Question

Решите систему уравнения:

{ X^2 + y^2 = 25
{ xy - x - y = 5

покажите решение , пожалуйста

Answer 1

$left { {{ x^{2} +y^2=25} atop {xy-x-y=5}} right.$ ; $left { {{ x^{2} +y^2-2xy+2xy=25} atop {xy-x-y=5}} right.$ ; $left { {{ (x+y)^2-2xy=25} atop {xy-(x+y)=5}} right.$ Введем замену пусть х+у=u, а ху=v
$left { {{u^2-2v=25} atop {v-u=5}} right.$ ; $left { {{u^2-2(5+u)=25} atop {v=5+u}} right.$ ; $left { {{u^2-10-2u=25} atop {v=5+u}} right.$ ; $left { {{u^2-2u-35=0} atop {v=5+u}} right.$ ; $D=2^2-4*1*35=4+140=144$ ; $u_{1}= frac{2-12}{2}=-5$ ; $u_{2}= frac{2+12}{2}=7$ ;
$left[begin{array}{ccc} left { {{ u_{1} =-5} atop { v_{1} =5+u=5-5=0}} right. \ left { {{ u_{2} =7} atop { v_{2} =5+7=12}} right. end{array}right$ ; $left[begin{array}{ccc} left { {{ x+y =-5} atop { xy =0}} right. \ left { {{x+y=7} atop { xy =12}} right. end{array}right$ ; $left[begin{array}{ccc} left { {{ x=-5-y} atop { y(-5-y) =0}} right. \ left { {{x=7-y} atop { y(7-y) =12}} right. end{array}right$ ;
$left[begin{array}{ccc} left { {{ x_{1} =-5-0=-5} atop { y_{1}=0}} right.\ left { {{ x_{2} =-5+5=0} atop { y_{2} =-5}} right. \ left { {{x=7-y} atop { 7y-y^2 =12}} right.end{array}right$
Отдельно решим уравнение
$y^2-7y+12=0$
$D=7^2-4*1*12=49-48=1$
$y_{3}= frac{7-1}{2}=3$ ; $y_{4}= frac{7+1}{2}=4$
$left[begin{array}{ccc} left { {{ x_{1} =-5} atop { y_{1}=0}} right.\ left { {{ x_{2} =0} atop { y_{2} =-5}} right. \ left { {{x_{3}=7-3}=4 atop { y_{3}=3 }} right.\ left { {{x_{4}=7-4=3} atop {y_{4}=4}} right. end{array}right$
Ответ:{(-5;0), (0;-5), (4;3), (3;4)}