Q:

How many integer pairs $(x,y)$ satisfy the equations $x+y-6=\sqrt{2x+y-1}$ and $x^2-x=3y+5$

How many integer pairs $(x,y)$ satisfy the equations
$x+y-6=\sqrt{2x+y-1}$ 
 and $x^2-x=3y+5$

A:

Plugging $$y=\frac{x^2-x-5}{3}$$ in the given equation we get
$$x-6+\frac{x^2-x-5}{3}=\sqrt{2x+\frac{x^2-x-5}{3}-1}$$ squaring and simplifying we obtain
$$x^4+4x^3-45x^2-107x+553=0$$ this equation has no integer Solutions.

