Linear Equations in Slope Intercept Form

Linear Equations in Slope Intercept Form

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

the equations of lines in
slope-intercept form.

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

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

and b is the y-intercept.

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

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

equal to negative 5.

And it has a y-intercept of 6.

So b is equal to 6.

So this is pretty
straightforward.

The equation of this line
is y is equal to

negative 5x plus 6.

That wasn't too bad.

Let's do this next
one over here.

The line has a slope of negative
1 and contains the

point 4/5 comma 0.

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

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

sure about where the y-intercept
is just yet.

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

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

y-intercept.

Now, we can use this coordinate
information, the

fact that it contains this
point, we can use that

information to solve for b.

The fact that the line contains
this point means that

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

this equation.

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

equal to 4/5.

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

I'll scroll down a little bit.

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

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

So we get add a 4/5 there.

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

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

You get b is equal to 4/5.

So we now have the equation
of the line.

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

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

Now we have this one.

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

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

explicitly.

But we could figure out both
of them from these

coordinates.

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

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

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

Let's start with this
one right here.

So we do 6 minus 0.

Let me do it this way.

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

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

Our change in x is 2 minus 5.

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

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

have to use this x term
first as well.

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

This is the coordinate
5 comma 0.

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

Then I would have gotten the
negative of the answer.

But what do we get here?

This is equal to
6 minus 0 is 6.

2 minus 5 is negative 3.

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

thing as negative 2.

So that's our slope.

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

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

plus our y-intercept.

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

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

We can use either one.

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

this equation.

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

you have a 0 there.

The math is a little
bit easier.

So let's put the 5
comma 0 there.

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

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

x is equal to 5 plus b.

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

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

both sides, these
two cancel out.

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

So you get b is equal to 10.

Now we know the equation
for the line.

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

to negative 2x plus b plus 10.

We are done.

Let's do another one of these.

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

negative 3 comma 0.

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

slope, which we will call m.

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

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

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

have to write all this.

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

all the same things.

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

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

to show you I could pick
either of these points.

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

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

as the endpoint.

Remember when I first learned
this, I would always be

tempted to do the x
in the numerator.

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

So that's the second
of the coordinates.

That is going to be over
negative 3 minus 3.

This is the coordinate
negative 3, 0.

This is the coordinate 3, 5.

We're subtracting that.

So what are we going to get?

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

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

negative 5 over negative 3
minus 3 is negative 6.

So the negatives cancel out.

You get 5/6.

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

is equal to 5/6 x plus b.

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

So let's do.

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

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

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

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

This must satisfy the equation
of the line.

Let's solve for b.

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

by 3, that becomes a 1.

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

So it becomes negative
5/2 plus b.

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

plus 5/2, plus 5/2.

I like to change my notation
just so you get

familiar with both.

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

equal to b.

b is 5/2.

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

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

We are done.

Let's do another one.

We have a graph here.

Let's figure out the equation
of this graph.

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

What's the slope?

Slope is change in y
over change it x.

So let's see what happens.

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

our change in x.

So change in x is 1.

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

What is the change in y?

It looks like y changes
exactly by 4.

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

when my delta x is equal to 1.

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

change in x is 1.

So the slope is equal to 4.

Now what's its y-intercept?

Well here we can just
look at the graph.

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

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

So we know that b is equal
to negative 6.

So we know the equation
of the line.

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

plus the y-intercept.

I should write that.

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

equation of our line.

Let's do one more of these.

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

negative 1 is 2.

What is that?

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

point when x is 1.5, when you
put 1.5 into the function, the

function evaluates
as negative 3.

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

on the line.

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

of x is equal to 2.

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

points are on the line,
nothing unusual.

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

with function notation, for you
to not get intimidated if

you see something like this.

If you evaluate the function
at 1.5, you get negative 3.

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

equal to f of x.

So this would be the
y-coordinate.

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

Anyway, I've said it
multiple times.

Let's figure out the
slope of this line.

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

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

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

1 minus this guy.

Let me write it this way,
negative 1 minus

that guy, minus 1.5.

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

1 and the 2 are both coming from
this, that's why I use

both of them first. If I used
these guys first, I would have

to use both the x and the y
first. If I use the 2 first, I

have to use the negative
1 first. That's why I'm

color-coding it.

So this is going to be equal
to 2 minus negative 3.

That's the same thing
as 2 plus 3.

So that is 5.

Negative 1 minus 1.5
is negative 2.5.

5 divided by 2.5
is equal to 2.

So the slope of this
line is negative 2.

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

matter what order
I do this in.

If I use this coordinate first,
then I have to use that

coordinate first. Let's
do it the other way.

If I did it as negative 3
minus 2 over 1.5 minus

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

negative 1.

This should give me
the same answer.

This is equal to what?

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

That's 1.5 plus 1.

That's over 2.5.

So once again, this is
equal the negative 2.

So I just wanted to show you,
it doesn't matter which one

you pick as the starting or
the endpoint, as long as

you're consistent.

If this is the starting y,
this is the starting x.

If this is the finishing
y, this has to be

the finishing x.

But anyway, we know that the
slope is negative 2.

So we know the equation is y is
equal to negative 2x plus

some y-intercept.

Let's use one of these
coordinates.

I'll use this one since it
doesn't have a decimal in it.

So we know that y
is equal to 2.

So y is equal to 2 when x
is equal to negative 1.

Of course you have
your plus b.

So 2 is equal to negative 2
times negative 1 is 2 plus b.

If you subtract 2 from both
sides of this equation, minus

2, minus 2, you're subtracting
it from both sides of this

equation, you're going to get
0 on the left-hand side is

equal to b.

So b is 0.

So the equation of our
line is just y is

equal to negative 2x.

Actually if you wanted to write
it in function notation,

it would be that f of x is
equal to negative 2x.

I kind of just assumed that
y is equal to f of x.

But this is really
the equation.

They never mentioned y's here.

So you could just write f of x
is equal to 2x right here.

Each of these coordinates
are the coordinates

of x and f of x.

So you could even view the
definition of slope as change

in f of x over change in x.

These are all equivalent ways
of viewing the same thing.