Video Transcript
Which table does not show 𝑥 varying directly with 𝑦?
And then we have five tables to choose from. So, let’s begin by reminding ourselves what it means if two variables are in direct variation or direct proportion to one another. 𝑥 and 𝑦 are in direct proportion to one another if their ratio is constant, in other words, if 𝑦 divided by 𝑥 equals 𝑘. 𝑘 is the constant of variation or the constant of proportionality, and multiplying through by 𝑥, we see that we can write this as 𝑦 equals 𝑘 times 𝑥. This is the general form we tend to use when describing direct variation between 𝑥 and 𝑦. So, in order to establish which tables do and don’t show 𝑥 varying directly with 𝑦, we’ll divide each of the values for 𝑦 by each of the values for 𝑥 and see if we do indeed get some constant value.
So, in our first table, we’re going to divide six by five. That, of course, is 1.2. Next, we’re going to divide 10 by three. Well, that’s 3.3 recurring. We’ve already shown that the ratio between the 𝑥- and 𝑦-values is not constant, and so the answer must be (A). This table must not show 𝑥 varying directly with 𝑦, but let’s double-check by looking at the rest of our tables.
With our second table, we can divide 12 by one, 24 by two, and 36 by three, and we’ll always get a value of 12. So 𝑥 and 𝑦 must be in direct proportion to one another and the constant of variation is 12. Similarly, in our third table, two divided by 10 equals four divided by 20 equals six divided by 30, which gives us a constant variation of 0.2. Since this remains unchanged no matter which pair of values we choose, we know 𝑥 and 𝑦 must be in direct proportion.
Similarly, let’s consider our third table. We’re not going to divide zero by zero since that’s undefined. But we do know that if 𝑥 is zero, 𝑦 is zero, so this pair of values does satisfy the equation 𝑦 equals 𝑘𝑥. However, dividing the remaining two 𝑦-values by the remaining two 𝑥-values gives us a constant of variation of four. And for our third table, we get a constant of variation of 0.75. So, we’ve verified that tables (B), (C), (D), and (E) do show 𝑥 varying directly with 𝑦 and the table that doesn’t is (A).
