Video: Combined Variation and Its Applications

We know that 𝑦 varies directly with π‘₯ and inversely with 𝑧. Given that 𝑦 = 5 when π‘₯ = 6 and 𝑧 = 3, find the value of 𝑦 when π‘₯ = 24 and 𝑧 = 15.

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Video Transcript

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

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

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

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

So our equation is 𝑦 equals 2.5π‘₯ divided by 𝑧. So we are supposed to find the value of 𝑦 when π‘₯ equals 24 and 𝑧 equals 15. So we will replace π‘₯ 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.

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