Video Transcript
Variable 𝑦 is inversely proportional to 𝑥. When 𝑥 equals three, 𝑦 equals six. Find the value of 𝑦 when 𝑥 equals eight.
To answer this question, we should begin by recalling what it means when 𝑦 is inversely proportional to 𝑥. If 𝑦 and 𝑥 are inversely proportional to one another, this tells us that as 𝑥 increases, 𝑦 decreases and vice versa. When 𝑦 and 𝑥 are inversely proportional to one another, this is the same as saying 𝑦 is directly proportional to one over 𝑥.
And the equation that we can use to link 𝑦 and 𝑥 with some constant 𝑘 is 𝑦 equals 𝑘 divided by 𝑥. Now that we look at this equation, it makes a lot of sense that as 𝑥 increases, the value of 𝑘 over 𝑥 decreases and so the value of 𝑦 also decreases.
So with this in mind, we’re going to substitute 𝑥 equals three and 𝑦 equals six into this equation and it will allow us to find the value of 𝑘. We call 𝑘 the constant of variation or the constant of proportionality. Substituting 𝑥 equals three and 𝑦 equals six into this equation, and we get six equals 𝑘 over three. To solve for 𝑘, we’ll multiply both sides by six. And we find 18 is equal to 𝑘.
And so, the equation that we can use to link 𝑦 and 𝑥 in this example is 𝑦 equals 18 over 𝑥. And this is really useful because we want to find the value of 𝑦 when 𝑥 is equal to eight. So we’re going to substitute 𝑥 equals eight into this equation. When we do, we find that 𝑦 is equal to 18 over eight. Since 18 divided by eight is two remainder two, this is the same as two and two-eighths which is equivalent to two and one-quarter.
So the value of 𝑦 when 𝑥 is equal to eight is two and one-quarter.
