Example 2. Find the real solutions of the equation

$$
(4+2 i) x+(5-3 i) y=13+i
$$
Solution. Let's separate the real and imaginary parts in the left side of the equation: $(4 x+5 y)+i(2 x-3 y)=13+i$. From here, according to the definition of equality of two complex numbers, we get

$$
\left\{\begin{array}{l}
4 x+5 y=13 \\
2 x-3 y=1
\end{array}\right.
$$

Solving this system, we find

$$
x=2, \quad y=1
$$

## Problems for Independent Solution

Find the real solutions of the equations:

2. $(3 x-i)(2+i)+(x-i y)(1+2 i)=5+6 i$.
3. $(x-i y)(a-i b)=i^{\text {s }}$, where $a, b$ are given real numbers, $|a| \neq|b|$.
4. $\frac{1}{z-i}+\frac{2+i}{1+i}=\sqrt{2}$, where $z=x+i y$.
5. Represent the complex number $\frac{1}{(a+i b)^{2}}+\frac{1}{(a-i b)^{2}}$ in algebraic form.
6. Prove that $\frac{\sqrt{1+x^{2}}+i x}{x-i \sqrt{1+x^{2}}}=i(x-$ real $)$.
7. Express $x$ and $y$ in terms of $u$ and $v$, if $\frac{1}{x+i y}+\frac{1}{u+i v}=1(x, y, u, v-$ real
numbers). 8. Find all complex numbers satisfying the condition $\bar{z}=z^{2}$.