Решебник по алгебре 9 класс Мерзляк Задание 689

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\[1)\left\{ \begin{matrix} 2x - y = 13\ \ \\ x^{2} - y^{2} = 23 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} y = 2x - 13\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ x^{2} - (2x - 13)^{2} = 23 \\ \end{matrix} \right.\ \]

\[x^{2} - 4x^{2} + 52x - 169 - 23 = 0\]

\[- 3x^{2} + 52x - 192 = 0\]

\[3x^{2} - 52x + 192 = 0\]

\[D = 2704 - 2304 = 400\]

\[x_{1} = \frac{52 - 20}{6} = \frac{16}{3}\]

\[x_{2} = \frac{52 + 20}{6} = 12\]

\[\left\{ \begin{matrix} x = \frac{16}{3}\text{\ \ \ } \\ y = - \frac{7}{3}\ \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }или\ \ \ \ \ \ \left\{ \begin{matrix} x = 12 \\ y = 11 \\ \end{matrix} \right.\ \]

\[Ответ:\left( \frac{16}{3};\ - \frac{7}{3} \right);\ \ (12;11).\]

\[2)\ \left\{ \begin{matrix} 2x^{2} - y^{2} = 23 \\ 2x^{2} + y^{2} = 41 \\ \end{matrix} \right.\ \text{\ \ }( + )\text{\ \ \ \ \ }\]

\[\left\{ \begin{matrix} 4x^{2} = 64\ \ \ \ \ \ \ \ \ \ \\ 2x^{2} - y^{2} = 23 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} x^{2} = 16\ \ \ \ \ \ \ \ \ \ \ \\ 2x^{2} - y^{2} = 23 \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} x = \pm 4\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ y² = 2 \cdot 16 - 23 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ }\left\{ \begin{matrix} x = \pm 4 \\ y = \pm 3 \\ \end{matrix} \right.\ \]

\[{\left\{ \begin{matrix} x = 4 \\ y = 3\ \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }или\ \ \ \ \ \ \left\{ \begin{matrix} x = 4\ \ \ \\ y = - 3 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ } }{или\ \ \ \ \ \ \left\{ \begin{matrix} x = - 4 \\ y = 3\ \ \ \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ }или\ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} x = - 4 \\ y = - 3 \\ \end{matrix} \right.\ \]

\[Ответ:(4;3);\ \ (4;\ - 3);\ \ \]

\[( - 4;3);\ \ ( - 4;\ - 3).\]