Задача 23. Решите систему уравнений: \(\left\{ \begin{array}{l}5{x^2} + {y^2} = 61,\\15{x^2} + 3{y^2} = 61x.\end{array} \right.\)
Ответ
ОТВЕТ: \(\left( {3;\,-4} \right),\,\,\,\;\left( {3;\,4} \right).\)
Решение
Разделим обе части второго уравнения на 3. Тогда система уравнений примет вид:
\(\left\{ \begin{array}{l}5{x^2} + {y^2} = 61,\\5{x^2} + {y^2} = \dfrac{{61x}}{3}\end{array} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left\{ \begin{array}{l}\dfrac{{61x}}{3} = 61,\\5{x^2} + {y^2} = 61\end{array} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left\{ \begin{array}{l}x = 3,\\5{x^2} + {y^2} = 61\end{array} \right.\,\,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,\left\{ \begin{array}{l}x = 3,\\5 \cdot {3^2} + {y^2} = 61\end{array} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left\{ \begin{array}{l}x = 3,\\{y^2} = 16\end{array} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left\{ \begin{array}{l}x = 3,\\\left[ \begin{array}{l}y = -4,\\y = 4\end{array} \right.\end{array} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left[ \begin{array}{l}\left\{ \begin{array}{l}x = 3,\\y = -4,\end{array} \right.\\\left\{ \begin{array}{l}x = 3,\\y = 4.\end{array} \right.\end{array} \right.\)
Ответ: \(\left( {3;\,-4} \right),\,\,\,\;\left( {3;\,4} \right).\)