Video transcript
- 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 variablesis always constant. So, for example, if I lookat y over x here, we see that y over x, here it's fourover 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 columnhere 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 actuallyuse this information, the ratio, the ratio betweeny and x is this constant four, to express the relationshipbetween y and x as an equation. In fact, in some ways thisis, or in a lot of ways, this is already anequation, but I can make it a little bit clearer, if Imultiply both sides by x. If I multiply both sides by x, if I multiply both sidesby x, I am left with, well, x divided by x, you'd justhave y on the left hand side. Y is equal to 4x andyou 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'remultiplying by four. We are multiplying by four,we are multiplying by four. And so, four, in thiscase, four, in this case, in this situation, this is ourconstant 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 andyou're like well, Sal, how is this a, how wouldfour be a rate of change? And, to make that a little bit clearer, let me actually do another example, but this time, I'll actuallyput some units there. So let's say that, let's saythat I have, let's say that x-- Let me do this, I alreadyused yellow, let me use blue. So let's x, let's saythat's a measure of time and y is a measure of distance. Or, let me put it this way, xis time in terms of seconds. Let me write it this way. So, x, x is going tobe 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'vetraveled 14 meters. After three seconds,we've traveled 21 meters, and you can verify that thisis a proportional relationship. The ratio between y and x is always seven. Seven over one, 14 overtwo, 21 over three. But, I wanna write thatin terms of it's units. So, y over x is going tobe, if we look at this point right over here, it's sevenmeters over one second. Seven meters over one second, or it's equal to seven meters per second. If you look at it right overhere, if you say y over x, it's 14 meters, 14 meters, intwo 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 havey over x is equal to seven, that this is also sometimesconsidered 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 theform, 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 manipulateit algebraically, multiply both sides by x andyou would have y is equal to, y is equal to kx, whereonce again k is our constant of proportionality orproportionality constant. So, this is a really, in someways it's a very simple idea, but in a lot of ways, you'llsee this showing up multiple, many, many times inyour mathematical career and it's neat to be able to recognize this as a proportional relationship.
