Suppose that the dependent variables $z$ and $w$ are functions of independent variables $x$ and $y$, defined by the equations $f(x,y,z(x),w(y))=0$ and $g(x,y,z(x),w(y))=0$, where $$f_zg_w-f_wg_z=1.\tag{*}$$ Then what is expression of $z_x$?
If I multiply $f_zg_w-f_wg_z=1$ with $z_x$, I get $f_zz_xg_w-f_wg_zz_x=z_x$ ie $f_xg_w-f_wg_x=z_x$.
But if I totally differentiate $f$ and $g$ respectively, then $f_z=-\frac{f_x}{z_x}$ and $g_z=-\frac{g_x}{z_z}$. Plugging this in equation $(*)$, I get $f_wg_x-f_xg_w=z_x$
Which one is correct answer? and why?
Since $f_x+f_zz_x=0$ and $g_x+g_zz_x=0$, your first try is missing two minus signs: $$f_zz_xg_w-f_wg_zz_x=z_x\implies -f_xg_w-(-f_wg_x)=z_x,$$ so your two results are the same.