Transcription
Welcome to the presentation on finding the equation of a line. Let's get started. Say I had two points. Let's say I have the point 1, 2, and I have the point 3, 4, and I want to figure out the equation of the line through these points. So let's at least figure out what that line looks like. So 1, 2 is here, and 2, 3; 3, 4. 3, 4 is here, and if I want to draw a line through them, it'll look something like that. So what we want to do is figure out the equation of this line.
Well, we know the form of an equation of a line is y = mx + b, where m is the slope, and that tells you how steep the line is, and b is the y-intercept. And I don't know why people chose m and b. We'll have to do some research on that. b is the y-intercept and the y-intercept is just where does it intersect the y-axis. And this problem, you could actually look at it and figure it out, but let's do it mathematically. So the equation for the slope m: it's rise over run. Another way to view that is for any amount that you run along the x-axis, how much do you rise?
Well, let's do that numerically. Rise is the same thing as change over y, and run is the same thing as change over x. Delta, this triangle, means change, change in y. Well, change in y, let's take the starting point to be 3, 4. Let's say we're going from 3, 4 to 2, 1. The change in y is 4 minus 2. We just took this 4 minus this 2 over 3 minus 1. My phone was ringing. And that's just this 3 minus this 1. So if we just solve for it, we get 4 minus 2 is 2, and 3 minus 1 is also 2, so we get the slope is equal to 1. And that makes sense because when we move over 1 in x, we go up exactly 1 in y. When we move to the left 1 in x, we move down exactly 1 in y.
So now we know the equation is y = 1x + b because we solved the m = 1. And this is, of course, the same thing as y = x + b. Now, all we have left to do is solve for b. Well, how do we do that because we have three variables here? Well, we could actually substitute one of these pairs of points in for y and x, and that makes sense, because these points have to satisfy this equation. So let's take this first pair. y is equal to 2. 2 is equal to x, which is 1 + b. It's a pretty easy equation to solve. We get b = 1, so that tells us that the equation of this line is y = x + 1. That's a pretty straightforward equation, and it makes sense. The y-intercept is 1, which is exactly here, 0, 1, and the slope is 1, and that's pretty obvious. For every amount that we move to the right, we move the same amount up, so the slope is 1.
Let's do another problem. Let's say I wanted to find the equation of the line between the points -3, 5 and 2, -6. Well, we do the same thing. m is equal to change in y over change in x. So let's take this as the starting point. So say -6 minus 5. So we just took -6 minus 5 over 2 minus -3. You've got to be real careful to get the signs right. So it's 2 minus -3. -6 minus 5 is -11, and 2 minus -3, well, that's the same thing as 2 + 3, so that's 5. So we have the slope is equal to -11/5. And notice that if on the numerator we use -6 as the starting point, that in the denominator, we have to use 2 as the starting point. We could have done it the other way around. We could have said 5 minus -6 over -3 minus 2, in which case we would have gotten-- this would have been 11 over -5. So as long as you-- if you use the -6 first, then you have to use the 2 first, or if you use the 5 first, then you have to use the -3 first. I hope I'm not completely confusing you guys.
Well, anyway, we know the slope is -11/5, so the equation of this line so far is y = -11/5x + b. Now we can take one of these pairs on the top and substitute back and solve for b. Let's take the first pair. So 5 is y. So we say 5 = -3, so it's -11/5 times -3, right? I just put the x in for x + b. So just simplifying that, I get 5 is equal to 33/5 + b, or b is equal to 5 - 33/5, and this equals 25 - 33/5. 25 - 33 is -8/5. So the equation of this line, and this one wasn't as clean as the other one, obviously, is-- let me do it in another color for emphasis-- y = -11/5x - 8/5. Hopefully, those two examples will give you enough of an idea to do the figuring out the equation of a line problems. And if you have problems with this, you might just want to try just the slope of the line problems or the y-intercept problems separately. I hope you have fun. Bye.