Parametrized complex equation What are the values that satisfy the following equation:
$$\frac{z+i}{\bar{z}-2}=-2i$$
I have tried substituting $z$ with $a+bi$, and multiplying the left hand side by $\frac{z-2}{z-2}$ but the equations became too complicated.

$$\frac{z+i}{\bar{z}-2}=-2i$$
Let's multiply by $\bar{z}-2$:
$$z+i=-2i(\bar{z}-2)$$
Let's substitute $z$ for $a+bi$:
$$a+bi+i=-2i(a-bi-2)$$
$$a+bi+i=-2ai+2bi^2+4i$$
$$a+(b+1)i=-2ai-2b+4i$$
$$a+(b+1)i=-2b+(-2a+4)i$$
$$
\left\{ 
\begin{array}{c}
a=-2b \\ 
b+1=-2a+4 \\ 
\end{array}
\right. 
$$
$$b+1=-2(-2b)+4$$
$$b+1=4b+4$$
$$3b=-3$$
$$b=-1$$
$$a=2$$
Thus $z=2-i$
