Welcome back everyone to part two in our Part 2 Series on Lines in a Plane.

We're just going to jump right in and talk about the example we finished with last time.

So we have this line here,

L. What we know about L is that it goes through the point (2,1) and it has sloped M=1.

So therefore, the point slope formula of the line is y-1= 1(x-2).

Any other point (x,y) that wants to be on the line has to satisfy this equation.

Okay, the point of this video is to show you a simpler formula called,

"the slope intercept formula".

Though it's probably more widely used,

but the reason I showed you the point slope formula first,

is that you can derive slope intercept

formula from the points of the formula fairly easily.

So first, here is a really key concept.

This line L has infinitely many points: (2,1) is 1,

(3,2) is 1, (4,3) is another.

But one point is really,

really important for some reason, this one here.

This is what we call, the Y intercept.

The Y intercept is the unique point where this line meets the y-axis.

So think about what the coordinates of this point are.

The coordinates of this point,

the X coordinate I can tell you right away is zero because

the Y-axis consists of all points (x,y) with X coordinate zero.

I don't know what the Y coordinate is.

What we often do in math,

if we don't know something but we want to compute with it

and do algebra with it, just give it a symbol.

So it's called the (0,b),

b is often what we use for Y intercept.

Okay, now let's be a little bit of a detective here and let's hunt for what b must be.

It turns out that if I know that the point (0,b) is on the line L,

and I know the equation for the line,

I can find out what b is, right?

Because here is (0,b) and (0,b) is on the line.

So over here is a little detective work.

I know that (0,b) is on the line and so,

if I plug in zero for X and b for Y, I have to make truth.

So b-1=1(0-2). Now it's just algebra,

arithmetic really, b-1=-2 or b=-1.

That passes the smell test, right?

That looks like that it could easily be drawn to scale but that

could easily be minus one, at least it's not positive or anything.

So in fact, let's erase this.

That has coordinates (0-1).

Great!

So what?

Well, let me now rewrite

the slope intercept form with the point for a formula line but using (0-1).

So if I write, and I know that (0-1) is on the line,

and I know the slope M=1,

the point slope formula tells me that y-(-1)=1(x-0)

is another perfectly good equation for that line.

This is y+1=1x or y=1x-1.

This is the slope intercept formula for a line.

Let's say that in general forms.

If a line L has slope

M and L hits the y-axis at the point (0,b),

then Y=Mx+b is an equation for the line.

M is the slope.

B is the Y intercept or the Y value of the line intercept.

That's often a nice,

pleasing way of describing Y because it

can pretty much allow you to draw a line just by seeing it.

So for example if I give you the equation Y=2x+1, let me draw that here.

I can immediately draw for you that line or at least sketch it.

I know the Y intercept is one, right about there.

I know the slope is two which means steeper than 45-degree angle,

so about like that.

Cool, now suppose someone tells me that

they want me to draw a line with the same Y intercept,

let's draw it in blue, the same Y intercept but sloped still positive but less positive.

Then that could be, say for example, this one.

Suppose someone tells me that in red,

they want me to draw a line with the same slope as L,

the slope of two, but negative Y intercept, let's say down here.

Now I'm gonna try my best to make it parallel to Y,

the line, there you go.

That's the point. The slope tells you in this formula Y=Mx+b,

the slope tells you how to angle the line,

the Y intercept tells you where to anchor it on the Y-axis.

So that's somehow much more pleasing than the point slope formula.

Okay, let's finish with this one example.

Let's give you the following problem:

Line_L has points (1,1)

and (3,0) on it.

Find an equation for L.

Okay, fine. First step is let's draw point to C. So here is (1,1),

here is about (3,0),

and there is the line between them.

All right, now let's find the equation for the line.

We can do it point slope formula,

we can do point intercept, whatever we want.

First let's figure out the slope.

So the slope of L is, of course,

the slope of this line segment between (1,1) and (3,0).

So M is equal to zero minus one divided by three minus one is negative 1/2.

So this line has slope -1/2.

And now to the point slope formula for the line.

So let's take (1,1) as our point.

So y-1=-1/2(x-1).

That's one equation for this line.

In a sense we're done. We would then try to find the Y intercept,

we could do any number or other things.

Here's a fun idea by the way,

this probably occurred to a lot of you, more skeptical-minded of you.

I used the point (1,1) but no one told me I had to.

I could've also use the point (3,0) and drawn the point slope formula that way.

So another equation for the same line is y-0=-1/2(x-3).

This seems like a contradiction, all right,

cause these are two very different looking equations for the same line.

Turns out they're the same.

You can do a little bit of algebraic manipulation to

turn this top equation into this bottom equation.

I'm going to challenge you in your own time to do so and to check that's true.

Okay. That concludes our video.