Question

4.Sa se determine z apartine lui c daca
a) z=i
b) z - 2z=-2+9i
c)2z +iz =1 +i
d) 3z =6+3i - 2z
e) iz + (1-i)*z=3+2i +4i

Answer 1

$\it a)\ \overline{z}=i \Rightarrow z=-i\\ \\ b)\ Fie\ \ z=a+bi \Rightarrow \overline{z}=a-bi\\ \\ \overline{z}-2z=-2+9i \Rightarrow a-bi-2a-2bi=-2+9i \Rightarrow -a-3bi=-2+9i\Rightarrow\\ \\ \Rightarrow a=2,\ b=-3 \Rightarrow z=2-3i$

$\it c)\ 2\overline{z}+iz=\overline{1+i} \Rightarrow 2(a-bi)+i(a+bi)=1-i \Rightarrow 2a-2bi+ai-b=1-i\Rightarrow\\ \\ \Rightarrow(2a-b)+(a-2b)i=1-i\Rightarrow\begin{cases}\it 2a-b=1\Rightarrow b=2a-1\ \ \ \ (*)\\ \\ \it a-2b=-1\stackrel{(*)}{\Longrightarrow}a-4a+2=-1\Rightarrow a=1\end{cases} \\ \\ \\ a=1\stackrel{(*)}{\Longrightarrow} a=2\cdot1-1 \Rightarrow a=1,\ \ Deci,\ z=1+i$