![\left \{ {{x-5y=9} \atop {x^{2}+3xy-y^{2}=3}} \right \left \{ {{x-5y=9} \atop {x^{2}+3xy-y^{2}=3}} \right](https://tex.z-dn.net/?f=%5Cleft+%5C%7B+%7B%7Bx-5y%3D9%7D+%5Catop+%7Bx%5E%7B2%7D%2B3xy-y%5E%7B2%7D%3D3%7D%7D+%5Cright+)
![\left \{ {{x=5y+9} \atop {x^{2}+3xy-y^{2}=3}} \right \left \{ {{x=5y+9} \atop {x^{2}+3xy-y^{2}=3}} \right](https://tex.z-dn.net/?f=%5Cleft+%5C%7B+%7B%7Bx%3D5y%2B9%7D+%5Catop+%7Bx%5E%7B2%7D%2B3xy-y%5E%7B2%7D%3D3%7D%7D+%5Cright)
![(5y+9)^{2}+3(5y+9)y-y^{2}=3 (5y+9)^{2}+3(5y+9)y-y^{2}=3](https://tex.z-dn.net/?f=%285y%2B9%29%5E%7B2%7D%2B3%285y%2B9%29y-y%5E%7B2%7D%3D3)
![(5y+9)^{2}+3y(5y+9)-y^{2}=3 (5y+9)^{2}+3y(5y+9)-y^{2}=3](https://tex.z-dn.net/?f=%285y%2B9%29%5E%7B2%7D%2B3y%285y%2B9%29-y%5E%7B2%7D%3D3)
![25y^{2}+90y+81+15y^{2}+27y-y^{2}=3 25y^{2}+90y+81+15y^{2}+27y-y^{2}=3](https://tex.z-dn.net/?f=25y%5E%7B2%7D%2B90y%2B81%2B15y%5E%7B2%7D%2B27y-y%5E%7B2%7D%3D3)
![25y^{2}+90y+81+15y^{2}+27y-y^{2}-3=0 25y^{2}+90y+81+15y^{2}+27y-y^{2}-3=0](https://tex.z-dn.net/?f=25y%5E%7B2%7D%2B90y%2B81%2B15y%5E%7B2%7D%2B27y-y%5E%7B2%7D-3%3D0)
![(25y^{2}+15y^{2}-y^{2})+(90y+27y)+(81-3)=0 (25y^{2}+15y^{2}-y^{2})+(90y+27y)+(81-3)=0](https://tex.z-dn.net/?f=%2825y%5E%7B2%7D%2B15y%5E%7B2%7D-y%5E%7B2%7D%29%2B%2890y%2B27y%29%2B%2881-3%29%3D0)
![39y^{2}+117y+78=0 39y^{2}+117y+78=0](https://tex.z-dn.net/?f=39y%5E%7B2%7D%2B117y%2B78%3D0)
![39(y^{2}+3y+2)=0 39(y^{2}+3y+2)=0](https://tex.z-dn.net/?f=39%28y%5E%7B2%7D%2B3y%2B2%29%3D0)
![y^{2}+3y+2=0 y^{2}+3y+2=0](https://tex.z-dn.net/?f=y%5E%7B2%7D%2B3y%2B2%3D0)
Cчитаем дискриминант:
![D=3^{2}-4\cdot1\cdot2=9-8=1 D=3^{2}-4\cdot1\cdot2=9-8=1](https://tex.z-dn.net/?f=D%3D3%5E%7B2%7D-4%5Ccdot1%5Ccdot2%3D9-8%3D1)
Дискриминант положительный
![\sqrt{D}=1 \sqrt{D}=1](https://tex.z-dn.net/?f=%5Csqrt%7BD%7D%3D1)
Уравнение имеет два различных корня:
![y_{1}=\frac{-3+1}{2\cdot1}=\frac{-2}{2}=-1 y_{1}=\frac{-3+1}{2\cdot1}=\frac{-2}{2}=-1](https://tex.z-dn.net/?f=y_%7B1%7D%3D%5Cfrac%7B-3%2B1%7D%7B2%5Ccdot1%7D%3D%5Cfrac%7B-2%7D%7B2%7D%3D-1)
![y_{2}=\frac{-3-1}{2\cdot1}=\frac{-4}{2}=-2 y_{2}=\frac{-3-1}{2\cdot1}=\frac{-4}{2}=-2](https://tex.z-dn.net/?f=y_%7B2%7D%3D%5Cfrac%7B-3-1%7D%7B2%5Ccdot1%7D%3D%5Cfrac%7B-4%7D%7B2%7D%3D-2)
Pешение разбивается на отдельные случаи
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Случай 1
![\left \{ {{x=5y+9} \atop {y=-1}} \right \left \{ {{x=5y+9} \atop {y=-1}} \right](https://tex.z-dn.net/?f=%5Cleft+%5C%7B+%7B%7Bx%3D5y%2B9%7D+%5Catop+%7By%3D-1%7D%7D+%5Cright)
![\left \{ {{x=5\cdot(-1)+9} \atop {y=-1}} \right \left \{ {{x=5\cdot(-1)+9} \atop {y=-1}} \right](https://tex.z-dn.net/?f=%5Cleft+%5C%7B+%7B%7Bx%3D5%5Ccdot%28-1%29%2B9%7D+%5Catop+%7By%3D-1%7D%7D+%5Cright)
![\left \{ {{x=-5+9} \atop {y=-1}} \right \left \{ {{x=-5+9} \atop {y=-1}} \right](https://tex.z-dn.net/?f=%5Cleft+%5C%7B+%7B%7Bx%3D-5%2B9%7D+%5Catop+%7By%3D-1%7D%7D+%5Cright)
![\left \{ {{x=4} \atop {y=-1}} \right \left \{ {{x=4} \atop {y=-1}} \right](https://tex.z-dn.net/?f=%5Cleft+%5C%7B+%7B%7Bx%3D4%7D+%5Catop+%7By%3D-1%7D%7D+%5Cright)
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Случай 2
![\left \{{{x=5y+9} \atop{y=-2}}\right \left \{{{x=5y+9} \atop{y=-2}}\right](https://tex.z-dn.net/?f=%5Cleft+%5C%7B%7B%7Bx%3D5y%2B9%7D+%5Catop%7By%3D-2%7D%7D%5Cright)
![\left \{{{x=5\cdot(-2)+9}\atop{y=-2}}\right \left \{{{x=5\cdot(-2)+9}\atop{y=-2}}\right](https://tex.z-dn.net/?f=%5Cleft+%5C%7B%7B%7Bx%3D5%5Ccdot%28-2%29%2B9%7D%5Catop%7By%3D-2%7D%7D%5Cright)
![\left \{{{x=-10+9}\atop{y=-2}}\right \left \{{{x=-10+9}\atop{y=-2}}\right](https://tex.z-dn.net/?f=%5Cleft+%5C%7B%7B%7Bx%3D-10%2B9%7D%5Catop%7By%3D-2%7D%7D%5Cright)
![\left \{{{x=-1}\atop{y=-2}}\right \left \{{{x=-1}\atop{y=-2}}\right](https://tex.z-dn.net/?f=%5Cleft+%5C%7B%7B%7Bx%3D-1%7D%5Catop%7By%3D-2%7D%7D%5Cright)
Ответ: (4;-1), (-1; -2)