The line through points three, negative three and eight, one has equation 𝑦 equals 𝑎𝑥 minus four. What is 𝑎?
So the first thing we look at is the fact that we have a line, so a straight line. And it has the equation 𝑦 equals 𝑎𝑥 minus four. Well, this is useful because actually the equation 𝑦 equals 𝑎𝑥 minus four is in the form 𝑦 equals 𝑚𝑥 plus 𝑐. And this is actually the general form for the equation of a straight line, where 𝑚 is the slope and 𝑐 is the 𝑦-intercept.
So how are we gonna use this to actually find 𝑎? Well, we can use this to find 𝑎 because we now know that 𝑎 is actually going to be the slope of our line. And as we got two points, there’s an equation we can use to actually help us find the slope.
So we can say that the slope or 𝑚 is equal to 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 one. Well, this can also be known or thought of as the change in 𝑦 divided by the change in 𝑥, so how much our line goes up or down divided by how much our line goes across.
Okay, so great, we’ve got the slope and we want to find that. So let’s use our points to actually calculate what the slope is going to be. So the first thing I’ve done is actually labelled our coordinates just so we know what they’re gonna be when we actually substitute them in.
So we’ve got 𝑥 one, 𝑦 one and 𝑥 two, 𝑦 two. So we’re gonna get 𝑚, our slope, is equal to one minus negative three divided by eight minus three, which would give us four over five. And we get that because we had one minus negative three. And if you minus a negative, it becomes positive and then divide it by eight minus three which is five. We get four-fifths.
So therefore, we get the answer that 𝑎 is gonna be equal to four-fifths. And that because as we said, 𝑎 was the coefficient of 𝑥, which is our slope.