Решебник по алгебре 9 класс Мерзляк Задание 468

\[\boxed{\text{468\ (468).}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]

\[1)\ \left\{ \begin{matrix} x^{2} - 12xy + 36y^{2} = 36 \\ x + 6y = 8\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} (x - 6y)^{2} = 36 \\ x + 6y = 8\ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} (x - 6y)^{2} = 36 \\ x = 8 - 6y\ \ \ \ \ \ \ \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} (8 - 12y)^{2} = 36 \\ x = 8 - 6y\ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} 64 - 192y + 144y^{2} - 36 = 0 \\ x = 8 - 6y\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]

\[144y^{2} - 192y + 28 = 0\ \ \ \ \ \ \ |\ :4\]

\[36y^{2} - 48y + 7 = 0\]

\[D = 1296\]

\[y_{1,2} = \frac{48 \pm 36}{72}\]

\[\left\{ \begin{matrix} y = \frac{7}{6} \\ x = 1 \\ \end{matrix} \right.\ \ \ \ или\ \ \ \left\{ \begin{matrix} y = \frac{1}{6} \\ x = 7 \\ \end{matrix} \right.\ \]

\[Ответ:\left( 1;\frac{7}{6} \right),\ \left( 7;\frac{1}{6} \right).\]

\[2)\ \left\{ \begin{matrix} y^{2} - 2xy = 32\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ x^{2} + 6xy + 9y^{2} = 100 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ \ }\]

\[\text{\ \ }\left\{ \begin{matrix} y^{2} - 2xy = 32 \\ (x + 3y)^{2} = 100 \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} y^{2} - 2xy = 32 \\ x + 3y = 10 \\ \end{matrix} \right.\ \ \ \ \ \ \ \ \ \ \ \ \ \ или\ \ \ \ \ \ \ \ \ \ \ \ \]

\[\ \left\{ \begin{matrix} y^{2} - 2xy = 32 \\ x + 3y = - 10 \\ \end{matrix} \right.\ \]

\[3)\ \left\{ \begin{matrix} x^{2} + y^{2} = 25\ \ \ \\ xy = 12\ \ \ \ \ \ \ | \cdot 2 \\ \end{matrix} \right.\ + \ \ \ \ \ \ \ \ \ \ \ \ \]

\[\left\{ \begin{matrix} x^{2} + y^{2} + 2xy = 49 \\ xy = 12\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} (x + y)^{2} = 49 \\ xy = 12\ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} x + y = 7 \\ xy = 12\ \ \\ \end{matrix} \right.\ \ \ \ \ или\ \ \ \ \left\{ \begin{matrix} x + y = - 7 \\ xy = 12\ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} x = 7 - y\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 7y - y^{2} - 12 = 0 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\]

\[\text{\ \ \ }\left\{ \begin{matrix} x = - 7 - y\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ - 7y - y^{2} - 12 = 0 \\ \end{matrix} \right.\ \]

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

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

\[Ответ:(3;4),\ (4;3),\ ( - 3;\ - 4),\ \]

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

\[4)\ \left\{ \begin{matrix} 9x^{2} + y^{2} = 10 \\ xy - 1\ \ \ \ \ \ | \cdot 6\ \ \\ \end{matrix} + \right.\ \text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} 9x^{2} + y^{2} + 6xy = 4 \\ xy = - 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} (3x + y)^{2} = 4 \\ xy = - 1\ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]

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

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

\[\left\{ \begin{matrix} y = 2 - 3x\ \ \ \ \ \ \ \ \ \ \ \ \ \\ 2x - 3x^{2} + 1 = 0 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\]

\[\text{\ \ \ }\left\{ \begin{matrix} y = - 2 - 3x\ \ \ \ \ \ \ \ \ \ \ \ \ \\ - 2x - 3x^{2} + 1 = 0 \\ \end{matrix} \right.\ \]