Worked example: slope from two points

find the slope of the line that goes through the ordered pairs 4 comma 2 and negative 3 comma 16 so just as a reminder slope slope is defined as rise over run or you could view that rise is just change in Y and run is just change in X the triangles here that's the Delta symbol it literally means change in or another way and you might see this formula and it tends to be really complicated but just remember it's just these two things over here sometimes slope will be specified with the variable M and they'll say that M is the same thing and this is really the same thing as change in Y they'll write y2 minus y1 over x2 minus x1 and this notation tends to be kind of complicated but all this means is is you take the y-value of your end point and subtract from the y-value of your starting point that'll essentially give you your change in Y and it says start to take the x-value of your end point and subtract from that the x-value of your starting point and that'll give you change in X so whatever whatever these work for you let's actually figure out the slope of the line that goes through these two points so let R ting at and actually we could do it both ways we could start at this point and go to that point and calculate the slope or we could start at this point and go to that point and calculate the slope so let's do it both ways so let's say that this our starting point our starting point is the point 4 comma 2 and let's say that our end point our endpoint is negative 3 comma 16 so what is the change in x over here what is the change in X in this scenario so we're going from 4 to negative 3 if something goes from 4 to negative 3 what was its change well it takes you have to go negative you have to go down 4 to get to 0 and then you have to go down another 3 to get to negative 3 so our change in X here our change in X here is negative 7 or let me write it even let me write it this way let me write it this way our change in X is equal to negative 3 - four which is equal to negative seven if I'm going from 4 to negative 3 I went down by 7 our change in X is negative 7 let's do the same thing for the change in Y and notice I implicitly use this formula over here our change in X our change in X was this value our end point our end of X x value minus our starting x-value let's do the same thing for our change in Y our change in Y if we're starting at 2 and we go to 16 that we that means we moved up 14 or another way you could say it you could take your ending Y value and subtract from that your starting Y value and you get 14 so what is the slope over here well the slope is just change in Y over change in X so the slope over here is change in Y over change in X which is our change in Y is 14 and our change in X is negative 7 and then if we want to simplify this 14 divided by negative 7 is negative is negative 2 now what I want to show you is is that we could have done it the other way around we could have made this the starting point and this the end point and what we would have gotten is the negative values of each of these but then they would have cancelled out and we would still get negative 2 let's try it out so let's say that our start point was negative 3 comma 16 and let's say that our end point is the 4 comma 2 4 comma 2 so in this situation what is our change in X our change in X if I start at negative 3 and I go to 4 that means I went up 7 or if you want to just calculate that you'd do 4 minus negative 3 4 minus negative 3 but needless to say we just went up 7 and what is our change in Y our change in Y over here all right we could say our rise if we start at 16 and we end at 2 that means we went down 14 or you could just say 2 minus 16 is negative 14 we went down by 14 this was our Run so if you say rise over run which is the same thing as change in Y over change in X our rise is negative 14 and our run here is 7 so notice these are just the negatives of these values from when we swap them so once again this is equal to negative 2 and let's just visualize this let me do a quick and let me just do a quick graph here just to show you what a downward slope would look like so let me draw our two points so this is my x-axis that is my y-axis so this point over here 4 comma 2 so let me graph it so we're going to go all the way up to 16 so let me save some space here so we have 1 2 3 4 it's 4 comma 1 2 so 4 comma 2 is right over here 4 comma 2 then we have the point negative 3 comma 16 so let me draw that over here so we have negative 1 2 3 and we have to go up 16 so this is 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 so it goes right over here so this is negative 3 comma 16 negative 3 comma 16 so the line that goes between them is going to look something like this try my best my best to draw a relatively straight line that line will keep going so the line will keep going so that's my best attempt and I notice it's downward sloping as you increase in X value the line goes down it's it's going from the top left to the bottom right as X gets bigger Y gets smaller that's what a downward sloping line looks like and just to visualize our change in X's and our change in Y is that we dealt with here when we started at 4 and we ended at or when we start at 4 comma 2 and edit at negative 3 comma 16 that was analogous to starting here and ending over there and we said our change in X our change in X was negative 7 we have to move back our run we had to move in the left direction by 7 that's why it was negative 7 and then we had to move in the y-direction we had to move in the Y direction positive 14 so that's why I rise was positive so it was 14 over negative seven or negative two when we did the other way we started at this point we started at this point and then ended at this point started at negative 3/16 and ended at that point so in that situation our run our run was positive seven and now we have to go down in the y-direction since we switched the starting in the endpoint and now we have to go down negative 14 our run is now positive 7 and our rise is now negative 14 either way we got the same slope