Equations for proportional relationships

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 1 Y is 4 and when X is 2 y is 8 and when X is 3 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 4 over 1 which is just 4 8 over 2 is just 4 8 halves is the same thing as 4 12 over 3 is the same thing as 4 y over X is always equal to 4 in fact I can make another column here I can make another column here where I have Y over X here it's 4 over 1 which is equal to 4 here it's 8 over 2 which is equal to 4 here's 12 over 3 which is equal to 4 and so you can actually use this information the ratio the ratio between y and X is this constant 4 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 just have Y on the left hand side Y is equal to 4x and you see that's the case X is 1 4 times that is 4 X is 2 4 times that is 8 so here you go we're multiplying by 4 we are multiplying by 4 we are multiplying by 4 and so for in this case for in this case in this situation this is our constant of proportionality constant constant sometimes people will say proportionality constant constant of proportionality portion ality now sometimes it might even be described as a rate of change well Sal how is this 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 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 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 7 7 over 1 14 over 2 21 over 3 but I want to write that in terms of its units so Y over X is going to be if we look at this point right over here it's 7 meters over one second 7 meters over one second or it's equal to 7 meters per second if you look at it right over here if you say Y over X it's 14 meters 14 meters in 2 seconds in 2 seconds well 14 over 2 is 7 14 over 2 is 7 and then the units are meters per second so that's why this constant the seven in all of these cases we have Y over X is equal to 7 that this is also sometimes considered a rate and over here it's very clear this is a 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 an 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 4 and this is second example K is equal to 7 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 multi in your mathematical career and it's neat to be able to recognize this as a proportional relationship