Задача 29. Решите систему уравнений: \(\left\{ \begin{array}{l}2{x^2} + 4{y^2} = 24,\\4{x^2} + 8{y^2} = 24x.\end{array} \right.\)
Ответ
ОТВЕТ: \(\left( {2;\,-2} \right),\,\,\,\;\left( {2;\,2} \right).\)
Решение
Разделим обе части второго уравнения на 2. Тогда система уравнений примет вид:
\(\left\{ \begin{array}{l}2{x^2} + 4{y^2} = 24,\\2{x^2} + 4{y^2} = 12x\end{array} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left\{ \begin{array}{l}12x = 24,\\2{x^2} + 4{y^2} = 24\end{array} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left\{ \begin{array}{l}x = 2,\\2{x^2} + 4{y^2} = 24\end{array} \right.\,\,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,\left\{ \begin{array}{l}x = 2,\\2 \cdot {2^2} + 4{y^2} = 24\end{array} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left\{ \begin{array}{l}x = 2,\\{y^2} = 4\end{array} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left\{ \begin{array}{l}x = 2,\\\left[ \begin{array}{l}y = -2,\\y = 2\end{array} \right.\end{array} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left[ \begin{array}{l}\left\{ \begin{array}{l}x = 2,\\y = -2,\end{array} \right.\\\left\{ \begin{array}{l}x = 2,\\y = 2.\end{array} \right.\end{array} \right.\)
Ответ: \(\left( {2;\,-2} \right),\,\,\,\;\left( {2;\,2} \right).\)