Decide if 𝑥 varies directly or inversely with 𝑦 and use this to find the value of 𝑦 when 𝑥 equals three.
If variables vary directly, then as one increases, the other increases or as one decreases, the other one decreases. Inversely, it’s just the opposite; as one variable increases, the other decreases or as one decreases, the other one increases. And 𝑘 is just a constant, so we’re talking about 𝑦 and 𝑥 when we’re talking about increasing and decreasing.
Here, we can see that 𝑥 is increasing. From two to four, it increases by two; from four to 70, it increases by 66. And 𝑦 is decreasing; from 70 to 35, we’re decreasing by 35 and from 35 to two, we’re decreasing by 33. So as the table shows, 𝑦 decreases as 𝑥 increases, meaning that 𝑦 varies inversely with 𝑥.
So if we need to find the value of 𝑦 when 𝑥 equals three, we first need to find that constant 𝑘. So we can solve for 𝑘 by multiplying both sides by 𝑥. So 𝑘 is equal to 𝑥 times 𝑦. So 𝑘 is equal to two times 70, which is equal to 140, four times 35 is also 140, and just for more reassurance, 70 times two is also 140.
So here is our equation 𝑦 equals 140 divided by 𝑥. So we want to know what is the value of 𝑦 when 𝑥 is three. So plugging in three, we’ll take 140 divided by three, which is 46 and two-thirds. Therefore, when 𝑥 equals three, 𝑦 equals 46 and two-thirds.