Linear Equations in Standard Form

Linear Equations in Standard Form

Let's do some examples dealing
with equations of lines in

standard form.

So, so far we've had two
other forms. We've had

slope-intercept, which
is of the form, y is

equal to mx plus b.

That's actually this
right here.

This is in slope-intercept
form.

We've seen point-slope form
in the last video.

That's of the form, y minus some
y-value on the line being

equal to the slope times x
minus some x-value on the

line, when you have
that y-value.

So the point x1, y1
is on the line.

This right here is an example
of point-slope form.

And now we're going to talk
about the standard form.

And the standard form-- let me
write it here-- standard form

is essentially putting all of
the x and y terms onto the

left-hand side of
the equation.

So you get ax plus
by is equal to c.

I want to really emphasize that
all of these are just

different ways of writing
the same equation.

If you're given this, you can
out algebraically manipulate

it to get to that or to that.

If you're given that, you
can get to that or that.

These are all different ways
of writing the exact same

relationship, the
exact same line.

So let's do a couple of
examples of this.

So here we have a
line right here.

We have an equation written
in slope-intercept form.

The slope is 3, the y-intercept
is negative 8.

Let's put it into
standard form.

So we just have to get
the 3x onto the

other side of the equation.

And the best way I can think of
doing that-- let me rewrite

the equation, y is equal to 3x
minus 8-- let's some subtract

3x from both sides
of the equation.

So if you subtract 3x from both
sides-- so you subtract

3x, subtract 3x-- what do the
left- and right-hand sides of

the equation become?

The left-hand side becomes
negative 3x plus y being equal

to-- the 3x and the negative 3x
cancel out-- being equal to

negative 8.

We're done.

That's standard form
right there.

Standard form, I guess people
like it because it has both

the coefficients on the
left-hand side.

But it's kind of useless in
trying to figure out slope and

y-intercept.

I don't know what the slope and
y-intercept is when I look

at it in standard form.

My favorite is slope-intercept
form, because it tells you

exactly the slope and
an intercept.

Point-slope, easy to get to,
and you can look at it and

figure out the slope.

But y-intercept, you have to
do a little bit of work to

figure it out.

But at least you can just go
immediately from the slope and

a point to it.

But anyway, let's go from this
equation, which is written in

point-slope form, and get
it to the standard form.

So we want to get it to the
standard form, to the same

type of standard form.

So a good thing to do, let's
just distribute things out. y

minus 7 is equal to negative 5
times x, negative 5x, plus

negative 5, times negative
12, which is positive 60.

Now, we want all of the variable
terms on the left,

all of the constant terms
on the right.

So let's add 7 to both sides
of this equation.

So plus 7 to both sides
of this equation.

What does it become?

Well, the minus 7 disappears,
because negative 7 plus 7.

So you're just left with
a y being equal to

negative 5x plus 67.

Now, if we want this x term on
the left-hand side, we could

add 5x to both sides.

So let's add 5x to both sides
of this equation.

And we will get y plus 5x
is equal to-- these

cancel out-- 67.

Now, this is pretty much
standard form.

If you really want to be a
stickler for it, you can

rearrange these two.

So it'd be 5x plus
y is equal to 67.

And you are done.

Let's do one more of these.

So this is in neither
point-slope nor in

slope-intercept form.

It's just in some type
of intermediary

mixed form right there.

This looks like some type of
point-slope, but this looks

like something different.

So it's really not
point-slope.

Let's see if we can
algebraically manipulate it to

the standard form.

So we get 3y plus 5.

Let's distribute out this 4.

So it's equal to 4x minus 36.

Let's do exactly what
we did in the last.

I'm using different notation on
purpose, to expose you to

different things.

So instead of doing it this way,
I'm going to subtract 5

from both sides, but I'm going
to do it on the same line.

So I'm going to subtract
5 from both sides.

And so the left-hand side of
this equation becomes 3y,

because these two guys cancel
out, and that is equal to 4x.

And then what is minus
36 minus 5?

That's minus 41.

And now we want the x terms
of the left-hand side.

So let's subtract 4x from both
sides of this equation.

So negative 4x plus,
and then minus 4x.

What does our equation become?

Well, the left-hand side just
stays negative 4x plus 3y.

And the right-hand, the reason
why we subtracted 4x is so it

cancels out with that.

You just have a negative 41.

And we're done.

We are in standard form.

Now, let's go the other way.

Let's start with some equations
in standard form and

figure out their slope
and y-intercept.

And the best way I know to
figure out the slope and

y-intercept is to put it into
slope-intercept form.

So we want to put these
equations right here into the

form, y is equal to mx plus b.

So we're essentially
solving for y.

Let's do that.

So the best thing to do here--
so let me rewrite it.

5x minus 2y is equal to 15.

Let's subtract 5x
from both sides.

So minus 5x plus, you
have a minus 5x.

These cancel out.

And so you're left with
negative 2y is

equal to 15 minus 5x.

And now, let's divide everything
by negative 2.

If you divide everything by
negative 2, what do we get?

The left-hand side
just becomes a y.

y is equal to-- 15 divided by
negative 2 is negative 7.5.

And then negative 5 divided by
negative 2-- you can imagine

I'm distributing the negative
1/2 if you will.

I'm dividing both of these
by negative 2.

So negative 5 divided by
negative 2 is positive 2.5x.

And if you really wanted to put
it in the slope-intercept

form, you could say that y is
equal to-- you could just

rearrange these--
2.5x minus 7.5.

You want the slope.

It's right here.

That is our slope.

You want the y-intercept.

Actually, let me be careful.

It is right there.

It is negative 7.5.

That is the y-intercept.

And now this would be a form
that's actually pretty

straightforward to
graph it in.

Let's do this one.

So once again, we just
need to solve for y.

So let's subtract 3x
from both sides.

So you get 6y is equal
to 25 minus 3x.

And then you can divide
both sides by 6.

So you're left with y is equal
to 25 over 6 minus 3 over 6,

or minus 1/2x.

If you really want it in this
from, you just rearrange this.

y is equal to negative
1/2x plus 25 over 6.

Where is the slope?

Here is the slope.

Negative 1/2, that
is the slope.

Where is the y-intercept?

That's the y-intercept.

That is our b.

The point 0, 25 over
6 is on the line.

Let's do one more of these.

So we get 9x minus
9y is equal to 4.

Just for fun, let's just start
off by dividing both sides of

the equation by 9.

You don't have to
do it that way.

But this is kind of a fun way
to do it, because the

coefficients here will
immediately become 1.

So if you divide both sides of
the equation by 9, if you

divide everything by 9, it
becomes-- actually, well,

let's divide everything.

Let's divide everything by
negative 9, even better.

I'm just doing this for fun.

So this first term will
become negative x.

The second term, you have a
negative 9 divided by a

negative 9, it will
be a plus y.

And then this last term
will just become a

negative 4 over 9.

Actually, let me write
this out here.

Negative 4 over 9.

I'm giving some space there.

Now, we want the x on the
right-hand side, so let's add

x to both sides of
this equation.

These cancel out.

And then the equation becomes
y is equal to x minus 4/9.

Where is the slope?

The slope is the coefficient
on the x term.

The slope is equal to 1.

Where is the y-intercept?

The y-intercept is
right there.

It is negative 4/9.