$\left \{ {{(x-3)(y+2)=xy-5} \atop {(x+2)(y-3)=xy}} \right.$
⇔$\left \{ {{xy+2x-3y-6=xy-5} \atop {xy-3x+2y-6=xy}} \right.$
⇔$\left \{ {{2x-3y=1} \atop {-3x+2y=6}} \right.$
⇔$\left \{ {{6x-9y=3} \atop {-6x+4y=12}} \right.$
⇔$\left \{ {{-5y=15} \atop {6x+4y=12}} \right.$
⇔$\left \{ {{y=-3} \atop {x=-4}} \right.$