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- 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.