Question Video: Identifying the Proportional Relationship between Two Quantities That Represents Inverse Variation Mathematics

Video Transcript

Which of the following relations represents an inverse variation between the two variables 𝑥 and 𝑦? Is it 𝑦 equals 𝑥 plus three? 𝑥 divided by 𝑦 is seven over two. Is it 𝑥𝑦 equals 14? Or is it 𝑦 equals six 𝑥?

Now, we’re interested in identifying an inverse relation between our two variables. But if we begin by identifying what it means for the two variables to have a direct variation, we’ll be able to disregard some of our options straightaway. If two variables 𝑥 and 𝑦 are in direct proportion or direct variation to one another, the ratio of these variables is constant. In other words, 𝑦 over 𝑥 is equal to some constant 𝑘. We quite often write this as 𝑦 equals 𝑘𝑥, where we define 𝑘 to be the constant of variation or the constant of proportionality.

Now, this definition allows us to identify at least one pair of variables out of our options that are in direct proportion to one another. Equation (D) is of the form 𝑦 equals six 𝑥. In this case, these variables are in direct proportion to one another. We get 𝑘 equals six. And so, we disregard option (D). Similarly, if we were to rearrange equation (B), we get 𝑦 equals two-sevenths 𝑥. 𝑘 is two-sevenths. So, option (B) shows two variables that are directly proportional to one another.

Now, generally, when two variables are inversely proportional to one another, their product gives us some constant 𝑘. Once again, 𝑘 is called the constant of proportionality, and we tend to represent this in the form 𝑦 equals 𝑘 over 𝑥. Now what this means for our variables is that as 𝑥 increases, 𝑦 decreases. So, we need to find a pair of variables for which this is true.

Consider the equation 𝑦 equals 𝑥 plus three. As 𝑥 increases, 𝑦 also increases. Similarly, as 𝑥 decreases, 𝑦 also decreases. In fact, equation (A) does not represent proportionality at all. So, we disregard option (A), leaving us only with option (C). Let’s check that we can write option (C) in the form given. In fact, it’s already of the form 𝑦 times 𝑥 equals 𝑘. It’s just 𝑥 times 𝑦 equals 14. So, 𝑘 is equal to 14. And option (C) represents an inverse variation between the variables.