Task B-2.1. Determine the absolute value of the complex number $z$ for which

$$
\frac{\bar{z}}{1+2 i}-\frac{2 z}{1-2 i}=5
$$
## Solution.

Let $z=x+y i$. After multiplying the given equation by $(1-2 i)(1+2 i)$, we get

$$
\begin{aligned}
& (x-y i)(1-2 i)-2(x+y i)(1+2 i)=25 \\
& -x-6 x i-3 y i+2 y=25
\end{aligned}
$$

thus we have

$$
\begin{aligned}
& -x+2 y=25 \\
& -6 x-3 y=0
\end{aligned}
$$

From this system, it follows that $y=10$ and $x=-5$.

Therefore, $z=-5+10 i$, which means $|z|=5 \sqrt{5}$.