WEBVTT

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In this video I'm going to do a
bunch of examples of finding

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the equations of lines in
slope-intercept form.

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Just as a bit of a review, that
means equations of lines

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in the form of y is equal to mx
plus b where m is the slope

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and b is the y-intercept.

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So let's just do a bunch of
these problems. So here they

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tell us that a line has a slope
of negative 5, so m is

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equal to negative 5.

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And it has a y-intercept of 6.

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So b is equal to 6.

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So this is pretty
straightforward.

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The equation of this line
is y is equal to

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negative 5x plus 6.

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That wasn't too bad.

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Let's do this next
one over here.

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The line has a slope of negative
1 and contains the

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point 4/5 comma 0.

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So they're telling us the slope,
slope of negative 1.

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So we know that m is equal to
negative 1, but we're not 100%

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sure about where the y-intercept
is just yet.

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So we know that this equation
is going to be of the form y

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is equal to the slope negative
1x plus b, where b is the

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y-intercept.

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Now, we can use this coordinate
information, the

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fact that it contains this
point, we can use that

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information to solve for b.

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The fact that the line contains
this point means that

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the value x is equal to 4/5, y
is equal to 0 must satisfy

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this equation.

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So let's substitute those in.
y is equal to 0 when x is

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equal to 4/5.

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So 0 is equal to negative
1 times 4/5 plus b.

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I'll scroll down a little bit.

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So let's see, we get a 0 is
equal to negative 4/5 plus b.

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We can add 4/5 to both sides
of this equation.

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So we get add a 4/5 there.

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We could add a 4/5 to
that side as well.

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The whole reason I did that is
so that cancels out with that.

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You get b is equal to 4/5.

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So we now have the equation
of the line.

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y is equal to negative 1 times
x, which we write as negative

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x, plus b, which is 4/5,
just like that.

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Now we have this one.

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The line contains the point
2 comma 6 and 5 comma 0.

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So they haven't given us the
slope or the y-intercept

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explicitly.

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But we could figure out both
of them from these

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coordinates.

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So the first thing we can do
is figure out the slope.

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So we know that the slope m is
equal to change in y over

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change in x, which is equal to--
What is the change in y?

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Let's start with this
one right here.

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So we do 6 minus 0.

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Let me do it this way.

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So that's a 6-- I want to make
it color-coded-- minus 0.

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So 6 minus 0, that's
our change in y.

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Our change in x is 2 minus 5.

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The reason why I color-coded
it is I wanted to show you

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when I used this y term first,
I used the 6 up here, that I

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have to use this x term
first as well.

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So I wanted to show you, this
is the coordinate 2 comma 6.

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This is the coordinate
5 comma 0.

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I couldn't have swapped
the 2 and the 5 then.

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Then I would have gotten the
negative of the answer.

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But what do we get here?

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This is equal to
6 minus 0 is 6.

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2 minus 5 is negative 3.

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So this becomes negative 6
over 3, which is the same

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thing as negative 2.

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So that's our slope.

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So, so far we know that the line
must be, y is equal to

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the slope-- I'll do that in
orange-- negative 2 times x

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plus our y-intercept.

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Now we can do exactly what we
did in the last problem.

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We can use one of these
points to solve for b.

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We can use either one.

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Both of these are on the line,
so both of these must satisfy

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this equation.

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I'll use the 5 comma 0 because
it's always nice when

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you have a 0 there.

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The math is a little
bit easier.

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So let's put the 5
comma 0 there.

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So y is equal to 0 when
x is equal to 5.

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So y is equal to 0 when you have
negative 2 times 5, when

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x is equal to 5 plus b.

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So you get 0 is equal
to -10 plus b.

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If you add 10 to both sides of
this equation, let's add 10 to

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both sides, these
two cancel out.

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You get b is equal to
10 plus 0 or 10.

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So you get b is equal to 10.

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Now we know the equation
for the line.

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The equation is y-- let me do it
in a new color-- y is equal

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to negative 2x plus b plus 10.

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We are done.

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Let's do another one of these.

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All right, the line contains
the points 3 comma 5 and

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negative 3 comma 0.

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Just like the last problem, we
start by figuring out the

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slope, which we will call m.

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It's the same thing as the rise
over the run, which is

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the same thing as the change
in y over the change in x.

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If you were doing this for your
homework, you wouldn't

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have to write all this.

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I just want to make sure that
you understand that these are

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all the same things.

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Then what is our change in
y over change in x?

