Решите систему уравнений 1/x+1/y=1/2; 3x-y=3.

Ответ:

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

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

\[\left\{ \begin{matrix} y = 3x - 3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 2(3x - 3) + 2x = (3x - 3)x \\ \end{matrix} \right.\ \ \]

\[6x - 6 + 2x = 3x^{2} - 3x\]

\[3x^{2} - 3x - 8x + 6 = 0\]

\[3x^{2} - 11x + 6 = 0\]

\[D = 121 - 72 = 49\]

\[x_{1} = \frac{11 + 7}{6} = 3;\ \ \ \ x_{2} = \frac{11 - 7}{6} = \frac{4}{6} = \frac{2}{3}\]

\[\left\{ \begin{matrix} x = 3 \\ y = 6 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ }\left\{ \begin{matrix} x = \frac{2}{3}\text{\ \ \ } \\ y = - 1 \\ \end{matrix} \right.\ \text{\ \ }\]

\[Ответ:(3;6);\left( \frac{2}{3}; - 1 \right).\]