Transcription
Hello. We're now going to do some more slope, and then maybe some y-intercept problems as well. Let's get started. So let me make up a problem. Let's say we have the points (2, 5). The other point, let's make that (-3, -3).
Well, first let's just graph those two points. I'm going to graph them in yellow. So (2, 5). Let's see, that's one, two. One, two, three, four, five. So (2, 5) is going to be right over there. OK. And then let me graph (-3, -3). So it's one, two, three. One, two, three. So (-3, -3) is right over there. And then now let me draw a line that will connect them. That's my new technique. I draw it in two pieces. I think that's good enough. OK.
So let's see if we can at least first figure out the slope of the line, and then if we have time we'll try to figure out the y-intercept. And then we'll know the whole equation for the line. Let me pick a slightly thinner color, and we'll get started. So the slope, if you saw the last module that just introduces how we calculate the slope, that's just rise over run. Or, change in y over change in x. This is y. So let's just do that real fast. So let's take this as our starting point. So change in y could be 5-- remember, y is the second coordinate-- 5 minus negative 3. And that's this one. Over-- now you do the change in x-- 2 minus, this is also negative 3. Well, 5 minus negative 3, that's 5 plus plus 3. So that equals 8. And then 2 minus negative 3. Once again, that's 2 plus plus 3, so that equals 5. So we figured out the slope of this equation. It's 8/5.
And let's see if that makes sense. Let's figure out what the rise and the run is. If we were to start at this point right here, let's see how much we have to rise to get to the same y-coordinate as the other point. So let's see. We're here, and the other point is up here. So let's figure out what this distance is. Actually, now is a good time to use the fat. Oh man, I have a shaky hand. OK. Let's figure out what that distance is. That distance is delta y, which is change in y. So it's one, two, three, four, five, six, seven, eight. That equals 8. And that makes sense, because if you think about it, what did we just do? We just took y = 5, which was up here, minus y = -3. And so obviously we just calculated that distance just by looking at the two coordinates 5 minus negative 3. When you do this calculation, it actually gives you this distance right here. So that's how we figure out how much we have to rise. So now let's do the run. Well, the run, to go from this point to the other point, we went this far. And let's count how far that is. Well, it's one, two, three, four, five units. So we can say delta x is equal to 5. And that's exactly what we did. delta y over delta x was equal to 8/5, or rise over run is equal to 8/5. And it would have been the same thing if we calculated run here or if we calculated rise here. But it's the same thing. Hope that's making sense to you. And I hope that also makes sense that if the rise for a given run becomes more, then the slope of the line is going to become steeper and it'll become a bigger number.
So let's see what we have so far for the equation of this line. So so far we know the equation of this line is equal to, y is equal to the slope 8/5 x + b. So we're almost done. We just have to figure out this b right here. Now that b, just so you remember, that's the y-intercept. And that's where we intersect the y-axis. And since this graph is pretty neat, we can actually inspect it and see that, well, it looks like we're intersecting the y-axis at 2. So my guess is we're going to come up with b = 2. But let's solve it, just in case we didn't have this neatly drawn graph here. So how can we solve for b? Well, we can substitute values that we know that work for x and y. Well, either of these points are on that line, so we can substitute them in for x and y. So let's use the first one. OK. So the y we get 5, will equal 8/5 times x. Well, x there is 2. Times 2 + b. Well, now we just get 5 is equal to 16/5 + b. And then we get b equals-- well, 5 is 25/5, right? 5 is 25/5 - 16/5 = 9/5. All right. See, so I was actually wrong. When I looked at this graph I said, oh that looks like almost 2, so yeah it's probably going to be 2. But when we actually did it using algebra, when we did it analytically, we actually saw that b is equal to 9/5. So it's almost 2. 9/5 is 1 and 4/5, or 1.8. So that's almost 2, but it actually turns out that it's not. It's at 1.8. And I can write it down as a decimal, 1.8. So the final equation for the line, I'm going to try to squeeze it in at the bottom of this page, it's going to be y is equal to-- well, we know the slope. 8/5 x. Now we just add the y-intercept. + 9/5. There. We solved it. Let's do another one. And so-- that's 9/5. I don't want to be too repetitive. Let's do another problem.
Time to do another problem, and let me put that graph back there again. There you go. All right. I'm going to think of two random numbers again. Let me try to do this fast, because YouTube puts a 10 minute limit on me. So let's say I had the points (2, -3). And I had the point (-4, 5). So (2, -3). Let's plot that sucker real fast. So x is 2, so it's here. And the -3. One, two, three. So (2, -3) is there. And (-4, 5). So that's one, two, three, four. One, two, three, four, five. I have to count like this because this graph is unlabeled. But if we actually were to draw in the coordinates you would see that this is 5, and this is -4, and so on. And this is 2, and this is -3. And now let's just draw a line. Let's draw it right there with my shaky hand. OK. There you go. Good line. And another good line. All right.
So first we need to figure out the slope. Well, we could just do that doing the algebra. So its slope is just delta-- I'm still using the line tool again-- delta y over delta x. Change in y over change in x. Let's take this y as the first point now. So we'll say 5 minus this y, -3. Over-- now since we used the 5 first we have to use the -4 first as well. -4 - 2. Well, 5 - negative 3, that equals 8. And -4 - 2, well that equals -6. And -8/6, well that equals-- they're both divisible by 2. So that equals -4/3. And let's see, does that make sense as the slope? Well, if we were to go down four from this point. So if the rise was negative 4-- one, two, three, four. So if we go down-- woops, I'm using white. So that's why you can't see it. We go down by four here, and then we go to the right three, positive 3. We still end up on the line. So it works. Looks good to me. Let's see if I can solve the y-intercept in 30 seconds. Otherwise, I'll start it on the next module. So we get y is equal to -4/3 x + b. And actually what we'll do is we'll leave off here, and I'm going to solve for b-- and you could try to do it on your own-- in the next installment of this presentation.