Question #141164
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.
1
Expert's answer
2020-11-05T17:03:24-0500

Let

z=x+iyz=xiyz = x + iy \Rightarrow z* = x - iy

Then

2(x+iy)i(xiy)=3(35i)2x+2iyix+i2y=915i2xy+i(2yx)=915i\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

{2xy=92yx=15\left\{ \begin{array}{l} 2x - y = 9\\ 2y - x = - 15 \end{array} \right.

{2xy=93x=3\left\{ \begin{array}{l} 2x - y = 9\\ 3x = 3 \end{array} \right.

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

Answer: z=17iz = 1 - 7i


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