Complexities

Equating the real and imaginary parts of the complex numbers gives us a system of linear equations to solve: $\begin{cases} 4x - 43 = 2y-x \\ 3x+y = 2x+2y+5 \\ \end{cases}$ Solving the first equation for $y$ gives us $y = \frac{5x-43}{2}$ . Substituting this expression in for $y$ in the second equation yields $3x + \frac{5x-43}{2} = 2x + 5x - 43 + 5$ , and multiplying both sides by $2$ to get rid of the fraction gives us $6x+5x-43 = 4x + 10x - 86 + 10$ . This equation simplifies to $33 = 3x$ , from which it follows that $x = 11$ . Hence, $y = \frac{55 - 43}{2} = 6$ , and the product $xy$ must be 66.