Решебник по алгебре 9 класс Мерзляк Задание 451

\[\boxed{\text{451\ (451).\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]

\[1)\left\{ \begin{matrix} y = x + 3\ \ \ \ \ \\ x^{2} - 2y = 9 \\ \end{matrix} \right.\ \text{\ \ \ \ \ }\]

\[\left\{ \begin{matrix} y = x + 3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ x^{2} - 2 \cdot (x + 3) = 9 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} y = x + 3\ \ \ \ \ \ \ \ \ \ \ \\ x^{2} - 2x - 6 = 9 \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} y = x + 3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ x^{2} - 2x - 15 = 0 \\ \end{matrix} \right.\ \]

\[x^{2} - 2x - 15 = 0\]

\[x_{1} + x_{2} = 2,\ \ x_{1} = 5\]

\[x_{1}x_{2} = - 15,\ \ \ \ \ \ \ \ \ \ \ \ x_{2} = - 3\]

\[\left\{ \begin{matrix} y = x + 3 \\ x = 5\ \ \ \ \ \ \ \ \\ x = - 3\ \ \ \ \\ \end{matrix} \right.\ \text{\ \ \ \ \ }\left\{ \begin{matrix} y = 8 \\ x = 5 \\ \end{matrix} \right.\ \text{\ \ \ \ }или\ \ \]

\[\left\{ \begin{matrix} y = 0\ \ \ \ \\ x = - 3 \\ \end{matrix} \right.\ \]

\[Ответ:(5;8);\ ( - 3;0).\]

\[2)\ \left\{ \begin{matrix} x + y = 5 \\ xy = 4\ \ \ \ \ \ \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ }\left\{ \begin{matrix} y = 5 - x\ \ \ \ \ \\ x(5 - x) = 4 \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} y = 5 - x\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ - x^{2} + 5x - 4 = 0 \\ \end{matrix} \right.\ \]

\[- x^{2} + 5x - 4 = 0\]

\[x_{1} + x_{2} = 5,\ \ x_{1} = 4\]

\[x_{1}x_{2} = 4,\ \ \ \ \ \ \ \ \ \ \ \ x_{2} = 1\]

\[\left\{ \begin{matrix} y = 5 - x \\ x = 4\ \ \ \ \ \ \ \ \\ y = 1\ \ \ \ \ \ \ \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\left\{ \begin{matrix} y = 1 \\ x = 4 \\ \end{matrix} \right.\ \text{\ \ \ \ }или\ \ \ \]

\[\left\{ \begin{matrix} y = 4 \\ x = 1 \\ \end{matrix} \right.\ \]

\[Ответ:(4;1);\ (1;4).\]

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

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

\[\left\{ \begin{matrix} y = 2 + x\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ x^{2} - 4x - 2x^{2} - 3 = 0 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} y = 2 + x\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ - x^{2} - 4x - 3 = 0 \\ \end{matrix} \right.\ \]

\[- x^{2} - 4x - 3 = 0\]

\[x_{1} + x_{2} = - 4,\ \ x_{1} = - 1\]

\[x_{1}x_{2} = 3,\ \ \ \ \ \ \ \ \ \ \ \ x_{2} = - 3\]

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

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

\[Ответ:( - 1;1);\ \ ( - 3;\ - 1).\]

\[4)\ \left\{ \begin{matrix} x - 4y = 2\ \ \\ xy + 2y = 8 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} x = 2 + 4y\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ (2 + 4y)y + 2y = 8 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\]

\[\ \left\{ \begin{matrix} x = 2 + 4y\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 2y + 4y^{2} + 2y = 8 \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} x = 2 + 4y\ \ \ \ \ \ \ \ \ \ \ \ \ \\ 4y^{2} + 4y - 8 = 0 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} x = 2 + 4y\ \ \ \ \ \ \ \ \\ y^{2} + y - 2 = 0 \\ \end{matrix} \right.\ \]

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

\[x_{1} + x_{2} = - 1,\ \ x_{1} = - 2\]

\[x_{1}x_{2} = - 2,\ \ \ \ \ \ \ \ \ \ \ \ x_{2} = 1\]

\[\left\{ \begin{matrix} x = 2 + 4y \\ y = - 2\ \ \ \ \ \ \\ y = 1\ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ }\left\{ \begin{matrix} x = - 6 \\ y = - 2 \\ \end{matrix} \right.\ \text{\ \ \ \ \ }или\]

\[\left\{ \begin{matrix} x = 6 \\ y = 1 \\ \end{matrix} \right.\ \]

\[Ответ:( - 6;\ - 2);\ \ (6;1).\]

\[5)\ \left\{ \begin{matrix} xy = 15\ \ \ \ \\ 2x - y = 7 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ }\left\{ \begin{matrix} xy = 15\ \ \ \ \\ y = 2x - 7 \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} x(2x - 7) = 15 \\ y = 2x - 7\ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} 2x^{2} - 7x - 15 = 0 \\ y = 2x - 7\ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]

\[2x^{2} - 7x - 15 = 0\]

\[D = 169\]

\[x_{1} = \frac{7 + 13}{4} = 5\]

\[x_{2} = \frac{7 - 13}{4} = - 1,5\]

\[\left\{ \begin{matrix} y = 2x - 7 \\ x = 5\ \ \ \ \ \ \ \ \ \ \ \\ x = - 1,5\ \ \ \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ }\left\{ \begin{matrix} y = 3 \\ x = 5 \\ \end{matrix} \right.\ \text{\ \ \ \ }или\ \ \ \]

\[\left\{ \begin{matrix} y = - 10 \\ x = - 1,5 \\ \end{matrix} \right.\ \]

\[Ответ:(5;3);\ ( - 1,5;\ - 10).\]

\[6)\ \left\{ \begin{matrix} x - y = 4\ \ \ \\ x^{2} + y^{2} = 8 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\]

\[\ \left\{ \begin{matrix} x = 4 + y\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ (4 + y)^{2} + y^{2} = 8 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} x = 4 + y\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 16 + 8y + y^{2} + y^{2} - 8 = 0 \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} x = 4 + y\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 2y^{2} + 8y + 8 = 0 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} x = 4 + y\ \ \ \ \ \ \ \ \ \ \ \ \\ y^{2} + 4y + 4 = 0 \\ \end{matrix} \right.\ \]

\[y^{2} + 4y + 4 = 0\]

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

\[y = - 2\]

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

\[Ответ:(2;\ - 2).\]