Решите систему уравнений 1/x-1/y=1/12; 5x-y=18.

Ответ:

\[\left\{ \begin{matrix} \frac{1}{x} - \frac{1}{y} = \frac{1}{12}\ \ | \cdot 12xy \\ 5x - y = 18\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} y = 5x - 18\ \ \ \ \ \ \ \\ 12y - 12x = xy \\ \end{matrix} \right.\ \ \]

\[\left\{ \begin{matrix} y = 5x - 18\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 12 \cdot (5x - 18) - 12x = x(5x - 18) \\ \end{matrix} \right.\ \ \]

\[60x - 216 - 12x = 5x^{2} - 18x\]

\[5x^{2} - 18x - 48x + 216 = 0\]

\[5x^{2} - 66x + 216 = 0\]

\[D = 1089 - 1080 = 9\]

\[x_{1} = \frac{33 + 3}{5} = \frac{36}{5} = 7,2;\ \ \ \ \]

\[x_{2} = \frac{33 - 3}{5} = 6.\]

\[\left\{ \begin{matrix} x = 7,2 \\ y = 18\ \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ }\left\{ \begin{matrix} x = 6\ \ \\ y = 12 \\ \end{matrix} \right.\ \text{\ \ }\]

\[Ответ:(7,2;18)\ и\ (6;12).\]