### Video Transcript

Circle the equation showing that π¦ is directly proportional to π₯, with π being a constant. π¦ equals ππ₯, π¦ equals π minus π₯, π¦ equals π over π₯, or π¦ equals π₯ plus π.

Well, the first thing we need to do is decide what does directly proportional mean in maths. Well, directly proportional means that if we have one amount increasing, another amount increases at exactly the same rate. And an example of this on a graph is shown here, cause we can see that as π₯ increases, π¦ increases at the same rate.

So what symbols do we use for proportionality? But weβve got a symbol here which means directly proportional. So, we can say that π¦ is directly proportional to π₯. However, if weβre looking to deal with this proportionality as an equation, the proportionality sign isnβt very useful to us. So, what we do is we put an equals in so we have an equation. And we also introduce the proportionality constant, and that proportionality constant is π. And as in this question, weβre told that we have π as a constant. So therefore, π¦ is proportional to π₯ becomes π¦ equals ππ₯. Because π, like we said, is our proportionality constant, and then π₯ is obviously our π₯-value. So therefore, we can see that the correct answer is π¦ equals ππ₯, so we can circle that. And weβve done that because thatβs our first answer.

Now, letβs have a look at a couple of the other answers to see why they would not be correct. Well, letβs consider π¦ equals π minus π₯. With π¦ equals π minus π₯, as one amount increases, the other amount would not increase at the same rate. Because if we consider that we doubled π¦, well if π¦ is π minus π₯ and we double it, weβve got two multiplied by π minus π₯. But this would not increase at the same rate. Because two π minus two π₯, which would be if we doubled π minus π₯, is not the same as π minus two π₯ which should be the expression if we just doubled the π₯. So therefore, they would not be directly proportional.

Well similarly, this would be the same with π¦ equals π₯ plus π. Because, again, here weβd be adding π onto π₯ so it would have the same result. Now, if we look at the third answer, π¦ equals π over π₯, this has something to do with proportionality. But itβs not directly proportional. This is inversely proportional because we can say that π¦ is inversely proportional to π₯. So, weβve got one over π₯ here. This means inversely proportional. So then, if we bring in π, our proportionality constant, we get π¦ is equal to π over π₯. So this is inversely proportional. And this would give us a different-shaped graph, as Iβve shown here with the curve. So, therefore, we can definitely say that the correct equation to show that π¦ is directly proportional to π₯, with π being a constant, is π¦ equals ππ₯.