Video: Pack 1 β€’ Paper 3 β€’ Question 10

Pack 1 β€’ Paper 3 β€’ Question 10


Video Transcript

Find the equation of the line 𝐿. Give your answer in the form 𝑦 equals π‘šπ‘₯ plus 𝑐.

So 𝑦 equals π‘šπ‘₯ plus 𝑐 is just a general form of the equation for a straight line. The letter π‘š represents a number, which is the gradient of the line. And the letter 𝑐 represents a number, which is the 𝑦-intercept of the line. So let’s just recall what those things mean.

The gradient, well one definition of the gradient is by how much does the 𝑦-coordinate change when I increase the π‘₯-coordinate by one. After this, we could just pick a point on the line, increase the π‘₯-coordinate by one, and then find our way back to the line.

And in finding our way back to the line, what happens to the corresponding 𝑦-coordinate? Well, in this case, it goes down from nine to seven. And that’s a decrease of two or negative two. So in this case, the gradient is negative two.

So what is the 𝑦-intercept? Well, we can interpret that as meaning what is the 𝑦-coordinate when the line cuts the 𝑦-axis. Now in this case, here’s the point where the line cuts the 𝑦-axis. And that’s the point zero, five. So the π‘₯-coordinate is zero. But the 𝑦-coordinate is five. So in our case, 𝑐 is equal to five.

Now we know the value of π‘š and the value of 𝑐, the value of the gradient, and the value of the 𝑦-intercept. We can write out our equation. So the answer is 𝑦 equals negative two π‘₯ plus five.

Now in this question, we’ve been given a nice graph that had a nice integer gradient of negative two. Now it’s not quite as simple as that in all cases. So just before we go, I’m gonna go through another method of working out the gradient. So an alternative definition for gradient is that the gradient is the difference in 𝑦-coordinates divided by the difference in π‘₯-coordinates between two points. And to use this method as accurately as possible, what you need to do is pick two points on the line, preferably as far away as possible, but both with preferably integer π‘₯- and 𝑦-coordinates. So that’s π‘₯- and 𝑦-coordinates which are whole numbers.

Now let’s think about it. In getting from the first point to the second point, my π‘₯-coordinate has increased by five and my 𝑦-coordinate has decreased by 10. So putting those numbers into our little formula, the difference in 𝑦-coordinates is negative 10. And the difference in π‘₯-coordinates is positive five. And negative 10 divided by positive five is negative two. So I’ve come up with the same gradient, negative two.

Now that sounds all well and good. But it seems a lot more complicated than the first method we used. So why would we ever choose to use it? Well, sometimes the gradients we get are not nice integers. For example, with this green line, if I move from my first point to my second point, again the π‘₯-coordinate increases by five. But this time the 𝑦-coordinate decreases by eight. So my gradient will be negative eight divided by positive five.

Now that wouldn’t have been so easy to calculate accurately using the original first method that I gave you. If I increase my π‘₯-coordinate by just one, in this case, the 𝑦-coordinate decreases by one and a bit. Is that 1.5? Is it 1.6? Is it 1.7? It’s very difficult to tell. But if I’ve picked nice integer values for my π‘₯- and 𝑦-coordinates here, it makes it much easier to calculate an accurate gradient. So going back to our original question then, the answer is 𝑦 equals negative two π‘₯ plus five.

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