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This is equal to, let's start
with this side first. It's just

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to show you I could pick
either of these points.

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So let's say it's 0 minus
5 just like that.

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So I'm using this coordinate
first. I'm kind of viewing it

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as the endpoint.

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Remember when I first learned
this, I would always be

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tempted to do the x
in the numerator.

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No, you use the y's
in the numerator.

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So that's the second
of the coordinates.

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That is going to be over
negative 3 minus 3.

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This is the coordinate
negative 3, 0.

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This is the coordinate 3, 5.

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We're subtracting that.

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So what are we going to get?

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This is going to be equal to--
I'll do it in a neutral

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color-- this is going to be
equal to the numerator is

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negative 5 over negative 3
minus 3 is negative 6.

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So the negatives cancel out.

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You get 5/6.

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So we know that the equation is
going to be of the form y

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is equal to 5/6 x plus b.

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Now we can substitute one of
these coordinates in for b.

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So let's do.

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I always like to use the one
that has the 0 in it.

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So y is a zero when x is
negative 3 plus b.

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So all I did is I substituted
negative 3 for x, 0 for y.

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I know I can do that because
this is on the line.

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This must satisfy the equation
of the line.

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Let's solve for b.

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So we get zero is equal to, well
if we divide negative 3

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by 3, that becomes a 1.

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If you divide 6 by 3,
that becomes a 2.

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So it becomes negative
5/2 plus b.

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We could add 5/2 to both
sides of the equation,

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plus 5/2, plus 5/2.

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I like to change my notation
just so you get

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familiar with both.

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So the equation becomes 5/2 is
equal to-- that's a 0-- is

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equal to b.

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b is 5/2.

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So the equation of our line is
y is equal to 5/6 x plus b,

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which we just figured out
is 5/2, plus 5/2.

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We are done.

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Let's do another one.

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We have a graph here.

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Let's figure out the equation
of this graph.

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This is actually, on some level,
a little bit easier.

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What's the slope?

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Slope is change in y
over change it x.

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So let's see what happens.

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When we move in x, when our
change in x is 1, so that is

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our change in x.

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So change in x is 1.

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I'm just deciding to change
my x by 1, increment by 1.

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What is the change in y?

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It looks like y changes
exactly by 4.

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It looks like my delta y, my
change in y, is equal to 4

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when my delta x is equal to 1.

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So change in y over change in
x, change in y is 4 when

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change in x is 1.

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So the slope is equal to 4.

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Now what's its y-intercept?

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Well here we can just
look at the graph.

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It looks like it intersects
y-axis at y is equal to

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negative 6, or at the
point 0, negative 6.

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So we know that b is equal
to negative 6.

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So we know the equation
of the line.

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The equation of the line is y is
equal to the slope times x

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plus the y-intercept.

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I should write that.

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So minus 6, that is plus
negative 6 So that is the

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equation of our line.

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Let's do one more of these.

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So they tell us that f of
1.5 is negative 3, f of

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negative 1 is 2.

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What is that?

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Well, all this is just a fancy
way of telling you that the

00:10:23.830 --> 00:10:30.530
point when x is 1.5, when you
put 1.5 into the function, the

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function evaluates
as negative 3.

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So this tells us that the
coordinate 1.5, negative 3 is

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on the line.

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Then this tells us that the
point when x is negative 1, f

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of x is equal to 2.

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This is just a fancy way of
saying that both of these two

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points are on the line,
nothing unusual.

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I think the point of this
problem is to get you familiar

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with function notation, for you
to not get intimidated if

00:10:56.870 --> 00:10:57.970
you see something like this.

00:10:57.970 --> 00:11:01.540
If you evaluate the function
at 1.5, you get negative 3.

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So that's the coordinate if
you imagine that y is

00:11:04.440 --> 00:11:06.020
equal to f of x.

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So this would be the
y-coordinate.

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It would be equal to negative
3 when x is 1.5.

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Anyway, I've said it
multiple times.

00:11:10.840 --> 00:11:13.280
Let's figure out the
slope of this line.

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The slope which is change in y
over change in x is equal to,

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let's start with 2 minus this
guy, negative 3-- these are

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the y-values-- over, all
of that over, negative

00:11:32.880 --> 00:11:40.140
1 minus this guy.

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Let me write it this way,
negative 1 minus

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that guy, minus 1.5.

