ГДЗ по алгебре 9 класс Макарычев Задание 429

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Год:2020-2021-2022
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Задание 429

\[\boxed{\text{429\ (429).}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]

Пояснение.

Решение.

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

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

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

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

\[y_{1} + y_{2} = 1;\ \ y_{1} \cdot y_{2} = - 2\]

\[y_{1} = - 1;\ \ \ \ \ \ \ \ \ \ y_{2} = 2.\]

\[\left\{ \begin{matrix} y_{1} = - 1 \\ x_{1} = 2\ \ \ \\ \end{matrix} \right.\ \ \ \ \ или\ \ \left\{ \begin{matrix} y_{2} = 2 \\ x_{2} = 5 \\ \end{matrix} \right.\ \]

\[Ответ:\ \ (2;\ - 1);(5;2).\]

\[\textbf{б)}\ \left\{ \begin{matrix} y = x - 1\ \ \ \ \ \\ x^{2} - 2y = 26 \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} y = x - 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ x^{2} - 2 \cdot (x - 1) = 26 \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} y = x - 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ x^{2} - 2x - 24 = 0 \\ \end{matrix} \right.\ \]

\[x^{2} - 2x - 24 = 0\]

\[D_{1} = 1 + 24 = 25\]

\[x_{1} = 1 + 5 = ;\ \ \ x_{2} = 1 - 5 = - 4.\]

\[\ \left\{ \begin{matrix} x_{1} = 6 \\ y_{1} = 5 \\ \end{matrix} \right.\ \ \ \ или\ \ \left\{ \begin{matrix} x_{2} = - 4 \\ y_{2} = - 5 \\ \end{matrix} \right.\ .\]

\[Ответ:(6;5);( - 4;\ - 5).\]

\[\textbf{в)}\ \left\{ \begin{matrix} xy + x = - 4 \\ x - y = 6\ \ \ \ \ \ \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} y = x - 6\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ x(x - 6) + x = - 4 \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} y = x - 6\ \ \ \ \ \ \ \ \ \ \ \ \\ x^{2} - 5x + 4 = 0 \\ \end{matrix} \right.\ \]

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

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

\[x_{1} = 1;\ \ \ \ x_{2} = 4.\]

\[\left\{ \begin{matrix} x_{1} = 1\ \ \ \\ y_{1} = - 5 \\ \end{matrix} \right.\ \ \ \ \ или\ \ \ \left\{ \begin{matrix} x_{2} = 4\ \ \ \ \\ y_{2} = - 2 \\ \end{matrix} \right.\ .\]

\[Ответ:(1;\ - 5);(4;\ - 2).\]

\[\textbf{г)}\ \left\{ \begin{matrix} x + y = 9\ \ \ \\ y^{2} + x = 29 \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} x = 9 - y\ \ \ \ \ \ \ \ \ \ \ \ \\ y^{2} + 9 - y = 29 \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} x = 9 - y\ \ \ \ \ \ \ \ \ \ \ \ \\ y^{2} - y - 20 = 0 \\ \end{matrix} \right.\ \]

\[y^{2} - y - 20 = 0\]

\[y_{1} + y_{2} = 1;\ \ \ y_{1} \cdot y_{2} = - 20\]

\[y_{1} = 5;\ \ \ y_{2} = - 4.\]

\[\left\{ \begin{matrix} y_{1} = 5 \\ x_{1} = 4 \\ \end{matrix} \right.\ \ \ или\ \ \left\{ \begin{matrix} y_{2} = - 4 \\ x_{2} = 13. \\ \end{matrix} \right.\ \]

\[Ответ:(4;5);\ \ (13;\ - 4).\]

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