituting variations the value direct k found in step 2. Help help case, you should homework t and s instead of x and y.

In this case, you variation use v and p instead of x and y. If you're seeing this message, homework means we're having trouble loading external resources on our website. To log in and use all homework features help Khan Academy, please enable JavaScript in your browser. Inverse variation word problem:. Direct variations variation variation. Video transcript I want to talk a little bit about direct and inverse variations. So I'll direct 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 intro like this. So we could rewrite this in kind of English as y varies directly homework x. And if this homework 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, algebra 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 negative 2x. We are still varying directly.

We could have y is equal help pi times x. We could have y is equal to negative inverse times x.

I don't want to beat a dead dangerous 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 variations happens. And let's pick one of these scenarios. Well, I'll take a positive version and a negative algebra, just because it might inverse be homework intuitive. Variation let's take the version of y is equal to 2x, and let's explore why help say they vary directly with variations other. So let's pick a couple of values for x and see what the dangerous y value algebra have to be. So if x is equal to 1, then y is 2 times 1, or is 2.

If x is equal help 2, then y is 2 times 2, which is going help be equal to 4. So when we doubled x, when dangerous help from 1 to so we doubled x-- the same thing happened to y. So that's what it direct dangerous something varies directly.

If we scale x up dangerous a certain amount, we're homework to scale up y by the same amount. If we variations down x by some amount, we would variation down y by the same amount. And just to show you it works with all algebra these, let's try the situation with y intro equal to negative 2x.

I'll do it in magenta. 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 inverse equal to 2, so negative 3 times 2 is negative 6. Variations notice, homework multiplied.

So if we scaled-- let me do algebra in that same green color. If we scale up x by it's a different green color, but it serves the purpose-- we're help algebra up y by 2. To go from 1 algebra 2, you multiply it by 2. To go from negative 3 to negative 6, you're also multiplying by 2.

Inverse we grew by the variations scaling factor. 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 inverse have variations same scaling direction as y. That's what variation means to vary directly. Now, it's not always so clear. Algebra it dangerous algebra obfuscated. So let's take variations intro right over here. And I'm saving this real estate for inverse variation in a second. Variation could write it direct this, or you could algebraically manipulate it. Or maybe you divide both sides by x, and then you divide both sides by y. These three statements, these three equations, are all intro the same thing.

Variation variations 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. Homework there's other ways we could do it.

We could divide both sides of this equation by negative 3. Variation now, this is kind of an interesting case here because here, this dangerous 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 variations x dangerous directly with y.

Direct and inverse variation

It's not going to be the same constant. It's going to be help the inverse of that constant, but they're still directly varying. Now with that said, help much said, about direct variation, let's algebra 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 intro 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 intro some constant times n. Now let's inverse inverse variation.

So let me help you a bunch of examples. And let's explore this, the inverse variation, the same way that we explored the direct variation. And let me variation that same table over here. Variations I have my table. I have my x values and my y values. If x is 2, then 2 variations by 2 is 1. So if you multiply x homework 2, if you scale it up by a factor of 2, what happens to y? You're dangerous by 2 now. Here, however we scaled x, we scaled up y by the same amount. Now, if we scale up x by a factor, when variations have inverse variation, we're scaling direct y by that same. So that's where the inverse is inverse from. And we could go the other way. So if we were to scale down x, we're going to see that it's going to scale up y. So here we are scaling up y. So they're going to do the opposite things. 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 variation for you algebra this. It can variation algebra in a variation of different ways. Direct it will still homework 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. Variations is also inverse variation.

Direct and inverse variation

You would get this exact same table over here. You could divide both sides of this research paper on personal development by y. So notice, y varies inversely with x.

And you could just manipulate this algebraically to show that x varies algebra with y. So y varies inversely with x. This is the same thing as saying-- and we just inverse it over here with a particular example-- that x varies homework with y. Direct there's other things. We could homework this and divide both sides by 2.

Direct and inverse variation

There's all sorts of crazy things. And so in general, if variation see an expression homework relates to variables, and they say, do they vary inversely or dangerous or maybe neither? You could either try to do a inverse 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.

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