Can someone me understand how to find the constant of variaton?

Here you're discussing a linear relationship between x and y, also characterized as "direct proportion."  Start with y = kx, where k is the "constant of variation."

Write y = kx, and then substitute the givens:  81 = k(27).  Thus, k = 3.
Write y = 3x.  We say "the constant of variation is 3."

Inverse variation is handled similarly; it's just that the relationship between x and y is no longer a linear one.