Transcription
Well, I had to stop that last presentation because I ran out of time. But now we can start up right where we left off. So far, we had figured out the slope, and it made sense to us because we figured out the slope was minus 4/3. You could look at it two ways. When the run is 3, the rise is negative 4. So, rise: we went down by negative 4, we ran 3, and you can keep doing that on the line, and you'll keep ending up on the line. You could've also gone down 8 and then moved over 6. Or you could move over 6 and go down 8. If you went down 8, you would've showed up here. Then you would've come down back here. Whatever the slope is, if you change the y by the rise, and you change the x by the run, you should end up on the line again. I know I'm doing this over and over again, but I think it'll slowly sink in.
But anyway, where we left off, we were going to solve for b so that we could figure out the final equation for this line. Well, just like we did the last time, we just have to substitute a y or an x that we know works for this line. We know either of these coordinates will work, and then we solve for b. Let's try this first one: 2, negative 3. So y is negative 3. Don't get confused between x and y. y is negative 3 equals minus 4/3 times x, which is 2, plus b. And then we get negative 3 equals negative 8/3 plus b. And then we get b is equal to-- well, 3 is equal to minus 9/3. And then I'm just going to put this on the other side. So that's plus 8/3. So we get minus 1/3 is the y-intercept. So the equation of this line is y is equal to minus 4/3 x and then this y-intercept is minus 1/3. I know this is extremely messy, but the equation of this line, once again, is y equals minus 4/3 x minus 1/3.
Now let's look at the graph and see if that makes sense. Well, let's see. The y-intercept is right here. That's where you intersect the y-axis at the point. And we would know exactly what that point is. It's 0, negative 1/3. That's the y-intercept, and it makes sense. And even when you look at an equation, it's pretty obvious that this is going to be the y-intercept because when x equals 0, this term gets crossed out. Because 0 times negative 4/3. And then y would equal negative 1/3.
Let's do one more just to bore you. And because my wife is a resident and she works 30 hours of time and I have nothing better to do. All right. Let me put that graph back there. I joke, but you don't realize that it's true. Let's put the graph back there, and I'm going to do another random points. I'm going to go a little faster this time. So let's say I had negative 8, 5 and I had-- let me think of a good one-- 2, comma-- I'm just going to make up a number. 2, let's say 0. That's interesting. 2, 0. So let's graph negative 8, 5. 1, 2, 3, 4, 5, 6, 7, 8. 1, 2, 3, 4, 5. Right there is minus 8, 5. And then 1, 2, and then 0 is right here. So that's 2, 0. And now let me draw the line. Oh my God. I thought I was using the line tool. That's horrendous. I wish there was an undo tool. That was horrendous. That was unacceptable. That's a little bit better. Just so you know that the line doesn't end there. Lines go on forever and ever in the coordinate axis. There you go. OK.
So let's figure out the slope of this line. Well, change in y over change in x. I'm using the line tool again. I'm a spaz today. Let's take this as a starting point. We'll get 5 minus 0. y sub 1 minus y sub 2. Over x sub 1-- minus 8 minus 2. So it equals 5 over minus 10. And that equals minus 1/2. So that means for every 2 we go over, we go down 1. Well, for every 1 we go down, we go over 2, which makes sense. If we go down 2, we'll go over 4. Because 2/4 is the same thing as 1/2. I hope that makes sense to you. I know this is a downwards sloping line. So that was fast. So we know that the equation of the line so far is y is equal to minus 1/2 x plus b. Now we just solve for b. Let's substitute some numbers in here. Well, let's use this one. This is interesting: 2, 0. So y is 0. Equals minus 1/2 times 2 plus b. Well, 0 is equal to and this is minus 1 plus b. And we get b equals 1. This is a pretty easy problem. So now we get y is equal to minus 1/2 x plus 1. OK, now let's see if this actually looks right on our problem. Well, this is telling us that the y-intercept is at the point 0, 1. 0, 1 is right here. And our algebra confirms our drawing. This is the y-intercept and we see that the slope is negative 1/2. It makes sense because it's a downward sloping line, but it's not too steep. For every 1 that we go down, we go over 2. So that's negative 1/2. Or you could say for every 2 we run, we rise negative 1. Either one, we end up on a line again. So I hope that helps. I think you are definitely ready to do a lot of these slope problems that we have on the Khan Academy. So I hope you have fun and if you have any questions you can attend one of the seminars. Have fun.