Find the gradient of the straight line which is parallel to the straight line passing through the points 𝐴, which is negative seven, eight, and 𝐵, one, one.
So when we look at this problem, the word that we need to consider really is “parallel” because this is the key to answering it, because if you have two lines that are actually parallel to one another, it means that actually they never are gonna get any closer or further away from each other.
So we’ve got two parallel lines down here just to show you that. What does it- what does that tell us? What we also know is that parallel lines have the same gradient. So what this means is that, actually, as we said before, if they have the same gradient, that means that they won’t get any closer together or further apart and also means that if we can find out the gradient of the straight line passing through the points 𝐴 and 𝐵, then therefore we’re gonna know the gradient of the straight line which is parallel to this line.
Well, to help us actually find this out, what we’ve actually got is a formula for the gradient of a line if we know two points. And that formula is 𝑦 two minus 𝑦 one divided by 𝑥 two minus 𝑥 one is equal to our gradient or 𝑚, cause that’s what we’ll use for our gradient.
What this actually means in practice is the change in 𝑦, so the difference between our 𝑦-coordinates, divided by the change in 𝑥, so the difference between our 𝑥-coordinates. So therefore, if we take a look at our two points, we’ve got 𝐴, which is negative seven, eight, and 𝐵, which is one, one.
And now what I’ve done to make this easier is actually labelled our coordinates. So we’ve got 𝑥 one, 𝑦 one and 𝑥 two, 𝑦 two. And now what we need to do is actually substitute it into our gradient formula. And when we do that, we get 𝑚, so our gradient, is equal to one minus eight, and that’s because our 𝑦 two is one and our 𝑦 one is eight, and then divided by one, cause that’s our 𝑥 two, minus negative seven; that’s because negative seven is our 𝑥 one.
Well then, as we know that if we subtract a negative this turns into a positive, we can now actually calculate the numerator and denominator. So therefore, we can say that our gradient or 𝑚 is gonna be equal to negative seven over eight. And this is because one minus eight is negative seven. And then one minus negative seven is like one add seven, which is eight.
So therefore, we can say that the gradient of the straight line which is parallel to the straight line passing through the points 𝐴, which is negative seven, eight, and 𝐵, which is one, one, is negative seven over eight, or negative seven-eighths. And this is because that was the gradient of the straight line which passed through the points 𝐴 and 𝐵 and we know that parallel lines have the same gradient.