Question 5, Exercise 1.1

Solutions of Question 5 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan.

Find the complex number $z$ if $4z-3\bar{z}=\dfrac{1-18i}{2-i}$


Suppose $z=x+iy$, then $\bar{z}=x-iy$. So we have \begin{align}&4z-3\bar{z}=\dfrac{1-18i}{2-i}\\ \implies &4(x+iy)-3(x-iy)=\dfrac{1-18i}{2-i}\times \dfrac{2+i}{2+i}\\ \implies &4x+4iy-3x+3iy=\dfrac{(1-18i)(2+i)}{2^2-i^2} \end{align} \begin{align} \implies x+7iy&=\dfrac{2-18i^2-36i+i}{4+1}\\ &=\dfrac{20-35i}{5}\\ &=\dfrac{5(4-7i)}{5}\\ \implies x+7iy&=4-7i\end{align} Equating real and imaginary parts. \begin{align}x=4 \quad \text{and}\quad 7y &=-7 \,\text{ i.e. }\,y=-1.\end{align} Thus we have $z=x+iy=4-i$.