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.