[tex]2i\overline{z} +z = 3-4i \\ \text{unde }~\overline{z} ~ \text{este } \texttt{conjugatul complex }\text{al lui }z \\ \\ z= a+bi ~~\Longrightarrow~~ \overline{z} = a-bi \\ \\ \text{Rescriem ecuatia: } \\ \\ 2i(a-bi) + (a+bi)=3-4i ~~~~~~~\text{unde a si b sunt necunoscutele} \\ 2ai - 2bi^2 + a + bi = 3 - 4i \\ \text{Dar } ~~i^2 = -1 \\ 2ai - 2b\times (-1) +a + bi = 3 - 4i \\ 2ai + 2b + a +bi = 3-4i [/tex]
[tex]\displaystyle \text{Descompunem ecuatia in doua ecuatii, una imaginara si una reala. } \\ \\
2ai +bi = -4i ~~~|:i \\
2b+a=3 \\
--- \\
2a+b = -4~~|\times (-2) \\
a+2b = 3 \\
--- \\
-4a-2b = 8\\
a+2b = 3 \\
--------- \\
-3a = 11 \\
\boxed{a= -\frac{11}{3}} \\ \\
2a+b = -4\\
a+2b = 3 ~~|\times (-2)\\
--- \\
2a+b = -4\\
-2a-4b = -6\\
--------- \\
-3b = -10 \\
\boxed{b = \frac{10}{3}} \\ \\
z=a+bi = \boxed{-\frac{11}{3} + \frac{10}{3}i }
[/tex]
