Решебник по алгебре и начала математического анализа 11 класс Колягин Задание 941

\[1)\ \left\{ \begin{matrix} x^{2} - y^{2} = 13 \\ x - y = 1\ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]

\[x - y = 1\]

\[x = 1 + y.\]

\[x^{2} - y^{2} = 13\]

\[(1 + y)^{2} - y^{2} - 13 = 0\]

\[1 + 2y + y^{2} - y^{2} - 13 = 0\]

\[2y - 12 = 0\]

\[2y = 12\]

\[y = 6;\]

\[x = 1 + 6 = 7.\]

\[Ответ:\ \ (7;\ 6).\]

\[2)\ \left\{ \begin{matrix} x^{2} - 3y = - 5 \\ 7x + 3y = 23\ \\ \end{matrix} \right.\ \]

\[7x + 3y = 23\]

\[3y = 23 - 7x\]

\[y = \frac{23 - 7x}{3}.\]

\[x^{2} - 3y = - 5\]

\[x^{2} - (23 - 7x) = - 5\]

\[x^{2} + 7x - 18 = 0\]

\[D = 49 + 72 = 121\]

\[x_{1} = \frac{- 7 - 11}{2} = - 9;\]

\[x_{2} = \frac{- 7 + 11}{2} = 2;\]

\[y_{1} = \frac{23 + 63}{3} = 28\frac{2}{3};\]

\[y_{2} = \frac{23 - 14}{3} = 3.\]

\[Ответ:\ \ \left( - 9;\ 28\frac{2}{3} \right);\ (2;\ 3).\]