Recognizing direct & inverse variation

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.