Hi everyone, it's Jenni with The How and Why of Mathematics, and today I want to talk
about how to find the slope or gradient of a line if you just know the
coordinates of two points on the line. And I don't want to just give you the
formula because that would be pretty boring and you'd forget it. Instead I
want to help you understand slope so well that you can work out the formula for
yourself. So slope or gradient as it's sometimes
called is just a measure of how steep a line is. So this line's quite steep; this
line's not as steep, so this one would have a higher slope than that line. And
slope is just how many units up or down a line goes when it goes across one
unit. So if I go across one unit and go up three units then that line has a
slope of 3. And slope is usually given the symbol m, so we would say m=3
for that line. Now if I go across two units and up six units,
well that triangle is just a scaled up version of this triangle, so this one
actually has the same slope - this one would also have m=3, and you
can work that out from dividing the rise which is how much it goes up, so 6
in this case, divide that by the run which is how much it goes across, 2, so
you can work that out by 6 divided by 2 and that makes 3. Now you can also have a
negative slope: if I go across one and I go down 2 then we consider the rise to
be negative 2 and so that line would have a slope of -2. Okay so let's
say you want to work out the slope of the line that passes through the points
(4, -1) and (2, 5). So I'm just going to pop these on some coordinate axes for
you so you can see what's going on. So (4,-1) is
around there, and (2,5) is approximately there. So we're trying to work out the slope of
that line. And if you remember from before, I said that slope was rise divided by
run so we need to work out how much across and down it is between those two
points so that's the run and that's the rise. If we can work those out then we
can work out the slope. So you might like to pause this video and see if you can
figure it out. Okay, let's start with the run. If the x coordinate here is 4 and
this one's at 2 then the distance from there to there is going to be 2
because four minus two equals two. And then let's work out that - the rise - as
well. So the y coordinate here was negative 1 and the y coordinate here was
5 so there's a distance of one going from there to there and then another 5, so
that distance is 6, but if it's going down we want to consider the rise to be
negative so that's negative 6, and you can also work that out from doing
negative 1 minus 5, would give you negative 6, so that y coordinate minus
that y coordinate. Okay so we've got the rise and the run and then we just need to
say m is minus 6 over 2 and that makes minus 3, if you simplify that. Now
let's just kind of deconstruct this process so that we can use it for other
coordinates because we're not always going to have a diagram to work from. You
might want to like just ignore all of that diagram and be able to work out the
slope just from the coordinates. So what did we actually do to figure out these
things? For the rise we did one of the y coordinates minus the other one, and for
the run we did one of the x-coordinates minus the
other one. So you often see slope written as y2 minus y1 over x2 minus x1 so
that's sort of the official formula for slope, but the way that I like to think
about it is that it's just the change in y - that's the Delta symbol, it's
often used for change - so it's the change in y divided by the change in x,
and that's really just another way of saying rise over run. Ok, let's have a go at
applying that formula to work out the slope between the points (3, -4)
and (2, -1). Okay so we need to do rise over run or Delta y over Delta x or
y2 minus y1 over x2 minus x1 - however you want to remember it. The
first thing we need to do is either -4 minus the -1 or the
other way around: you can do the -1 minus -4. I'll show you both ways.
Let's just start with -4... minus -1 is the same as plus 1 and
then on the bottom if I started with the -4 on the top then I have to
start with the 3 on the bottom, so 3 minus 2 and then that makes -3
over 1 which simplifies to negative 3. Now you could also have started with the
negative 1... you can do negative 1 minus the negative 4 is plus 4 and then on the
bottom because we started with that one we have to do the 2 first and that will
make 3 over negative 1 which also simplifies to make -3. Either one of
those ways is fine, but a really common mistake is if you mix up the order and
let's say I do the minus 4 on top first and then accidentally if I do the 2
first on the bottom that's not gonna work because that's gonna give us
-3 over -1 which is positive 3 so that's wrong; you have to
be consistent about which point is point 1 and which one is point 2, so if the minus
4 is first on the top then the 3 has to be first on the bottom because this is
your y2 and your x2 and they come first. Okay, I'll do another example:
let's find the slope of the line passing through the points (-1, 7) and (2, 0).
So you might like to pause this video and give it a go and then come back and
check your answer. So the first thing we need to do is work out the rise, so
that's either the 7 minus 0 or 0 minus 7, either way. I'll do 7-0. And then for
the run, on the bottom, if I did 7 first then I gotta do the negative 1 first
so -1 minus the 2, and that makes 7 over negative 3.
Now you don't need to pop that in a calculator or anything; you can just
leave it as a fraction, but what we do need to do is bring the negative out the
front and check that there isn't any simplification that we need to do, like
if it was 9 over 3 we'd have to simplify, but there's no common
factor so we can just leave it like that. Or alternatively you could have said
0 minus 7 over 2 minus the -1 is the same as +1, but you can
write it as like 2 minus and then negative 1 in brackets, if you
prefer; I like to sort of simplify that as I'm going. And then that gives us
negative 7 over 3 and then again we just need to pull the negative out the front
and that's it. All right I'll just do one more example, and I thought I'd sort of
throw in some algebra with this one as well. So let's say we want to work out
the slope through those two points. So we go... one of the y's minus the other one. Let's
do 3y minus 0 and then on the bottom, so
if I pick 3y first then I have to do this one first on the bottom so it's
gonna be 2x minus the negative x which is the same as plus x and
that makes 3y over 3x that simplifies to that, and then you
can cancel the threes, so we get y/x, that would be the slope. You could also do it
the other way around: you can do that one minus that one and then on the bottom it
would be the -x minus the 2x so that would give you negative 3y
over -3x oops x. And then you can cancel the negative threes, so that makes
y over x, so either one of those y's - those *ways* is fine. Now if you
don't practice this process then you're likely to forget it, so I have a link down
below for a website where you can go and get practice questions for slope. So
it'll ask the question, you figure out the answer, so you can do it in your head
or you can work it out on paper if you like, and then you click on "Show answer"
and it will tell you, so you compare what you thought it was to that answer,
and then you rate how easy that was, so if you got it wrong or if you completely
forgot how to do it, you click on "Again", if it was like way too easy then you
click on "Easy" and if it was just all right then you click on "Good" and the
idea with this is that it will use your ratings to work out when you need to
revise this again. So the idea with spaced repetition which is the most
efficient way to memorise things is that, you know, maybe you do a couple of practice
problems one day then you do a couple the next day and then a couple more a
couple days later and then maybe some more a week later, and that's much more
efficient than doing just one huge block of practice. All right that's it for
slope. Let me know in the comments if there was anything confusing in this
video and stay tuned for the next one because I'm gonna be talking about how
to find the y intercept of a straight line. And if you like this video then
don't forget to subscribe because there's lots more coming up
about the How and Why of Mathematics.