Equations for proportional relationships

- So, let's set up a relationship between the variables x and y. So, let's say, so this is x and this is y, and when x is one, y is four, and when x is two, y is eight, and when x is three, y is 12. Now, you might immediately recognize that this is a proportional relationship. And remember, in order for it to be a proportional relationship, the ratio between the two variables is always constant. So, for example, if I look at y over x here, we see that y over x, here it's four over one, which is just four. Eight over two is just four. Eight halves is the same thing as four. 12 over three it's the same thing as four. Y over x is always equal to four. In fact, I can make another column here. I can make another column here where I have y over x, here it's four over one, which is equal to four. Here it's eight over two, which is equal to four. Here it's 12 over three, which is equal to four. And so, you can actually use this information, the ratio, the ratio between y and x is this constant four, to express the relationship between y and x as an equation. In fact, in some ways this is, or in a lot of ways, this is already an equation, but I can make it a little bit clearer, if I multiply both sides by x. If I multiply both sides by x, if I multiply both sides by x, I am left with, well, x divided by x, you'd just have y on the left hand side. Y is equal to 4x and you see that's the case. X is one, four times that is four. X is two, four times that is eight. So, here you go, we're multiplying by four. We are multiplying by four, we are multiplying by four. And so, four, in this case, four, in this case, in this situation, this is our constant of proportionality. Constant, constant, sometimes people will say proportionality constant. Constant of proportionality, portionality. Now sometimes, it might even be described as a rate of change and you're like well, Sal, how is this a, how would four be a rate of change? And, to make that a little bit clearer, let me actually do another example, but this time, I'll actually put some units there. So let's say that, let's say that I have, let's say that x-- Let me do this, I already used yellow, let me use blue. So let's x, let's say that's a measure of time and y is a measure of distance. Or, let me put it this way, x is time in terms of seconds. Let me write it this way. So, x, x is going to be in seconds and then, y is going to be in meters. So, this is meters, the units, and this right over here is seconds. So, after one second, we have traveled, oh, I don't know, seven meters. After two seconds, we've traveled 14 meters. After three seconds, we've traveled 21 meters, and you can verify that this is a proportional relationship. The ratio between y and x is always seven. Seven over one, 14 over two, 21 over three. But, I wanna write that in terms of it's units. So, y over x is going to be, if we look at this point right over here, it's seven meters over one second. Seven meters over one second, or it's equal to seven meters per second. If you look at it right over here, if you say y over x, it's 14 meters, 14 meters, in two seconds, in two seconds. Well, 14 over two is seven, 14 over two is seven, and then the units are meters per second. So, that's why this constant, this seven, in all of these cases we have y over x is equal to seven, that this is also sometimes considered a rate. And over here it's very clear, this is my distance per time. Now, if I wanted to write it generally, I could say that, look, if I'm dealing with a proportional relationship, it's going to be of the form, I can always construct and equation of the form, of the form, either y over x is equal to k, where k is some constant. In this first example, k was equal to four and in this second example, k is equal to seven. Or, you can just manipulate it algebraically, multiply both sides by x and you would have y is equal to, y is equal to kx, where once again k is our constant of proportionality or proportionality constant. So, this is a really, in some ways it's a very simple idea, but in a lot of ways, you'll see this showing up multiple, many, many times in your mathematical career and it's neat to be able to recognize this as a proportional relationship.