Решите систему уравнений 1/y-1/x=1/12; 2x-y=18.

Ответ:

\[\left\{ \begin{matrix} \frac{1}{y} - \frac{1}{x} = \frac{1}{12}\ \ \ | \cdot 12xy \\ 2x - y = 18\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} 12x - 12y = xy \\ y = 2x - 18\ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \ \]

\[\left\{ \begin{matrix} 12x - 12 \cdot (2x - 18) = (2x - 18)x \\ y = 2x - 18\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \ \]

\[12x - 24x + 216 = 2x^{2} - 18x\]

\[2x^{2} - 18x + 12x - 216 = 0\]

\[2x^{2} - 6x - 216 = 0\ \ \ \ \ \ \ |\ :2\]

\[x^{2} - 3x - 108 = 0\]

\[D = 9 + 432 = 441\]

\[x_{1} = \frac{3 + 21}{2} = 12;\ \ \ \ \]

\[x_{2} = \frac{3 - 21}{2} = - 9\]

\[\left\{ \begin{matrix} x = 12 \\ y = 6\ \ \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ }\left\{ \begin{matrix} x = - 9\ \ \\ y = - 36 \\ \end{matrix} \right.\ \text{\ \ }\]

\[Ответ:(12;6);\ \ ( - 9;\ - 36).\]