(i) First, calculate \(iw = i(5 + 3i) = -3 + 5i\).

Now, divide by \(z = 4 + i\):

\(\frac{-3 + 5i}{4 + i}\).

Multiply numerator and denominator by the conjugate of the denominator:

\(\frac{(-3 + 5i)(4 - i)}{(4 + i)(4 - i)} = \frac{-12 + 3i + 20i - 5i^2}{16 + 1}\).

Simplify: \(\frac{-12 + 23i + 5}{17} = \frac{-7 + 23i}{17}\).

Thus, \(x = -\frac{7}{17}\) and \(y = \frac{23}{17}\).

(ii) Multiply \(w\) and \(z\):

\(w z = (5 + 3i)(4 + i) = 20 + 5i + 12i + 3i^2 = 20 + 17i - 3 = 17 + 17i\).

The argument of \(w\) is \(\arctan \left( \frac{3}{5} \right)\) and the argument of \(z\) is \(\arctan \left( \frac{1}{4} \right)\).

The argument of \(wz\) is the sum of the arguments: \(\arctan \left( \frac{3}{5} \right) + \arctan \left( \frac{1}{4} \right)\).

Since \(wz = 17 + 17i\), its argument is \(\arctan(1) = \frac{\pi}{4}\).

Thus, \(\arctan \left( \frac{3}{5} \right) + \arctan \left( \frac{1}{4} \right) = \frac{\pi}{4}\).