z is a complex number, z=a+ib and (2+i)z=1-z, what is z in terms of i ?

Mar 9, 2016

$z = \frac{3}{10} - \frac{1}{10} i$

Explanation:

$\left(2 + i\right) z = 1 - z$

$\implies \left(2 + i\right) z + z = 1$

$\implies \left(2 + i + 1\right) z = 1$

$\implies \left(3 + i\right) z = 1$

$\implies z = \frac{1}{3 + i}$

$= \frac{3 - i}{\left(3 + i\right) \left(3 - i\right)}$

$= \frac{3 - i}{10}$

$= \frac{3}{10} - \frac{1}{10} i$

$\therefore z = \frac{3}{10} - \frac{1}{10} i$