Chapter 14.10, Problem 12E

To determine

To prove: That ${\left(\frac{\partial z}{\partial x}\right)}_y=-\frac{\frac{\partial f}{\partial x}\frac{\partial g}{\partial w}-\frac{\partial f}{\partial w}\frac{\partial g}{\partial x}}{\frac{\partial f}{\partial z}\frac{\partial g}{\partial w}-\frac{\partial f}{\partial w}\frac{\partial g}{\partial z}}$ and ${\left(\frac{\partial w}{\partial y}\right)}_x=-\frac{\frac{\partial f}{\partial z}\frac{\partial g}{\partial y}-\frac{\partial f}{\partial y}\frac{\partial g}{\partial z}}{\frac{\partial f}{\partial z}\frac{\partial g}{\partial w}-\frac{\partial f}{\partial w}\frac{\partial g}{\partial z}}$ if $f\left(x,y,z,w\right)=0$ and $g\left(x,y,z,w\right)=0$.