Solve k_{m}=frac{k_{F}*A_{F}+k_{W}*A_{W}}{A_{F}+A_{W}}

$$k _ { m } = \frac { k _ { F } \cdot A _ { F } + k _ { W } \cdot A _ { W } } { A _ { F } + A _ { W } }$$

Solve for A_F

$$\left\{\begin{matrix}A_{F}=-\frac{A_{W}\left(k_{m}-k_{W}\right)}{k_{m}-k_{F}}\text{, }&A_{W}\neq 0\text{ and }k_{F}\neq k_{W}\text{ and }k_{m}\neq k_{F}\\A_{F}\neq -A_{W}\text{, }&\left(A_{W}=0\text{ and }k_{m}=k_{F}\right)\text{ or }\left(k_{m}=k_{W}\text{ and }k_{F}=k_{W}\right)\end{matrix}\right.$$

Solve for A_W

$$\left\{\begin{matrix}A_{W}=-\frac{A_{F}\left(k_{m}-k_{F}\right)}{k_{m}-k_{W}}\text{, }&A_{F}\neq 0\text{ and }k_{F}\neq k_{W}\text{ and }k_{m}\neq k_{W}\\A_{W}\neq -A_{F}\text{, }&k_{m}=k_{W}\text{ and }\left(k_{F}=k_{W}\text{ or }A_{F}=0\right)\end{matrix}\right.$$