Question

In this problem, x cubed = 12y...how do you solve to determine if this is direction variation or not?

Answer 1

This can be reduced to 12y=x³, or
y=x³/12

Direct variation have two properties:
1. it passes through the origin, i.e. if x=0, y=0.
This property is satisfied by the above equation.
2. x and y are both linear, i.e. if you multiply x by any finite number, y increases by the same ratio.
For example, for the above,
x=1, y=1/12
x=2, y=8/12 (y is increased 8 times instead of 2), so the equation is not a direct variation.

Example of a direct variation:
y=2x
x=1, y=2
x=2, y=4
x=7, y=14
So the ratios are always 2:
2/1 = 4/2 = 14/7 =2 (equals the coefficient of x).