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CCSS Math: 8.F.B.4, HSF.IF.C.7, HSF.IF.C.7a, HSF.LE.A.2

Find the slope of the line
that goes through the ordered pairs 7, negative 1 and
negative 3, negative 1. Let me just do a
quick graph of these just so we can visualize
what they look like. So let me draw a
quick graph over here. So our first point
is 7, negative 1. So 1, 2, 3, 4, 5, 6, 7. This is the x-axis. 7, negative 1. So it's 7, negative 1
is right over there. 7, negative 1. This, of course, is the y-axis. And then the next point
is negative 3, negative 1. So we go back 3 in the
horizontal direction. Negative 3 for the y-coordinate
is still negative 1. So the line that connects these
two points will look like this. It will look like that. Now, they're asking us to find
the slope of the line that goes through the ordered pairs. Find the slope of this line. And just to give a little
bit of intuition here, slope is a measure of
a line's inclination. And the way that
it's defined-- slope is defined as rise over run, or
change in y over change in x, or sometimes you'll see it
defined as the variable m. And then they'll
define change in y as just being the
second y-coordinate minus the first
y-coordinate and then the change in x as the
second x-coordinate minus the first x-coordinate. These are all different
variations in slope, but hopefully you'll
appreciate that these are measuring inclination. If I rise a ton when
I run a little bit, if I move a little bit in the x
direction, and I rise a bunch, then I have a very steep line. I have a very steep
upward-sloping line. If I don't change at
all when I run a bit, then I have a very low slope. And that's actually
what's happening here. I'm going from-- you
could either view this as the starting
point or view this as the starting point. But let's view this
as the starting point. So this negative 3, 1. If I go from negative 3,
negative 1 to 7, negative 1, I'm running a good bit. I'm going from negative 3. My x value is negative 3 here,
and it goes all the way to 7. So my change in x here is 10. To go from negative 3 to 7,
I changed my x value by 10. But what's my change in y? Well, my y value
here is negative 1, and my y value over here
is still negative 1. So my change in y is a 0. My change in y is going to be 0. My y value does not
change no matter how much I change my x value. So the slope here
is going to be-- when we run 10,
what was our rise? How much did we change in y? Well, we didn't rise at all. We didn't go up or down. So the slope here is 0. Or another way to think about
is this line has no inclination. It's a completely flat-- it's
a completely horizontal line. So this should make sense. This is a 0. The slope here is 0. And just to make
sure that this gels with all of these other
formulas that you might know-- but I want to make
it very clear. These are all just
telling you rise over run or change in y over change in
x, a way to measure inclination. But let's just
apply them just so, hopefully, it all
makes sense to you. So we could also say slope is
change in y over change in x. If we take this to be
our start and if we take this to be our
end point, then we would call this over here x1. And then this is over here. This is y1. And then we would
call this x2 and we would call this y2, if
this is our start point and that is our end point. And so the slope here, the
change in y, y2 minus y1. So it's negative 1
minus negative 1, all of that over x2, negative
3, minus x1, minus 7. So the numerator, negative
1 minus negative 1, that's the same thing
as negative 1 plus 1. And our denominator
is negative 3 minus 7, which is negative 10. So once again, negative 1
plus 1 is 0 over negative 10. And this is still going to be 0. And the only reason why
we got a negative 10 here and a positive 10
there is because we swapped the starting
and the ending point. In this example right over here,
we took this as the start point and made this coordinate
over here as the end point. Over here, we
swapped them around. 7, negative 1 was
our start point, and negative 3, negative
1 is our end point. So if we start over
here, our change in x is going to be negative 10. But our change in y is
still going to be 0. So regardless of how you do it,
the slope of this line is 0. It's a horizontal line.