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# Slope-intercept form from a table

Sal writes an equation of the line that matches up to a table of values. Created by Sal Khan.

Video transcript

A line goes through
the following points, and the equation of that line
is written in y equals mx plus b form. Also known as
slope-intercept form. What is the equation
of the line? So the first thing we
want to think about, what is the slope of this line? What is m here? So what is our change in
y for given change in x? So this is an
interesting example here. And I encourage you to pause the
video and try it out yourself. Because no matter how much we
change x, y is not changing. y is a constant, 2. So your change in y between any
two points is going to be 0. It doesn't matter
what your change in x is, your change in x could be
1, your change in x could be 4, your change in y is always 0. So y is not changing
as you change x. So your slope for this
relationship is actually 0. Y is equal to 0x
plus-- and then, you could just realize that
the equation of this is just that y is
always equal to 2. So it's 0x plus 2, which is the
same thing as y is equal to 2. You could substitute back in. You could say OK, well, if
y is equal to 0x plus b, that means that y is equal to b. Well, y is always equal to 2,
no matter what thing you pick, so b is equal to 2. So either way, this
just boils down to y is equal to 0x plus 2,
or y is just equal to 2. Let's do another one of these. Maybe one where the y
is actually changing. So here, the y is clearly
actually changing. So let me copy and paste this. I want to put on my scratch pad. We can work it out. So we'll stick it
right over here. And then we are told a line
goes through the-- OK, so same thing. The line goes
through these points with the equation of a line. So the main idea
here is, you only need 2 points for
an equation of line. They've given us
more than necessary. So I'd like to pick
the two points that make things a
little bit simpler. So I'll pick the
point 4, 2 and 7, 0. I just picked those
two points because they have nice, clean numbers
associated with it. So what is our change in x here? So our change in x here, if we
go from 4 to 7, our change in x is equal to 3. And what's our change in y here? So we went up from 4 to 7. We increased by 3. Our y decreased by 2. Change in y is
equal to negative 2. So our slope, which is equal to
change in y over change in x, is equal to negative 2/3. And if you wanted to
relate that to the formulas that you normally
see for slope, you're just looking at your end point. So this is y2 minus
y1, which is negative 2 over x2 minus x1,
which is 7 minus 4. But that just boils
down to negative 2/3. And so our equation is going
to be y is equal to negative 2/3 x plus b. So let's substitute one
of these points in here, to figure out what
our b must be. And once again, I
want to figure out something where this is going
to become nice and clean. But this isn't going to
be really clean for any of these numbers
right over here. If we had a 3 for x, or a
6 for x, or a 0 for x, then things would work out nicely. But they don't give
us any of those. So let's just try
the 7 and the 0. So when x is equal to 0--
sorry, when x is equal to 7-- y is equal to 0. So when x is equal to 7, I'll
just do it in the same color, y is equal to 0. So 0 is equal to negative
2/3 times 7 plus b, or 0 is equal to negative
14/3 plus b. Add 14/3 to both sides,
you get 14/3 is equal to b. So this is going to be
y is equal to negative-- I'm going to go back
to the other screen-- so y is equal to
negative 2/3 x plus 14/3. So let me do that. So y is equal to
negative 2/3 x plus 14/3. Let's check our answer. We got it right.