Algebra: Slope

Welcome to the presentation
on figuring out the slope.

Let's get started.

So, let's say I
have two points.

And, as we learned in previous
presentations, that all

you need to define a
line is two points.

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

Let's say we have two points.

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

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

Sometimes this thing
acts a little finicky.

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

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

So, let's see.

Where do we graph that?

So, this is 0, 0.

We go negative 1, this
is negative 1 here.

And then we're
going to go 3 up.

1, 2, 3.

Because this is 3 right here.

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

OK, so that's the first point.

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

The second point is 2, 1.

Let's see where we
would put that.

We would count 1, 2.

This is 2, 1.

Because this is 1.

So the point's
going to be here.

So we've graphed
our two points.

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

something thing like this.

And I hope I can draw it well.

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00:01:36,3 --> 00:01:39,078
Through that point.

Like that.

Then I'm going to do it.

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

That might be the
best technique.

Something like that.

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00:01:57,68 --> 00:01:58,57
So, let's look at that line.

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

I think will help you.

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

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

inclination of this line.

And we can already
see that this is a

downward sloping line.

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

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

So you know that immediately.

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

to figure out the slope.

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

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

Because m, clearly, does
not stand for slope.

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

you might hear.

Change in y over change in x.

That triangle, which is
pronounced, delta just a Greek

letter, that means change.

The change in y
over change in x.

And that also is equal
to rise over run.

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

So let's start at one
of these points.

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

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

to get to the second
point, 2, 1?

So let's do the rise first.

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

So the rise is
equal to minus 2.

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

as this yellow point.

And then we have to
run right there.

We have to run plus 3.

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

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

here to actually draw on?

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

as a starting point.

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

we take the first y
point, which is 3.

And we subtract the
second y point, which

is 1, you see that?

We just took 3 minus 1.

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

Negative 1, minus the
second x point, minus

2, so 3 minus 1 is 2.

And negative 1 minus 2
is equal to minus 3.

So, same thing.

We got minus 2 over 3.

Now we could have done
it the other way.

And I'm running out
of space here.

But we could've made
this the first point.

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

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

so to confuse you.

Change in y would be this y.

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

Well, 1 minus 3 is minus 2.

And 2 minus negative 1 is 3.

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

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

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

coordinate first.

Let's do some more problems.

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

algebra without even
graphing it first.

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00:05:22,45 --> 00:05:24,56
So, let's say I wanted to
figure out the slope between

the points 5, 2, and 3, 5.

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

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

change in y would be this 5.

5 minus this 2.

Over this 3 minus this 5.

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

Equals minus 3/2.

Let's do another one.

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

more self-explanatory.

Say, it's 1, 2.

That's the first point.

And then the second
point is 4, 3.

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

y over change in x.

Well, in y.

We take the first y.

Let's start here.

And we'll call that y1.

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

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

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

And this equals 3
minus 2, is 1.

And 4 minus 1 is 3.

So the slope in this
example is 1/3.

And we could have actually
switched it around.

We could have also
done it other way.

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

In which case we would
have gotten negative

1 over negative 3.

Well, that just
equals 1/3 again.

Because the negatives
cancel out.

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

out to the same thing.

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

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

use the 4 first for the x.

That's a common mistake.

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

signs when you do these
type of problems.

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

you could start the
slope problems.

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

out the y intercept.

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

y is equal to m x plus b.

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

Where m is the slope.

So if you know the
slope of a line.

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

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

the equation of a line, and
figure out other points

that are on it.

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

I hope I haven't
confused you too much.

And try some of those
the slope modules.

You should be able to do them.

And I hope you have fun.