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Welcome to the presentation on figuring out the slope.

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Let's get started.

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So, let's say I have two points.

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And, as we learned in previous presentations, that all

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you need to define a line is two points.

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And I think if you think about that, that makes sense.

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Let's say we have two points.

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And let me write down the two points we're going to have.

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Let's say one point is, why isn't it writing.

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Sometimes this thing acts a little finicky.

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Oh, that's because I was trying to write in black.

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Let's say that one point is, negative 1, 3.

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So, let's see.

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Where do we graph that?

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So, this is 0, 0.

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We go negative 1, this is negative 1 here.

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And then we're going to go 3 up.

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1, 2, 3.

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Because this is 3 right here.

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So, negative 1, 3 is going to be right over there.

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OK, so that's the first point.

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The second point, I'm going to do it in a different color.

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The second point is 2, 1.

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Let's see where we would put that.

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We would count 1, 2.

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This is 2, 1.

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Because this is 1.

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So the point's going to be here.

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So we've graphed our two points.

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And now the line that connects them, it's going to look

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something thing like this.

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And I hope I can draw it well.

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Through that point.

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Like that.

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Then I'm going to do it.

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And then I'm just going to try to continue the line from here.

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That might be the best technique.

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Something like that.

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So, let's look at that line.

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So what we want to do in this presentation is, figure out

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the slope of that line.

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So let's write out a couple of things that

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I think will help you.

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So, there's a couple ways to view slope.

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I think, intuitively, you know that the slope is the

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inclination of this line.

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And we can already see that this is a

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downward sloping line.

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Because it comes from the top left to the bottom right.

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So it's going to be a negative number, the slope.

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So you know that immediately.

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And we'll have -- what we're going to do is figure out how

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to figure out the slope.

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So the slope, let me write this down, slope and -- oftentimes

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they'll use the variable m, for slope, I have no idea why.

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Because m, clearly, does not stand for slope.

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That is equal to -- there's a couple of things

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you might hear.

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Change in y over change in x.

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That triangle, which is pronounced, delta just a Greek

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letter, that means change.

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The change in y over change in x.

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And that also is equal to rise over run.

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And I'm going to explain what all of this means in a second.

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So let's start at one of these points.

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Let's start at this green point, negative 1, 3.

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So how much do we have to rise and how much do we have to run

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to get to the second point, 2, 1?

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So let's do the rise first.

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Well, we have to go minus 2, so that's the rise.

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So the rise is equal to minus 2.

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Because we have to go down 2 to get to the same y

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as this yellow point.

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And then we have to run right there.

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We have to run plus 3.

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So rise divided by run is equal to minus 2 over 3.

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Well, how would we do that if we didn't have this nice graph

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here to actually draw on?

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Well, what we can do is, we can say let's take this

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as a starting point.

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Change in y, change in y, over change in x, is equal to

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we take the first y point, which is 3.

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And we subtract the second y point, which

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is 1, you see that?

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We just took 3 minus 1.

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So that's the change in y over, and we take the first x point.

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Negative 1, minus the second x point, minus

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2, so 3 minus 1 is 2.

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And negative 1 minus 2 is equal to minus 3.

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So, same thing.

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We got minus 2 over 3.

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Now we could have done it the other way.

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And I'm running out of space here.

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But we could've made this the first point.

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If we made that the first point, then the change in y

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would have been -- I want to make it really cluttered,

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so to confuse you.

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Change in y would be this y.

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1 minus 3 over change in x, would be 2, minus minus 1.

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Well, 1 minus 3 is minus 2.

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And 2 minus negative 1 is 3.

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So, once again, we got minus 2/3, So it doesn't matter which

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point we start with, as long as, if we use the y in this

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coordinate first, then we have to use the x in that

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coordinate first.

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Let's do some more problems.

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Actually, I'm going to do a couple just so you see the

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algebra without even graphing it first.

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So, let's say I wanted to figure out the slope between

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the points 5, 2, and 3, 5.

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Well, let's take this as our starting point.

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So, change in y over change in x, or rise over run, well,

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change in y would be this 5.

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5 minus this 2.

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Over this 3 minus this 5.

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And that gets us 3, this is a 5, over minus 2.

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Equals minus 3/2.

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Let's do another one.

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This time I'm going to try to make it color-coded so it'll

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more self-explanatory.

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Say, it's 1, 2.

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That's the first point.

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And then the second point is 4, 3.

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So, once again, we say slope is equal to change in

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y over change in x.

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Well, in y.

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We take the first y.

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Let's start here.

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And we'll call that y1.

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So that's 3 minus the second y, which is that 2.

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And then all of that over, once again, the first x.

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Which is 4, minus the second x, which is that 1.

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And this equals 3 minus 2, is 1.

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And 4 minus 1 is 3.

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So the slope in this example is 1/3.

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And we could have actually switched it around.

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We could have also done it other way.

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We could have said, 2 minus 3 over 1 minus 4.

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In which case we would have gotten negative

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1 over negative 3.

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Well, that just equals 1/3 again.

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Because the negatives cancel out.

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So I'll let you think about why this and this come

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out to the same thing.

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But the important thing to realize is, if we use the 3

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first, if we use the 3 first for the y, we also have to

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use the 4 first for the x.

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That's a common mistake.

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And also, you always have to be very careful with the negative

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signs when you do these type of problems.

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But I think that will give you at least enough of a sense that

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you could start the slope problems.

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The next module, I'll actually show you how to figure

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out the y intercept.

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Because, as we said, before the equation of any line is,

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y is equal to m x plus b.

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And I'm going to go into some more detail.

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Where m is the slope.

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So if you know the slope of a line.

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And you know the y intercept of a line, you know everything you

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need to know about the line, and you can actually write down

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the equation of a line, and figure out other points

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that are on it.

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So I'm going to do that in future modules.

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I hope I haven't confused you too much.

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And try some of those the slope modules.

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You should be able to do them.

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And I hope you have fun.

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