Question Video: Combined Variation and Its Applications

Video Transcript

We know that 𝑦 varies directly with 𝑥 and inversely with 𝑧. Given that 𝑦 equals five when 𝑥 equals six and 𝑧 equals three, find the value of 𝑦 when 𝑥 equals 24 and 𝑧 equals 15.

We’re told that 𝑦 varies directly with 𝑥. So then we will have the equation 𝑦 equals 𝑘𝑥, where 𝑘 is a constant. And we also know that 𝑦 varies inversely with 𝑧. So we have the equation 𝑦 equals 𝑘 divided by 𝑧, where 𝑘 is a constant. So how can we put these together to make one equation?

Well, we can have 𝑦 equals 𝑘, and 𝑘 needs to be multiplied by 𝑥 and divided by 𝑧. So we have something like this. And we are given that when 𝑦 equals five and 𝑥 equals six and 𝑧 equals three that we’re asked to find a value of 𝑦 when 𝑥 equals 24 and 𝑧 equals 15.

So in order to do that, we first must solve for 𝑘. So we simplify the right-hand side of the equation. Six divided by three is two. And now to solve for 𝑘, we must divide both sides of the equation by two. And five divided by two gives us that 𝑘 is equal to 2.5. So we must replace 𝑘 with 2.5 in our equation.

So our equation is 𝑦 equals 2.5𝑥 divided by 𝑧. So we are supposed to find the value of 𝑦 when 𝑥 equals 24 and 𝑧 equals 15. So we will replace 𝑥 with 24 and 𝑧 with 15. So we need to take 2.5 times 24, which is 60, and 60 divided by 15 is four. Therefore, our final answer will be four.