Question

If $(1+i)z=(1-i)\overline{z},$
then show that$z=-i\overline{z}.$

Answer 1

Given that-
$(1+i)z=(1-i)\overline{z},$
$\Rightarrow z=\frac{(1-i)}{(1+i)}\overline{z}$
$\Rightarrow z=(\frac{1-i}{1+i}\times \frac{1-i}{1-i})\overline{z}$
$\Rightarrow z=\left(\frac{1+i^2-2i}{1^2-i^2}\right)\overline{z}$
$\Rightarrow z=\left(\frac{-2i}{2}\right)\overline{z}$
$\Rightarrow z=-i\overline{z}$
Hence, proved.