Решить симметрическую систему уравнений:[latex] \left \{ {{ x^{2} + xy + y^{2} = 3} \atop {xy( x^{2} + y^{2}) = 2 }} \right. [/latex]

[latex] \left \{ {{ x^{2} + xy + y^{2} = 3} \atop {xy( x^{2} + y^{2}) = 2 }} \right. \\\ x^2+y^2=a \\\ xy=b \\\ \left \{ {{ a+b= 3} \atop {ab= 2 }} \right. \\\ a=3-b \\\ (3-b)b=2 \\\ b^2-3b+2=0 \\\ D=9-8=1 \\\ b_1=2 \\\ b_2=1 \\\ a_1=1 \\\ a_2=2[/latex] [latex]x^2+y^2=2 \\\ xy=1 \\\ x= \frac{1}{y} \\\ \frac{1}{y^2} +y^2-2=0 \\\ y^4-2y^2+1=0 \\\ (y^2-1)^2=0 \\\ y^2=1 \\\ y_1=1 \\\ y_2=-1 \\\ x_1=1 \\\ x_2=-1[/latex] [latex]x^2+y^2=1 \\\ xy=2 \\\ x= \frac{2}{y} \\\ \frac{4}{y^2} +y^2-2=0 \\\ 4y^4-2y^2+1=0 \\\ 4z^2-2z+1=0 \\\ D_1=1-4<0[/latex] Ответ: (1; 1); (-1; -1)