Partial derivative using Jacobian shortcut?

Question

Let $x,y$ and $z$ be determined by $x+y^2=u, y+z^2=v$ and $z+x^2=w$. By writing $F=x+y^2-u, G=y+z^2-v$ and $H=z+x^2-w$ and using the quotient of Jacobian's 'shortcut' calculate. \begin{equation} \frac{\partial x}{\partial u} \end{equation} Confused what Jacobian's 'shortcut' is and why I cannot just rearrange the first equation to yield $x=u-y^2$ and then find the partial derivative of that.