Решите систему уравнений 9x^2-12xy+4y^2=9; x+2y=9.

Ответ:

\[\left\{ \begin{matrix} 9x^{2} - 12xy + 4y^{2} = 9 \\ x + 2y = 9\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]

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

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

\[(27 - 6y - 2y)^{2} = 9\]

\[(27 - 8y)^{2} = 9\]

\[27 - 8y = 3\ \ \ \ \ \ \ \ \ \ \ \ \ 27 - 8y = - 3\]

\[- 8y = - 24\ \ \ \ \ \ \ \ \ \ \ \ \ - 8y = - 30\]

\[y = 3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ y = \frac{30}{8} = \frac{15}{4} = 3,75\]

\[\left\{ \begin{matrix} y = 3 \\ x = 3 \\ \end{matrix} \right.\ \Longrightarrow \left\{ \begin{matrix} y = 3,75 \\ x = 1,5\ \\ \end{matrix} \right.\ \]

\[Ответ:(3;3);\ \ (1,5;3,75).\]