Решите пожалуйста систему методом замены переменной:) с подробностями
xy+2x+2y=5
x^2+y^2+3x+3y=8
буду благодарна:)
Ответы
Ответ дал:
1
Пусть xy = v; x+y=u, получаем
![\left \{ {{v+2u-5=0} \atop {u^2-2v+3u-8=0}} \right. =\ \textgreater \ \left \{ {{v=5-2u} \atop {u^2-2(5-2u)+3u-8=0}} \right. \\ u^2-10+4u+3u-8=0\\ u^2+7u-18=0\\ u_1=-9;\,\,\,v_1=23\\ u_2=2;\,\,\,v_2=1 \left \{ {{v+2u-5=0} \atop {u^2-2v+3u-8=0}} \right. =\ \textgreater \ \left \{ {{v=5-2u} \atop {u^2-2(5-2u)+3u-8=0}} \right. \\ u^2-10+4u+3u-8=0\\ u^2+7u-18=0\\ u_1=-9;\,\,\,v_1=23\\ u_2=2;\,\,\,v_2=1](https://tex.z-dn.net/?f=+%5Cleft+%5C%7B+%7B%7Bv%2B2u-5%3D0%7D+%5Catop+%7Bu%5E2-2v%2B3u-8%3D0%7D%7D+%5Cright.+%3D%5C+%5Ctextgreater+%5C++%5Cleft+%5C%7B+%7B%7Bv%3D5-2u%7D+%5Catop+%7Bu%5E2-2%285-2u%29%2B3u-8%3D0%7D%7D+%5Cright.+%5C%5C+u%5E2-10%2B4u%2B3u-8%3D0%5C%5C+u%5E2%2B7u-18%3D0%5C%5C+u_1%3D-9%3B%5C%2C%5C%2C%5C%2Cv_1%3D23%5C%5C+u_2%3D2%3B%5C%2C%5C%2C%5C%2Cv_2%3D1)
Возвращаемся к замене
![\left \{ {{xy=23} \atop {x+y=-9}} \right. =\ \textgreater \ \left \{ {{y(-y-9)=23} \atop {x=-y-9}} \right. \\ -y^2-9y=23\\ y^2+9y+23=0 \\ D=9^2-4\cdot 23\ \textless \ 0 \left \{ {{xy=23} \atop {x+y=-9}} \right. =\ \textgreater \ \left \{ {{y(-y-9)=23} \atop {x=-y-9}} \right. \\ -y^2-9y=23\\ y^2+9y+23=0 \\ D=9^2-4\cdot 23\ \textless \ 0](https://tex.z-dn.net/?f=+%5Cleft+%5C%7B+%7B%7Bxy%3D23%7D+%5Catop+%7Bx%2By%3D-9%7D%7D+%5Cright.+%3D%5C+%5Ctextgreater+%5C++%5Cleft+%5C%7B+%7B%7By%28-y-9%29%3D23%7D+%5Catop+%7Bx%3D-y-9%7D%7D+%5Cright.++%5C%5C+-y%5E2-9y%3D23%5C%5C+y%5E2%2B9y%2B23%3D0+%5C%5C+D%3D9%5E2-4%5Ccdot+23%5C+%5Ctextless+%5C+0)
D<0, значит уравнение корней не имеет
![\left \{ {{xy=1} \atop {x+y=2}} \right. =\ \textgreater \ \left \{ {{y(2-y)=1} \atop {x=2-y}} \right. \\ 2y-y^2=1\\ y^2-2y+1=0\\ (y-1)^2=0\\ y=1 \\ x=1 \left \{ {{xy=1} \atop {x+y=2}} \right. =\ \textgreater \ \left \{ {{y(2-y)=1} \atop {x=2-y}} \right. \\ 2y-y^2=1\\ y^2-2y+1=0\\ (y-1)^2=0\\ y=1 \\ x=1](https://tex.z-dn.net/?f=+%5Cleft+%5C%7B+%7B%7Bxy%3D1%7D+%5Catop+%7Bx%2By%3D2%7D%7D+%5Cright.+%3D%5C+%5Ctextgreater+%5C++%5Cleft+%5C%7B+%7B%7By%282-y%29%3D1%7D+%5Catop+%7Bx%3D2-y%7D%7D+%5Cright.+%5C%5C+2y-y%5E2%3D1%5C%5C+y%5E2-2y%2B1%3D0%5C%5C+%28y-1%29%5E2%3D0%5C%5C+y%3D1+%5C%5C+x%3D1)
Ответ: (1;1)
Возвращаемся к замене
D<0, значит уравнение корней не имеет
Ответ: (1;1)
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