Решите систему уравнений: (x+3y)/(2x-y)+(6*(2x-y))/(x+3y)=5; x^2-xy-y^2=1.

Ответ:

\[\left\{ \begin{matrix} \frac{x + 3y}{2x - y} + \frac{6 \cdot (2x - y)}{x + 3y} = 5 \\ x^{2} - xy - y^{2} = 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix}\text{\ \ \ \ \ \ } \right.\ \]

\[Пусть\ \ \ \ \ \ \frac{x + 3y}{2x - y} = t:\]

\[t + \frac{6}{t} - 5 = 0\ \ \ \ \ \ | \cdot t\]

\[t^{2} - 5t + 6 = 0\]

\[t_{1} + t_{2} = 5,\ \ t_{1} \cdot t_{2} = 6\]

\[t = 2,\ \ t = 3\]

\[1.\ \left\{ \begin{matrix} \frac{x + 3y}{2x - y} = 2\ \ \ \ \ \ \ \ \ \ \ \\ x^{2} - xy - y^{2} = 1 \\ \end{matrix}\text{\ \ \ } \right.\ \ \]

\[\left\{ \begin{matrix} 4x - 2y = x + 3y \\ x^{2} - xy - y^{2} = 1 \\ \end{matrix}\text{\ \ \ \ } \right.\ \text{\ \ \ }\]

\[\left\{ \begin{matrix} 3x - 5y = 0\ \ \ \ \ \ \ \ \ \\ x^{2} - xy - y^{2} = 1 \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} x = \frac{5y}{3}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\ \frac{25y^{2}}{9} - \frac{5y^{2}}{3} - y^{2} - 1 = 0 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} 25y^{2} - 15y^{2} - 9y^{2} - 9 = 0 \\ x = \frac{5y}{3}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\ \end{matrix}\text{\ \ } \right.\ \]

\[\left\{ \begin{matrix} y^{2} = 9 \\ x = \frac{5y}{3} \\ \end{matrix}\text{\ \ \ \ \ \ \ \ } \right.\ \]

\[\left\{ \begin{matrix} y = 3 \\ x = 5 \\ \end{matrix}\text{\ \ \ \ \ } \right.\ \]

\[\left\{ \begin{matrix} y = - 3 \\ x = - 5 \\ \end{matrix} \right.\ \]

\[2.\ \left\{ \begin{matrix} \frac{x + 3y}{2x - y} = 3\ \ \ \ \ \ \ \ \ \ \ \ \ \\ x^{2} - xy - y^{2} = 1 \\ \end{matrix}\text{\ \ \ \ \ \ \ \ \ \ } \right.\ \]

\[\left\{ \begin{matrix} x = \frac{6y}{5}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\ \frac{36y^{2}}{25} - \frac{6y^{2}}{5} - y^{2} = 1 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} 36y^{2} - 30y^{2} - 25y^{2} = 25 \\ x = \frac{6y}{5}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\ \end{matrix}\text{\ \ \ \ \ \ \ } \right.\ \]

\[\left\{ \begin{matrix} - 19y^{2} = 25 \\ x = \frac{6y}{5}\text{\ \ \ \ \ \ \ \ \ \ \ } \\ \end{matrix}\text{\ \ \ \ } \right.\ \text{\ \ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} y^{2} = - \frac{25}{19} \\ x = \frac{6y}{5}\text{\ \ \ \ \ \ } \\ \end{matrix} \right.\ \ \ \ \ \Longrightarrow нет\ корней.\]

\[Ответ:(5;3);\ \ ( - 5;\ - 3)\text{.\ }\]