A rough way of interpreting direct variation is to say that, if y varies directly with x, scaling x by some amount will scale y by that same amount. If we double x, we double y, too. If we cut x in half, y gets cut in half, too.
We can capture this idea mathematically with some equality with x on one side and y on the other. For example, if we say y = x, doubling x would have to double y in order to keep the equality true - 2y = 2x. x and y aren't always going to start out equal, though, so to account for this, we can say that y = kx, where k is some scaling factor that tells us that y might be different from x.
Dividing both sides of this equation by y gives us the equivalent, crucial fact that y/x = k. In other words, the ratio between y and x is constant; it never changes. This fits nicely with how fractions work: if we double x and y, we end up with the fraction 2y/2x, the 2's in the numerator and denominator cancel, and we're simply back at the fraction y/x again.
This gives us a way to test whether y varies directly with x: we can take the ratio between them at two points, and if we come up with the same value, we know that they're indeed varying directly.
Let's look at the first and second rows of the table. The first row gives us the values x = 4 and y = 6.4. Setting up a ratio between y and x, we find that y/x = 6.4/4 = 1.6.
Now, let's compare that to the second row, where x = 7 and y = 11.2. The ratio between these two is y/x = 11.2/7 = 1.6. Since our two ratios are the same, we can say that yes, y does vary directly with x, and their constant of variation k = 1.6
