Task B-2.1. Solve the equation in the set of complex numbers

$$
z \cdot(4+i)+\bar{z} \cdot(2-3 i)=1-i^{11}
$$
## Solution.

Let $z=x+y i$. Then the given equation has the form

$(x+y i)(4+i)+(x-y i)(2-3 i)=1+i$ or $6 x-4 y+i(-2 x+2 y)=1+i$.

By equating the real and imaginary parts, we obtain the system of equations:

$$
\left\{\begin{array}{l}
6 x-4 y=1 \\
-2 x+2 y=1
\end{array}\right.
$$

The solution to the system is $x=\frac{3}{2}, y=2$.