Задача 26. Решите систему уравнений: \(\left\{ \begin{array}{l}3{x^2} + 2{y^2} = 45,\\9{x^2} + 6{y^2} = 45x.\end{array} \right.\)
Ответ
ОТВЕТ: \(\left( {3;\,-3} \right),\,\,\,\;\left( {3;\,3} \right).\)
Решение
Разделим обе части второго уравнения на 3. Тогда система уравнений примет вид:
\(\left\{ \begin{array}{l}3{x^2} + 2{y^2} = 45,\\3{x^2} + 2{y^2} = 15x\end{array} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left\{ \begin{array}{l}15x = 45,\\3{x^2} + 2{y^2} = 45\end{array} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left\{ \begin{array}{l}x = 3,\\3{x^2} + 2{y^2} = 45\end{array} \right.\,\,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,\left\{ \begin{array}{l}x = 3,\\3 \cdot {3^2} + 2{y^2} = 45\end{array} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left\{ \begin{array}{l}x = 3,\\{y^2} = 9\end{array} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left\{ \begin{array}{l}x = 3,\\\left[ \begin{array}{l}y = -3,\\y = 3\end{array} \right.\end{array} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left[ \begin{array}{l}\left\{ \begin{array}{l}x = 3,\\y = -3,\end{array} \right.\\\left\{ \begin{array}{l}x = 3,\\y = 3.\end{array} \right.\end{array} \right.\)
Ответ: \(\left( {3;\,-3} \right),\,\,\,\;\left( {3;\,3} \right).\)