$ \frac {x - 2}{y} + \frac {5}{xy} = \frac {4 - y}{x} - \frac {|y - 2x|}{xy}$ $ x^2 - 2x + 5 = 4y - y^2 - |y - 2x|$ $ x^2 - 2x + 1 + y^2 - 4y + 4 = - |y - 2x|$ $ (x - 1)^2 + (y - 2)^2 = - |y - 2x|$ Since $ (x - 1)^2 + (y - 2)^2 \ge 0$ and $ - |y - 2x| \le 0$, the only solution occurs when: $ (x - 1)^2 + (y - 2)^2 = - |y - 2x| = 0 \Rightarrow (x,y) = (1,2)$.