Pair of non-linear equations

Equation 1: $x(x - y - 1) = 22$ Equation 2: $y(y - x + 1) = 34 \implies -y(x - y - 1) = 34$

Subtracting Equation 2 from Equation 1: $x^2 - xy - x - (y^2 - xy + y) = 22 - 34$ $x^2 - y^2 - (x + y) = -12$ $(x + y)(x - y) - (x + y) = -12$ $(x + y)(x - y - 1) = -12$

Also, adding Equation 1 and Equation 2: $x(x - y - 1) - y(x - y - 1) = 22 + 34 = 56$ $(x - y)(x - y - 1) = 56$

Let $k = x - y$. Then: $k(k - 1) = 56$ $k^2 - k - 56 = 0$ $(k - 8)(k + 7) = 0$

Since $x > y$, $k = x - y > 0$, so $x - y = 8$.