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I do the colors because I want
to show you that the negative

00:11:50.340 --> 00:11:54.060
1 and the 2 are both coming from
this, that's why I use

00:11:54.060 --> 00:11:57.500
both of them first. If I used
these guys first, I would have

00:11:57.500 --> 00:12:00.495
to use both the x and the y
first. If I use the 2 first, I

00:12:00.495 --> 00:12:02.080
have to use the negative
1 first. That's why I'm

00:12:02.080 --> 00:12:03.390
color-coding it.

00:12:03.390 --> 00:12:08.360
So this is going to be equal
to 2 minus negative 3.

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That's the same thing
as 2 plus 3.

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So that is 5.

00:12:16.480 --> 00:12:20.040
Negative 1 minus 1.5
is negative 2.5.

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5 divided by 2.5
is equal to 2.

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So the slope of this
line is negative 2.

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Actually I'll take a little
aside to show you it doesn't

00:12:32.130 --> 00:12:34.480
matter what order
I do this in.

00:12:34.480 --> 00:12:36.180
If I use this coordinate first,
then I have to use that

00:12:36.180 --> 00:12:38.140
coordinate first. Let's
do it the other way.

00:12:38.140 --> 00:12:54.180
If I did it as negative 3
minus 2 over 1.5 minus

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negative 1, this should be minus
the 2 over 1.5 minus the

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negative 1.

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This should give me
the same answer.

00:13:04.780 --> 00:13:06.130
This is equal to what?

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Negative 3 minus 2 is negative
5 over 1.5 minus negative 1.

00:13:12.860 --> 00:13:14.520
That's 1.5 plus 1.

00:13:14.520 --> 00:13:16.610
That's over 2.5.

00:13:16.610 --> 00:13:18.840
So once again, this is
equal the negative 2.

00:13:18.840 --> 00:13:20.340
So I just wanted to show you,
it doesn't matter which one

00:13:20.340 --> 00:13:23.090
you pick as the starting or
the endpoint, as long as

00:13:23.090 --> 00:13:23.980
you're consistent.

00:13:23.980 --> 00:13:26.650
If this is the starting y,
this is the starting x.

00:13:26.650 --> 00:13:28.370
If this is the finishing
y, this has to be

00:13:28.370 --> 00:13:29.500
the finishing x.

00:13:29.500 --> 00:13:33.100
But anyway, we know that the
slope is negative 2.

00:13:33.100 --> 00:13:36.540
So we know the equation is y is
equal to negative 2x plus

00:13:36.540 --> 00:13:39.170
some y-intercept.

00:13:39.170 --> 00:13:40.720
Let's use one of these
coordinates.

00:13:40.720 --> 00:13:43.430
I'll use this one since it
doesn't have a decimal in it.

00:13:43.430 --> 00:13:47.450
So we know that y
is equal to 2.

00:13:47.450 --> 00:13:52.630
So y is equal to 2 when x
is equal to negative 1.

00:13:55.140 --> 00:13:57.290
Of course you have
your plus b.

00:13:57.290 --> 00:14:02.710
So 2 is equal to negative 2
times negative 1 is 2 plus b.

00:14:02.710 --> 00:14:06.390
If you subtract 2 from both
sides of this equation, minus

00:14:06.390 --> 00:14:10.370
2, minus 2, you're subtracting
it from both sides of this

00:14:10.370 --> 00:14:12.480
equation, you're going to get
0 on the left-hand side is

00:14:12.480 --> 00:14:14.520
equal to b.

00:14:14.520 --> 00:14:15.670
So b is 0.

00:14:15.670 --> 00:14:18.430
So the equation of our
line is just y is

00:14:18.430 --> 00:14:19.680
equal to negative 2x.

00:14:22.040 --> 00:14:23.870
Actually if you wanted to write
it in function notation,

00:14:23.870 --> 00:14:28.190
it would be that f of x is
equal to negative 2x.

00:14:28.190 --> 00:14:30.810
I kind of just assumed that
y is equal to f of x.

00:14:30.810 --> 00:14:32.420
But this is really
the equation.

00:14:32.420 --> 00:14:33.990
They never mentioned y's here.

00:14:33.990 --> 00:14:37.890
So you could just write f of x
is equal to 2x right here.

00:14:37.890 --> 00:14:40.190
Each of these coordinates
are the coordinates

00:14:40.190 --> 00:14:42.610
of x and f of x.

00:14:46.960 --> 00:14:49.960
So you could even view the
definition of slope as change

00:14:49.960 --> 00:14:53.320
in f of x over change in x.

00:14:53.320 --> 00:14:57.090
These are all equivalent ways
of viewing the same thing.