Video Transcript
Find the Slope of a Line from Coordinates
The general form of a straight line is 𝑦 equals 𝑚𝑥 plus 𝑐, where 𝑚
is the slope of the line, or I like to call it sometimes the gradient. Now, to find the gradient or the
slope, there’s a nice formula.
The informal way we can say it is,
the gradient or the slope is equal to the change in 𝑦 divided by the change in 𝑥
between the two points, or another nice way to think of it, rise over run, so it
goes up what it goes up on the numerator and what it goes across on the denominator,
exactly the same as saying change in 𝑦 over change in 𝑥. We just have to be careful with our
negatives when we use something like this. Now, the proper formula is 𝑦 two
minus 𝑦 one all divided by 𝑥 two minus 𝑥 one, and this looks a lot more confusing
that it is. It’s just essentially saying the 𝑥
two and the 𝑦 two are saying our second set of coordinates and then 𝑦 one and 𝑥
one are referring to our first set of coordinates. So, let’s use this formula to help
us find the slope between two points.
Find the slope between three six
and five eight. So, first, we remember the formula
where change in 𝑦, 𝑦 two minus 𝑦 one, is all divided by change in 𝑥, which is 𝑥
two minus 𝑥 one. So, looking at each set of
coordinates, we’ve got the first set of coordinates; we can say that is an 𝑥 one as
three and then 𝑦 one is six.
And in our second set of
coordinates, we’ve got 𝑥 two is five and 𝑦 two is eight. So then just substituting those in,
we see that 𝑦 two is eight. So, on the numerator, we’ll have
eight minus what 𝑦 one is, and that’s six, all divided by 𝑥 two, which is five,
minus what 𝑥 one is, and that’s three.
So then, on the numerator, we’ve
got eight minus six, which gives us two, and then we’ll divide that by five minus
three, which is two as well. Two divided by two we of course
know is one.
Now, if we’d forgotten the proper
formula for it, we would think back to rise over run or change in 𝑦 over change in
𝑥, so what we could do is just look at the change in the 𝑦-coordinates. We would say, to get from six to
eight, I would have to add two, so that gives us two as the numerator and then the
change in 𝑥 to get from three to five. Again, I’d have to add two, so then
that would give me two as the denominator, and then I’d have one.
Now, if we chose to use this method
that we’ve just done here, with change in 𝑦 over change in 𝑥, we have to be
careful to make sure that we go from one set of coordinates to the other set of
coordinates. Don’t, for example, go from six to
eight but then five to three. You have to make sure you go from
one whole set to the other whole set. There are positives with using the
formula of 𝑦 two minus 𝑦 one all divided by 𝑥 two minus 𝑥 one as it helps us
remember another formula that we’ll learn later down the line.
Find the slope between six,
negative three and two, negative two. So, we’re gonna do exactly the same
thing as we did last time. We’ll see that our 𝑥 one and 𝑦
one are six and negative three, respectively, and 𝑥 two and 𝑦 two are two and
negative two, respectively.
It doesn’t really matter which set
of coordinates you choose to be the first and which set you choose to be the second
as long as you make sure it’s 𝑥 one 𝑦 one, 𝑥 two 𝑦 two as a set, which either
way it is. So now the formula.
The slope is 𝑦 two minus 𝑦 one
all divided by 𝑥 two minus 𝑥 one, where 𝑦 two is equal to negative two, and we’re
subtracting from that negative three. And that’s all divided by 𝑥 two,
which is two, minus 𝑥 one, which is six, so two minus six.
Then, on the numerator, we’ve got
negative two minus minus three, so that’s the same as saying negative two plus
three. Negative two plus three is one, and
that’s divided by two minus six, which gives us negative four.
Well, obviously, we don’t write
fractions like that, so we’ll take the negative out in front. And we have that the slope between
these two points is negative one over four or a quarter.
Again, if we wanted to just look at
change in 𝑦 and change in 𝑥, looking at the 𝑦’s, we seem to get from negative
three to negative two we add one, and to get from six to two we subtract four, so
that gives us the same as one over minus four, but we don’t write that; we’ll take
the minus out in front, giving us negative a quarter.
So, we have it, find the slope
between two points we just simply must use the formula or use the fact that the — to
find the slope it’s change in 𝑦 divided by change in 𝑥.