Задача 24. Решите систему уравнений: \(\left\{ \begin{array}{l}{x^2} + 3{y^2} = 31,\\2{x^2} + 6{y^2} = 31x.\end{array} \right.\)
Ответ
ОТВЕТ: \(\left( {2;\,-3} \right),\,\,\,\;\left( {2;\,3} \right).\)
Решение
Разделим обе части второго уравнения на 2. Тогда система уравнений примет вид:
\(\left\{ \begin{array}{l}{x^2} + 3{y^2} = 31,\\{x^2} + 3{y^2} = \dfrac{{31x}}{2}\end{array} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left\{ \begin{array}{l}\dfrac{{31x}}{2} = 31,\\{x^2} + 3{y^2} = 31\end{array} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left\{ \begin{array}{l}x = 2,\\{x^2} + 3{y^2} = 31\end{array} \right.\,\,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,\left\{ \begin{array}{l}x = 2,\\{2^2} + 3{y^2} = 31\end{array} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left\{ \begin{array}{l}x = 2,\\{y^2} = 9\end{array} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left\{ \begin{array}{l}x = 2,\\\left[ \begin{array}{l}y = -3,\\y = 3\end{array} \right.\end{array} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left[ \begin{array}{l}\left\{ \begin{array}{l}x = 2,\\y = -3,\end{array} \right.\\\left\{ \begin{array}{l}x = 2,\\y = 3.\end{array} \right.\end{array} \right.\)
Ответ: \(\left( {2;\,-3} \right),\,\,\,\;\left( {2;\,3} \right).\)