Решите систему уравнений 1/x-1/y=1/6; 5x-y=9.

\[\left\{ \begin{matrix} \frac{1}{x} - \frac{1}{y} = \frac{1}{6}\ \ \ \ | \cdot 6xy \\ 5x - y = 9\ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ \ }\left\{ \begin{matrix} 6y - 6x = xy \\ y = 5x - 9\ \ \ \\ \end{matrix} \right.\ \text{\ \ \ \ \ }\]

\[\left\{ \begin{matrix} (6 - x)(5x - 9) - 6x = 0 \\ y = 5x - 9\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \ \]

\[30x - 5x^{2} - 54 + 9x - 6x = 0\]

\[- 5x^{2} + 33x - 54 = 0\ \ \ \ \ \ |\ :( - 1)\]

\[5x^{2} - 33x + 54 = 0\]

\[D = 1089 - 1080 = 9\]

\[x_{1} = \frac{33 - 3}{10} = 3;\ \ \ \ x_{2} = \frac{33 + 3}{10} = 2,6.\]

\[\left\{ \begin{matrix} x = 3 \\ y = 6 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ }\left\{ \begin{matrix} x = 3,6 \\ y = 9\ \ \ \\ \end{matrix} \right.\ \text{\ \ }\]

\[Ответ:(3;6)\ и\ (3,6;9).\]