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CCSS.Math: , , ,

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.