Circle the equation showing that 𝑦 is directly proportional to 𝑥, with 𝑘 being a constant. 𝑦 equals 𝑘𝑥, 𝑦 equals 𝑘 minus 𝑥, 𝑦 equals 𝑘 over 𝑥, or 𝑦 equals 𝑥 plus 𝑘.
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 𝑥 increases, 𝑦 increases at the same rate.
So what symbols do we use for proportionality? But we’ve got a symbol here which means directly proportional. So, we can say that 𝑦 is directly proportional to 𝑥. However, if we’re looking to deal with this proportionality as an equation, the proportionality sign isn’t 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 𝑘. And as in this question, we’re told that we have 𝑘 as a constant. So therefore, 𝑦 is proportional to 𝑥 becomes 𝑦 equals 𝑘𝑥. Because 𝑘, like we said, is our proportionality constant, and then 𝑥 is obviously our 𝑥-value. So therefore, we can see that the correct answer is 𝑦 equals 𝑘𝑥, so we can circle that. And we’ve done that because that’s our first answer.
Now, let’s have a look at a couple of the other answers to see why they would not be correct. Well, let’s consider 𝑦 equals 𝑘 minus 𝑥. With 𝑦 equals 𝑘 minus 𝑥, as one amount increases, the other amount would not increase at the same rate. Because if we consider that we doubled 𝑦, well if 𝑦 is 𝑘 minus 𝑥 and we double it, we’ve got two multiplied by 𝑘 minus 𝑥. But this would not increase at the same rate. Because two 𝑘 minus two 𝑥, which would be if we doubled 𝑘 minus 𝑥, is not the same as 𝑘 minus two 𝑥 which should be the expression if we just doubled the 𝑥. So therefore, they would not be directly proportional.
Well similarly, this would be the same with 𝑦 equals 𝑥 plus 𝑘. Because, again, here we’d be adding 𝑘 onto 𝑥 so it would have the same result. Now, if we look at the third answer, 𝑦 equals 𝑘 over 𝑥, this has something to do with proportionality. But it’s not directly proportional. This is inversely proportional because we can say that 𝑦 is inversely proportional to 𝑥. So, we’ve got one over 𝑥 here. This means inversely proportional. So then, if we bring in 𝑘, our proportionality constant, we get 𝑦 is equal to 𝑘 over 𝑥. So this is inversely proportional. And this would give us a different-shaped graph, as I’ve shown here with the curve. So, therefore, we can definitely say that the correct equation to show that 𝑦 is directly proportional to 𝑥, with 𝑘 being a constant, is 𝑦 equals 𝑘𝑥.