Example 1. Let $\sim$, $w$, $\lambda$ be complex numbers, and $|\lambda| \neq 1$, solve the equation $\bar{z} - \lambda z = w$ for $z$.
Analysis: We consider the equation $\bar{z}-\lambda z=w$ as a linear equation in two variables $z$ and $\bar{z}$. Therefore, to find $z$, we must construct a new equation involving $z$ and $\bar{z}$.

Solution $\because \bar{z}-\lambda z=w, \therefore(\bar{z}-\lambda z)=\bar{w}$, i.e., $z-\bar{\lambda} \bar{z}=\bar{w}$.
Solve the system of equations $\left\{\begin{array}{r}\bar{\lambda} z-z=-\bar{w} \\ z-\lambda z=w\end{array}\right.$,
we get: $\quad z=\frac{\bar{\lambda} w+w}{1-|\lambda|^{2}}$.