12. Suppose that f(x, y, z. w) = 0 and g(x, y, z, w) = 0 determine z and w as differentiable functions of the independent variables x and y,

and suppose that \frac{\partial f}{\partial z} \frac{\partial g}{\partial w}-\frac{\partial f}{\partial w} \frac{\partial g}{\partial z} \neq 0 Show that \left(\frac{\partial z}{d x}\right)_{y}=-\frac{\frac{\partial f}{\partial x} \frac{\partial g}{\partial v}-\frac{\partial f}{\partial w} \frac{\partial g}{\partial x}}{\frac{\partial f}{\partial z} \frac{\partial g}{\partial h}-\frac{\partial f}{\partial h v} \frac{\partial g}{d z}} and \left(\frac{\partial w}{\partial y}\right)_{\bar{x}}=-\frac{\frac{\partial f}{d z} \frac{\partial g}{\partial y}-\frac{\partial f}{\partial y} \frac{\partial g}{d z}}{\frac{\partial f}{\partial z} \frac{\partial g}{\partial w}-\frac{\partial f}{\partial w} \frac{\partial g}{\partial z_{z}}} .