In the paper of G.Frey there is a link between stable elliptic curves and certain Diophantine equations. The Frey curve of the equation $A-B=C$  is 

 $$E :\;y^2=x(x-A)(x-B)$$

 where  $A=a^p$, $B=b^p$, $C=c^p$. 

And he define also the minimal equation of $E$ by the the change of variable $x=4X$ and $y=4X+8Y$ and the the equation of $E$ becomes 

$$ Y^2+XY=X^3+ \frac{A+B-1}{4}X^2+\frac{AB}{16}X$$ 

And the discriminant is  $\Delta= \frac{(ABC)^2 }{2^8}$. 
 
My question is why Frey make this change of variable and define two equations of the curve? and why he takes specifically this change $x=4X$, $y=4X+8Y$ what is the aim of all this ?