Solve simultaneous equations without a graphing calculator

Label the equations

$\begin{aligned}z+(i-1)w&=0 &&(1),\\3z+(1-i)w&=1 &&(2).\end{aligned}$

From (1), $z=-(i-1)w=(1-i)w$ (3). Substitute (3) into (2):

$\begin{aligned}3(1-i)w+(1-i)w&=1,\\4(1-i)w&=1,\\w&=\frac1{4(1-i)}\cdot\frac{1+i}{1+i}\\&=\frac18(1+i).\end{aligned}$

This is (4). Substitute (4) into (3):

$\begin{aligned}z&=(1-i)w\\&=(1-i)\frac18(1+i)\\&=\frac18(1-i^2)\\&=\frac14.\end{aligned}$

The substitution and conjugate multiplication give exact values without a graphing calculator.