Yes. Say that C is a cubic. We start by picking a rational point O on the cubic. Then we take the tangent line of C at the rational point O. This will intersect the cubic in another rational point. We take the X-axis to be the tangent line at that other rational point. And we let the Y-axis to be any line through O.
By changing coordinates, you get an equation of the form
Multiply by x and change into the variable u=xy. This will get you something of the form
Change the variable again by setting [itex]v^2=u^2+(ax+b)[/itex].
Now change the variables once more to obtain that c=1.