Question

Решите систему уравнений!) x^2 + xy = 10 y^2 + xy = 15

Answer 1

Добавим и вычтем уравнения друг из друга
$\left \{ {{x^2+2xy+y^2=10+15} \atop {x^2-y^2=-5}} \right.$

$\left \{ {{(x+y)^2-5^2=0} \atop {(x-y)(x+y)=-5}} \right. \left \{ {{(x+y-5)(x+y+5)=0} \atop {(x-y)(x+y)=-5}} \right.$
Два случая:
1)
$\left \{ {{x+y-5=0} \atop {(x-y)(x+y)=-5}} \right. \left \{ {{x+y=5} \atop {(x-y)*5=-5}} \right. \left \{ {{x+y=5} \atop {x-y=-1}} \right. \left \{ {{2x=5-1} \atop {y=5-x}} \right. \left \{ {{x=2} \atop {y=3}} \right.$

2)

$\left \{ {{x+y+5=0} \atop {(x-y)(x+y)=-5}} \right. \left \{ {{x+y=-5} \atop {(x-y)*(-5)=-5}} \right. \left \{ {{x+y=-5} \atop {x-y=1}} \right. \left \{ {{2x=-5-1} \atop {y=-5-x}} \right. \left \{ {{x=-3} \atop {y=-2}} \right.$

Ответ: (2;3) и (-3;-2).