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Hello.

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We're now going to do some more slope, and then maybe some

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y-intercept problems as well.

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Let's get started.

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So let me make up a problem.

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Let's say we have the points 2, 5.

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The other point, let's make that negative 3, negative 3.

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Well, first let's just graph those two points.

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I'm going to graph them in yellow.

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So 2, 5.

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Let's see that's one two.

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One, two, three, four, five.

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So 2, 5 is going to be right over there.

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OK.

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And then let me graph negative 3, negative 3.

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So it's one, two, three.

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One, two, three.

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So negative 3, negative 3 is right over there.

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And then now let me draw a line that will connect them.

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That's my new technique.

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I draw it in two pieces.

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I think that's good enough.

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OK.

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So let's see if we can at least first figure out the slope of

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the line, and then if we have time we'll try to figure

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out the y-intercept.

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And then we'll know the whole equation for the line.

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Let me pick a slightly thinner color, and we'll get started.

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So the slope, if you saw the last module that just

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introduces how we calculate the slope, that's

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just rise over run.

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Or, change in y over change in x.

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This is y.

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So let's just do that real fast.

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So let's take this as our starting point.

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So change in y could be 5-- remember, y is the

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second coordinate-- 5 minus negative 3.

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And that's this one.

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Over-- now you do the change in x-- 2 minus, this

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is also negative 3.

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Well 5 minus negative 3, that's 5 plus plus 3.

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So that equals 8.

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And then 2 minus negative 3.

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Once again that's 2 plus plus 3, so that equals 5.

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So we figured out the slope of this equation.

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It's 8/5.

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And let's see if that makes sense.

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Let's figure out what the rise and the run is.

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If we were to start at this point right here, let's see how

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much we have to rise to get to the same y-coordinate

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as the other point.

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So let's see.

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We're here, and the other point is up here.

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So let's figure out what this distance is.

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Actually, now is a good time to use the fat.

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Oh man, I have a shaky hand.

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OK.

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Let's figure out what that distance is.

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That distance is delta y, which is change in y.

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So it's one, two, three, four, five, six, seven, eight.

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That equals 8.

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And that makes sense, because if you think about it

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what did we just do?

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We just took y equals 5, which was up here, minus

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y equals negative 3.

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And so obviously we just calculated that distance

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just by looking at the two coordinates 5 minus negative 3.

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When you do this calculation it actually gives you

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this distance right here.

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So that's how we figure out how much we have to rise.

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So now let's do the run.

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Well the run, to go from this point to the other

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point, we went this far.

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And let's count how far that is.

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Well, it's one, two, three, four, five units.

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So we can say delta x is equal to 5.

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And that's exactly what we did. delta y over delta x was equal

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to 8/5, or rise over run is equal to 8/5.

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And it would have been the same thing if we calculated run here

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or if we calculated rise here.

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But it's the same thing.

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Hope that's making sense to you.

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And I hope that also makes sense that if the rise for a

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given run becomes more, then the slope of the line is going

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to become steeper and it'll become a bigger number.

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So let's see what we have so far for the

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equation of this line.

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So so far we know the equation of this line is equal to, y is

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equal to the slope 8/5 x plus b.

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So we're almost done.

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We just have to figure out this b right here.

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Now that b, just so you remember, that's

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the y-intercept.

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And that's where we intersect the y-axis.

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And since this graph is pretty neat, we can actually inspect

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it and see that, well, it looks like we're intersecting

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the y-axis at 2.

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So my guess is we're going to come up with b equals 2.

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But let's solve it, just in case we didn't have this

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neatly drawn graph here.

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So how can we solve for b?

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Well, we can substitute values that we know

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that work for x and y.

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Well either of these points are on that line, so we can

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substitute them in for x and y.

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So let's use the first one.

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OK.

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So the y we get 5, will equal 8/5 times x.

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Well, x there is 2.

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Times 2 plus b.

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Well, now we just get 5 is equal to 16/5 plus b.

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And then we get b equals-- well 5 is 25/5, right?

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5 is 25/5 minus 16/5 equals 9/5.

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All right.

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See, so I was actually wrong.

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When I looked at this graph I said, oh that looks like

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almost 2, so yeah it's probably going to be 2.

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But when we actually did it using algebra, when we did it

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analytically, we actually saw that b is equal to 9/5.

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So it's almost 2.

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9/5 is 1 and 4/5, or 1.8.

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So that's almost 2, but it actually turns

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out that it's not.

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It's at 1.8.

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And I can write it down as a decimal.

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1.8.

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So the final equation for the line, I'm going to try to

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squeeze it in at the bottom of this page, it's going to be y

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is equal to-- well, we know the slope.

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8/5 x.

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Now we just add the y-intercept.

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Plus 9/5.

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There.

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We solved it.

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Let's do another one.

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And so-- that's 9/5.

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I don't want to be too repetitive.

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Let's do another problem.

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Time to do another problem, and let me put that

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graph back there again.

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There you go.

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All right.

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I'm going to think of two random numbers again.

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Let me try to do this fast, because YouTube puts a

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10 minute limit on me.

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So let's say I had the points 2, negative 3.

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And I had the point negative 4, 5.

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So 2, negative 3.

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Let's plot that sucker real fast.

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So x is 2, so it's here.

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And the negative 3.

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One, two, three.

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So 2, negative 3 is there.

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And negative 4, 5.

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So that's one, two, three, four.

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One, two, three, four, five.

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I have to count like this because this

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graph is unlabeled.

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But if we actually were to draw in the coordinates you would

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that see this is 5, and this is negative 4, and so on.

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And this is 2, and this is negative 3.

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And now let's just draw a line.

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Let's draw it right there with my shaky hand.

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OK.

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There you go.

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Good line.

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And another good line.

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All right.

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So first we need to figure out the slope.

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Well we could just do that doing the algebra.

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So its slope is just delta-- I'm still using the line tool

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again-- delta y over delta x.

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Change in y over change in x.

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Let's take this y as the first point now.

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So we'll say 5 minus this y, negative 3.

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Over-- now since we used the 5 first we have to use the

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negative 4 first as well.

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Negative 4 minus 2.

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Well 5 minus negative 3, that equals 8.

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And negative 4 minus 2, well that equals negative 6.

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And negative 8/6, well that equals-- they're

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both divisible by 2.

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So that equals minus 4/3.

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And let's see, does that make sense as the slope?

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Well, if we were to go down four from this point.

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So if the rise was negative 4-- one, two, three, four.

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So if we go down-- woops, I'm using white.

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So that's why you can't see it.

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We go down by four here, and then we go to the right

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three, positive 3.

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We still end up on the line.

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So it works.

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Looks good to me.

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Let's see if I can solve the y-intercept in 30 seconds.

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Otherwise, I'll start it on the next module.

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So we get y is equal to minus 4/3 x, plus b.

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And actually what we'll do is we'll leave off here, and I'm

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going to solve for b-- and you could try to do it on your

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own-- in the next installment of this presentation.

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