Answer to Question #141164 in Complex Analysis for Hayri

Question

The complex number z satisfies the equation 2z-iz* = 3(3-5i) where z* denotes the complex conjugate of z. Determine the value of z giving your answer in the form x+yi where x and y are real numbers.

Answer 1

Let

$z = x + iy \Rightarrow z* = x - iy$

Then

$\begin{array}{l} 2(x + iy) - i(x - iy) = 3(3 - 5i)\\ 2x + 2iy - ix + {i^2}y = 9 - 15i\\ 2x - y + i(2y - x) = 9 - 15i \end{array}$

Then

$\left\{ \begin{array}{l} 2x - y = 9\\ 2y - x = - 15 \end{array} \right.$

$\left\{ \begin{array}{l} 2x - y = 9\\ 3x = 3 \end{array} \right.$

$\left\{ \begin{array}{l} x = 1\\ y = -7 \end{array} \right.$

Answer: $z = 1 - 7i$