设函数 $z=z(x, y)$ 由方程 $x-a z=e^{y+a z}$（ $a$ 是非零常数）确定，则（ ）
A. $\frac{\partial z}{\partial x}-\frac{\partial z}{\partial y}=\frac{1}{a}$
B. $\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}=\frac{1}{a}$
C. $\frac{\partial z}{\partial x}-\frac{\partial z}{\partial y}=-\frac{1}{a}$
D. $\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}=-\frac{1}{a}$