HCI Complex Numbers Tutorial Q1

Solve the simultaneous equations

\(\left( 1+\mathbf{i} \right)z+w=3+2\,\mathbf{i}\)

\(\mathbf{i}\,z+\left( 3+\mathbf{i} \right)w=5\mathbf{i}\),

giving \(z\) and \(w\) in the form \(x+y\,\mathbf{i}\), where \(x\) and \(y\) are real.[3]

Find, in the form \(x+y\,\mathbf{i}\), the complex numbers \(z\) satisfying both of the equations \(\frac{z}{z^*} = \frac{3}{5} + \frac{4}{5}\mathbf{i}\) and \(zz^* = 5\).[4]