what is a direct variation?

Direct variation is a linear function in the form of y=kx; where k is the constant of variation (the slope) and the y-intercept equals 0. 

*all direct variation is linear, and passes through the origin.

It can also be written as a proportion: x1/x2 = y1/y2

$$x\:\:\alpha\:\: y\:\:\mbox{direct variation}\\

x\:\:\alpha\:-y\\

x\:\:\alpha\:\: \frac{1}{y}\:\:\mbox{inverse variation}\\

x\:\:\alpha\:\: \frac{-1}{y}$$

I get confused with variation - I think the 1st and 3rd are right but I am not too sure about the others.

Does someone want to give me a lesson please?

Melody, the second one is essentially the same as the first and the fourth is the same as the third.  The minus sign would be part of the constant that you need to turn the proportionality into an equality.