Video Transcript
Given that 𝑦 varies inversely as 𝑥, write an equation for 𝑦 in terms of 𝑥 using 𝑘 as a nonzero constant.
Before we think about what it means for two items to vary inversely to one another, let’s remind ourselves what it means if they vary directly. In this case, we say that these two items, let’s call them 𝑥 and 𝑦, are in direct proportion to one another. And this symbol that looks a little bit like the Greek letter 𝛼 means is proportional to; 𝑦 is proportional to 𝑥. When 𝑦 and 𝑥 are directly proportional to one another, their ratio is constant. Another way to represent this is 𝑦 is equal to 𝑘 times 𝑥 for some real constant 𝑘. And we think about this as meaning that when 𝑥 varies, 𝑦 also varies at the same factor of 𝑘 each time.
Now, when two items are in inverse proportion to one another, we say that as 𝑥 increases, 𝑦 decreases. More specifically, 𝑦 is proportional to one over 𝑥. So what does this mean for the corresponding equation that links 𝑦 and 𝑥 when they are inversely proportional to one another? Well, the corresponding equation is 𝑦 equals 𝑘 over 𝑥. In this case, we notice that as 𝑥 increases, 𝑘 divided by 𝑥 will decrease meaning 𝑦 itself will also decrease. And so, given that 𝑦 varies inversely as 𝑥, the equation for 𝑦 in terms of 𝑥 using 𝑘 as a nonzero constant is 𝑦 equals 𝑘 over 𝑥.
