Решите систему уравнений: а) [latex]\begin{cases} x+y+z=9\\\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1\\xy+xz+yz=27 \end{cases} [/latex] б) [latex]\begin{cases} x^2+y^2-2z^2=0\\x+y+z=8\\xy=-z^2\end{cases} [/latex]

[latex]x+y+z=9\\ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1\\ xy+xz+yz=27\\ \\ x+y+z=9\\ \frac{yz+xz+xy}{xyz}=1\\ xy+xz+yz=27\\ \\ x+y=9-z\\ (y+x)z+xy=xyz\\ xy+z(x+y)=27\\ \\ x+y=9-z\\ (9-z)z+xy=xyz\\ xy+z(9-z)=27\\ \\ 9z-z^2=xy(z-1)\\ xy=27-z(9-z)\\ 9z-z^2=(27-z(9-z))(z-1)\\ z=3\\ x+y=6\\ xy+3(x+y)=27\\ xy=9\\ x=3\\ y=3\\ [/latex] Ответ x=y=z=3 2)[latex]x^2+y^2-2z^2=0\\ x+y+z=8\\ xy=-z^2\\ \\ x^2+y^2=2z^2\\ x+y+z=8\\ xy=-z^2\\ \\ x^2+y^2=-2xy\\ (x+y)^2=0\\ x=y\\ x=-8\\ y=8\\ z=8\\ Libo\\ x=8\\ y=-8\\ z=8 [/latex]