-
-
So what I've drawn here
in yellow is a line.
-
And let's say we know two
things about this line.
-
We know that it
has a slope of m,
-
and we know that the point
a, b is on this line.
-
And so the question that we're
going to try to answer is,
-
can we easily come
up with an equation
-
for this line using
this information?
-
Well, let's try it out.
-
So any point on this line,
or any x, y on this line,
-
would have to
satisfy the condition
-
that the slope
between that point--
-
so let's say that this
is some point x, y.
-
It's an arbitrary
point on the line--
-
the fact that it's
on the line tells us
-
that the slope between a, b
and x, y must be equal to m.
-
So let's use that knowledge to
actually construct an equation.
-
So what is the slope
between a, b and x, y?
-
Well, our change in y--
remember slope is just
-
change in y over change in x.
-
Let me write that.
-
Slope is equal to change
in y over change in x.
-
This little triangle character,
that's the Greek letter Delta,
-
shorthand for change in.
-
Our change in y--
well let's see.
-
If we start at y is equal
to b, and if we end up
-
at y equals this arbitrary
y right over here,
-
this change in y right over
here is going to be y minus b.
-
-
Let me write it in
those same colors.
-
So this is going to be y
minus my little orange b.
-
And that's going to be
over our change in x.
-
And the exact same logic--
we start at x equals a.
-
We finish at x
equals this arbitrary
-
x, whatever x we
happen to be at.
-
So that change in x is
going to be that ending
-
point minus our starting
point-- minus a.
-
And we know this is the slope
between these two points.
-
That's the slope between
any two points on this line.
-
And that's going
to be equal to m.
-
So this is going
to be equal to m.
-
And so what we've
already done here
-
is actually create an equation
that describes this line.
-
It might not be in any form
that you're used to seeing,
-
but this is an
equation that describes
-
any x, y that satisfies this
equation right over here
-
will be on the line because
any x, y that satisfies this,
-
the slope between that x, y
and this point right over here,
-
between the point a, b,
is going to be equal to m.
-
So let's actually now
convert this into forms
-
that we might
recognize more easily.
-
So let me paste that.
-
So to simplify this expression
a little bit, or at least
-
to get rid of the x minus
a in the denominator, let's
-
multiply both
sides by x minus a.
-
So if we multiply both sides
by x minus a-- so x minus a
-
on the left-hand side and
x minus a on the right.
-
-
Let me put some
parentheses around it.
-
So we're going to multiply
both sides by x minus a.
-
The whole point of that is you
have x minus a divided by x
-
minus a, which is just
going to be equal to 1.
-
And then on the
right-hand side, you just
-
have m times x minus a.
-
So this whole thing
has simplified
-
to y minus b is equal
to m times x minus a.
-
And right here, this is
a form that people, that
-
mathematicians, have
categorized as point-slope form.
-
So this right over here
is the point-slope form
-
of the equation that
describes this line.
-
Now, why is it called
point-slope form?
-
Well, it's very easy to
inspect this and say, OK.
-
Well look, this is the
slope of the line in green.
-
That's the slope of the line.
-
And I can put the two points in.
-
If the point a, b
is on this line,
-
I'll have the slope times x
minus a is equal to y minus b.
-
Now, let's see
why this is useful
-
or why people like to
use this type of thing.
-
Let's not use just a, b
and a slope of m anymore.
-
Let's make this a little
bit more concrete.
-
Let's say that someone tells you
that I'm dealing with some line
-
where the slope is equal
to 2, and let's say
-
it goes through the
point negative 7, 5.
-
-
So very quickly, you
could use this information
-
and your knowledge
of point-slope form
-
to write this in this form.
-
You would just say,
well, an equation that
-
contains this point and has this
slope would be y minus b, which
-
is 5-- y minus the
y-coordinate of the point
-
that this line contains--
is equal to my slope times x
-
minus the x-coordinate
that this line contains.
-
So x minus negative 7.
-
And just like that, we have
written an equation that
-
has a slope of 2
and that contains
-
this point right over here.
-
And if we don't like the x minus
negative 7 right over here,
-
we could obviously
rewrite that as x plus 7.
-
But this is kind of the
purest point-slope form.
-
If you want to simplify
it a little bit,
-
you could write it as y minus
5 is equal to 2 times x plus 7.
-
And if you want to see that this
is just one way of expressing
-
the equation of this line--
there are many others,
-
and the one that we're
most familiar with
-
is y-intercept form--
this can easily
-
be converted to
y-intercept form.
-
To do that, we just have
to distribute this 2.
-
So we get y minus 5 is equal
to 2 times x plus 2 times 7, so
-
that's equal to 14.
-
And then we can get rid of
this negative 5 on the left
-
by adding 5 to both
sides of this equation.
-
And then we are left with,
on the left-hand side, y
-
and, on the right-hand
side, 2x plus 19.
-
So this right over here
is slope-intercept form.
-
You have your slope
and your y-intercept.
-
So this is slope-intercept form.
-
And this right up here
is point-slope form.