Direct, Inverse, Joint and Combined Variation – She Loves Math
In an inverse relationship, as x increases the absolute value of y decreases, a situation can be considered an example of a direct or an inverse relationship. Here are some examples of direct and inverse variation: . there is an extra fixed constant, so we'll have an equation like, which is our typical linear equation.). Direct variation describes a simple relationship between two variables. We say y varies directly with x (or as x, in some textbooks) if.
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 to pi times x. We could have y is equal to negative pi times x.
Direct and inverse relationships - Math Central
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 so we doubled x-- the same thing happened to 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.
Intro to direct & inverse variation
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. 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 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. 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. 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. Or maybe you divide both sides by x, and then you divide both sides by 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 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.
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. This would be direct variation. 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. So this actually meets the direct variation pattern. It's some constant times n. So this right over here is direct.
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. 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. 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. Once again, this is an inverse, y and x vary inversely. Let's do this one over here.
What are the different types of mathematical relationships?
So this one's actually already done for us. And it might be a little bit clearer if we just flip this around. 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.
- Recognizing direct & inverse variation
- Inverse Variation Word Problems
- Direct, Inverse, Joint and Combined Variation
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 you say, hey, maybe they're opposites, or whatever. And it actually turns out that this is neither. 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 actually, I should this with m and n.