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Welcome to the presentation on finding the equation of a line.

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Let's get started.

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Say I had two points.

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Let's say I have the point 1 comma 2, and I have the point 3

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comma 4, and I want to figure out the equation of the

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line through these points.

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So let's at least figure out what that line looks like.

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So 1 comma 2 is here, and 2, 3; 3, 4.

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3 comma 4 is here, and if I want to draw a line

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through them, it'll look something like that.

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So what we want to do is figure out the equation of this line.

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Well, we know the form of an equation of a line is y equals

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mx plus b, where m is the slope, and that tells you how

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steep the line is, and b is the y-intercept.

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And I don't know why people chose m and b.

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We'll have to do some research on that.

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b is the y-intercept and the y-intercept is just where does

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it intersect the y-axis.

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And this problem, you could actually look at it and

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figure it out, but let's do it mathematically.

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So the equation for the slope m: it's rise over run.

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Another way to view that is for any amount that you

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run along the x-axis, how much do you rise?

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Well, let's do that numerically.

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Rise is the same thing as change over y, and run is the

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same thing as change over x.

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Delta, this triangle, means change, change in y.

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Well, change in y, let's take the starting

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point to be 3 comma 4.

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Let's say we're going from 3 comma 4 to 2 comma 1.

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The change in y is 4 minus 2.

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We just took this 4 minus this 2 over 3 minus 1.

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My phone was ringing.

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And that's just this 3 minus this 1.

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So if we just solve for it, we get 4 minus 2 is 2, and 3

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minus 1 is also 2, so we get the slope is equal to 1.

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And that makes sense because when we move over 1 in x,

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we go up exactly 1 in y.

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When we move to the left 1 in x, we move down exactly 1 in y.

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So now we know the equation is y equals 1x plus b because

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we solved the m equals 1.

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And this is, of course, the same thing as

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y equals x plus b.

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Now, all we have left to do is solve for b.

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Well, how do we do that because we have three variables here.

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Well, we could actually substitute one of these pairs

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of points in for y and x, and that makes sense, because

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these points have to satisfy this equation.

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So let's take this first pair. y is equal to 2.

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2 is equal to x, which is 1 plus b.

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It's a pretty easy equation to solve.

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We get b equals 1, so that tells us that the equation of

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this line is y equals x plus 1.

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That's a pretty straightforward equation, and it makes sense.

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The y-intercept is 1, which is exactly here, 0 comma

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1, and the slope is 1, and that's pretty obvious.

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For every amount that we move to the right, we move the same

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amount up, so the slope is 1.

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Let's do another problem.

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Let's say I wanted to find the equation of the line between

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the points negative 3 comma 5 and 2 comma negative 6.

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Well, we do the same thing.

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m is equal to change in y over change in x.

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So let's take this as the starting point.

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So say negative 6 minus 5.

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So we just took negative 6 minus 5 over 2

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minus negative 3.

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You've got to be real careful to get the signs right.

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So it's 2 minus negative 3.

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Negative 6 minus 5 is minus 11, and 2 minus negative 3, well,

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that's the same thing as two plus plus 3, so that's 5.

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So we have the slope is equal to negative 11/5.

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And notice that if on the numerator we use negative 6 as

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the starting point, that in the denominator, we have to use

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2 as the starting point.

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We could have done it the other way around.

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We could have said 5 minus negative 6 over negative 3

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minus 2, in which case we would have gotten-- this would have

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been 11 over negative 5.

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So as long as you-- if you use the negative 6 first, then you

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have to use the 2 first, or if you use the 5 first, then

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you have to use the negative 3 first.

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I hope I'm not completely confusing you guys.

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Well, anyway, we know the slope is negative 11/5, so the

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equation of this line so far is y equals minus 11/5x plus b.

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Now we can take one of these pairs on the top and substitute

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back and solve for b.

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Let's take the first pair.

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So 5 is y.

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So we say 5 equals negative 3, so it's negative 11/5

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times negative 3, right?

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I just put the x in for x plus b.

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So just simplifying that, I get 5 is equal to 33/5 plus b, or b

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is equal to 5 minus 33/5, and this equals 25 minus 33/5.

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25 minus 33 is minus 8/5.

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So the equation of this line, and this one wasn't as clean as

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the other one, obviously, is-- let me do it in another color

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for emphasis-- y equals minus 11/5x minus 8/5.

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Hopefully, those two examples will give you enough of an idea

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to do the figuring out the equation of a line problems.

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And if you have problems with this, you might just want to

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try just the slope of the line problems or the y-intercept

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problems separately.

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I hope you have fun.

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Bye.

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