Video transcript
Find the slope ofthe line that goes through the ordered pairs4 comma 2 and negative 3 comma 16. So just as a reminder, slopeis defined as rise over run. Or, you could view that riseis 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 youmight see this formula, and it tends to bereally complicated. But just remember it's justthese two things over here. Sometimes, slope will bespecified with the variable m. And they'll say thatm is the same thing-- and this is really thesame thing as change in y. They'll write y2 minusy1 over x2 minus x1. And this notation tendsto be kind of complicated, but all this meansis, is you take the y-value of your endpointand subtract from it the y-value of yourstarting point. That will essentiallygive you your change in y. And it says take thex-value of your endpoint and subtract from that thex-value of your starting point. And that'll giveyou change in x. So whatever ofthese work for you, let's actually figure outthe slope of the line that goes through these two points. So we're startingat-- and actually, we could do it both ways. We could start at thispoint and go to that point and calculate the slope orwe could start at this point and go to that pointand calculate the slope. So let's do it both ways. So let's say that our startingpoint is the point 4 comma 2. And let's say that our endpointis negative 3 comma 16. So what is the changein x over here? What is the change inx in this scenario? So we're going from4 to negative 3. If something goesfrom 4 to negative 3, what was it's change? You have to godown 4 to get to 0, and then you have to go downanother 3 to get to negative 3. So our change in xhere is negative 7. Actually, let mewrite it this way. Our change in x is equalto negative 3 minus 4, which is equal to negative 7. If I'm going from 4 tonegative 3, I went down by 7. Our change in x is negative 7. Let's do the same thingfor the change in y. And notice, I implicitlyuse this formula over here. Our change in x was this value,our endpoint, our end x-value minus our starting x-value. Let's do the same thingfor our change in y. Our change in y. If we're starting at2 and we go to 16, that means we moved up 14. Or another way youcould say it, you could take your endingy-value and subtract from that your starting y-valueand you get 14. So what is the slope over here? Well, the slope is justchange in y over change in x. So the slope overhere is change in y over change in x, whichis-- our change in y is 14. And our change inx is negative 7. And then if we want to simplifythis, 14 divided by negative 7 is negative 2. Now, what I wantto show you is, is that we could have doneit the other way around. We could have madethis the starting point and this the endpoint. And what we would havegotten is the negative values of each of these, but thenthey would've canceled out and we would stillget negative 2. Let's try it out. So let's say that our startpoint was negative 3 comma 16. And let's say that ourendpoint 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 negative3 and I go to 4, that means I went up 7. Or if you want tojust calculate that, you would 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,or we could say our rise. If we start at 16 and we end at2, that means we went down 14. Or you could just say 2minus 16 is negative 14. We went down by 14. This was our run. So if you say riseover run, which is the same thing as changein y over change in x, our rise is negative 14and our run here is 7. So notice, these arejust the negatives of these values fromwhen we swapped them. So once again, this isequal to negative 2. And let's just visualize this. Let me do a quickgraph here just to show you what a downwardslope would look like. So let me draw our two points. So this is my x-axis. That is my y-axis. So this point overhere, 4 comma 2. So let me graph it. So we're going to goall 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 pointnegative 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 thatgoes between them is going to looksomething like this. Try my best to draw arelatively straight line. That line will keep going. So the line will keep going. So that's my best attempt. And now notice, it'sdownward sloping. As you increase an x-value,the line goes down. It's going from the topleft to the bottom right. As x gets bigger,y gets smaller. That's what a downward-slopingline looks like. And just to visualize ourchange in x's and our change in y's that we dealt withhere, when we started at 4 and we ended at-- or whenwe started at 4 comma 2 and ended atnegative 3 comma 16, that was analogous to startinghere and ending over there. And we said our changein x was negative 7. We had to move back. Our run we had to move inthe left direction by 7. That's why it was a negative 7. And then we had to movein the y-direction. We had to move in they-direction positive 14. So that's why ourrise was positive. So it's 14 over negative7, or negative 2. When we did it the other way,we started at this point. We started at this point,and then ended at this point. Started at negative 3, 16and ended at that point. So in that situation,our run was positive 7. And now we have to godown in the y-direction since we switched thestarting and the endpoint. And now we had togo down negative 14. Our run is now positive 7 andour rise is now negative 14. Either way, we gotthe same slope.
