Video Transcript
Which of the following represents a direct variation between the two variables 𝑥 and 𝑦? Is it (A) 𝑦 equals 𝑥 plus two? (B) 𝑥 over five equals 𝑦 over four. (C) 𝑥 over six equals three over 𝑦. Or is it (D) 𝑥𝑦 equals six?
In order to answer this question, we’re going to need to remind ourselves what it means if we have direct variation between a pair of variables. Two variables are said to be in direct variation or direct proportion to one another if the ratio between 𝑦 and 𝑥 is equal to some constant 𝑘, in other words if 𝑦 divided by 𝑥 equals 𝑘. We call 𝑘 the constant of variation or the constant of proportionality. And we more generally see this written in the form 𝑦 equals 𝑘 times 𝑥.
So our job is to identify which of the equations (A), (B), (C), and (D) can be written in either of these forms. Now, in fact, we’re going to quite quickly disregard option (A). When 𝑥 and 𝑦 represent direct variation, we do not write it in the form 𝑦 equals 𝑘𝑥 plus 𝑐, where 𝑐 is a second constant. And this is because if 𝑥 and 𝑦 are directly proportional to one another, when 𝑥 is zero, 𝑦 is zero. With our first equation, when 𝑥 is zero, 𝑦 is zero plus two, which is two. So equation (A) cannot represent direct variation between our two variables.
So what about equation (B)? Let’s look to make 𝑦 the subject. And to do so, we’ll multiply both sides of the equation by four. When we do, we get four 𝑥 over five equals 𝑦. And that can alternatively be written as 𝑦 equals four-fifths 𝑥. This is of the form 𝑦 equals 𝑘𝑥, where 𝑘 is equal to four-fifths. So equation (B) must represent direct variation between 𝑥 and 𝑦.
We will just double-check that (C) indeed do not represent direct variation. In fact, if we rearrange equation (C) to make 𝑦 the subject, we get 𝑦 equals one over two 𝑥. This is an example of two variables in inverse proportion to one another. Similarly, when we rearrange equation (D), we get 𝑦 equals six over 𝑥. Once again, this represents inverse variation between the variables 𝑥 and 𝑦. So we’ve demonstrated that the answer is (B). (B) represents direct variation between the two variables 𝑥 and 𝑦.
