 In this video, I want to introduce one last very general application of linear functions. It's the idea of variation or specifically we call this direct variation in contrast to inverse variation, which is something we'll talk about later on. Variation describes how two or more variables relate to each other as one or more variables is allowed to vary. So we say that variable y varies directly with a variable x if y equals kx, where capital K is just some nonzero constant called the constant of variation. So we will say that two variables vary directly with each other. This is called direct variation. It's a very simple linear function, but it turns out things like this happen all the time. So for example, like in chemistry, if you take a gas and if you increase the temperature of the gas and it's inside some rigid container, as you increase the temperature, the pressure will go up as well. And therefore, the temperature and the pressure of the gas are directly proportional to each other. As one goes up, the other one goes up. And as the other goes down, the other goes down. And this affects each other at a constant rate. So we're not going to do chemistry or other type of scientific applications right now. Let's just do generic numbers. Let's say that y varies directly with x. This then tells us immediately we have a variation equation, y equals some constant times x. We have to figure out what is that constant of variation. And so they were going to tell us like, okay, if y is 15, then x equals five. If that happens, then what is y when x is seven? So there's some information going on here. So this first thing gives us a data point. When x is five, y is 15. And then we have to figure out what is y when x is equal to seven. That's what we're trying to figure out right here. What is the y-coordinate when x is seven? So using the data point here, we can actually use this to solve the coefficient of variation. So when x equals five, we're going to get k times five equals 15. That's the y-coordinate. Divide both sides by five. We see very quickly that the coefficient is going to be three. So we have k equals three. And so now we have to solve the equation y when k, we have to solve the equation y equals kx when x is seven here. Well, since we know the k value is always going to be three, we have to solve the equation y equals three x. And so if x is seven, we get that y is going to equal 21. And so we can see that, oh, it's very nice there. Variation questions, direct variation are very simple equations to work with. They're just very basic linear equations. But it turns out that direct variation, direct proportions happen all the time in the physical sciences, life sciences, social sciences, you name it. Direct variation is a very common thing. I do want to give you a quick example of how this works. It just comes down to we have a linear function which has a y-interceptive zero. And there's this, we have to first find out what is the constant of variation? What's that coefficient? And then once you have the constant variation, then you can make predictions all the time. If x is this, what is y? If y is this, what is this? You know, we can solve for these variation questions. This gives us a few examples in this section of using linear models, linear functions to model data. It's very useful. Are all data sets linear? Absolutely not. But it turns out that linear data modeling is extremely useful because there are many sets that we do in fact use linear models. Direct variation, of course, being such an example of that.