$$\left\lbrace \begin{array}{l} 2x^2y+3xy=4x^2+9y \\ 7y+6=2x^2+9x \end{array} \right.$$

Đề bài:

Giải hệ phương trình sau:
\begin{equation}
\label{eq:4.I} \left\lbrace \begin{array}{l} 2x^2y+3xy=4x^2+9y \\ 7y+6=2x^2+9x \end{array} \right.
\end{equation}

Lời giải:

\begin{equation}
\label{eq:4.II} \eqref{eq:4.I} \Leftrightarrow \left\lbrace \begin{array}{l} 2x^2.7y+3x.7y-28x^2-9.7y=0 \\ 7y=2x^2+9x-6 \end{array} \right.
\end{equation}
Ta đánh số các phương trình của \eqref{eq:4.II}:
\begin{align}
\label{eq:4.1} 2x^2.7y+3x.7y-28x^2-9.7y=0 \\
\label{eq:4.2} 7y=2x^2+9x-6
\end{align}
Thế \eqref{eq:4.2} vào \eqref{eq:4.1} ta được
\begin{align}
& \phantom{\Leftrightarrow} 2x^2(2x^2+9x-6)+3x(2x^2+9x-6)-28x^2-9(2x^2+9x-6)=0 \nonumber \\
& \Leftrightarrow 4x^4+24x^3-31x^2-99x+54=0 \nonumber \\
& \Leftrightarrow (2x-1)(x+2)(4x^2+18x-54)=0 \nonumber \\
& \Leftrightarrow \left[ \begin{array}{l} x= \dfrac{1}{2} \\ x=-2 \\ 4x^2+18x-54=0 \quad (\Delta'=9^2+4.54=297)\end{array} \right. \nonumber \\
\label{eq:4.4} & \Leftrightarrow \left[ \begin{array}{l} x= \dfrac{1}{2} \\ x=-2 \\ x= \dfrac{-9+3\sqrt{33}}{4} \\ x= \dfrac{-9-3\sqrt{33}}{4} \end{array} \right.
\end{align}
\[\eqref{eq:4.2},\eqref{eq:4.4} \Leftrightarrow \left[ \begin{array}{l} \left\lbrace \begin{array}{l} x= \dfrac{1}{2} \\ y= \dfrac{2x^2+9x-6}{7} \end{array} \right. \\ \left\lbrace \begin{array}{l} x=-2 \\ y= \dfrac{2x^2+9x-6}{7} \end{array} \right. \\ \left\lbrace \begin{array}{l} x= \dfrac{-9+3\sqrt{33}}{4} \\ y= \dfrac{2x^2+9x-6}{7} \end{array} \right. \\ \left\lbrace \begin{array}{l} x= \dfrac{-9-3\sqrt{33}}{4} \\ y= \dfrac{2x^2+9x-6}{7} \end{array} \right. \\ \end{array} \right. \Leftrightarrow \left[ \begin{array}{l} \left\lbrace \begin{array}{l} x= \dfrac{1}{2} \\ y= \dfrac{-1}{7} \end{array} \right. \\ \left\lbrace \begin{array}{l} x=-2 \\ y= \dfrac{-16}{7} \end{array} \right. \\ \left\lbrace \begin{array}{l} x= \dfrac{-9+3\sqrt{33}}{4} \\ y=3 \end{array} \right. \\ \left\lbrace \begin{array}{l} x= \dfrac{-9-3\sqrt{33}}{4} \\ y=3 \end{array} \right. \\ \end{array} \right.\]
\[S=\left\lbrace \left( \dfrac{1}{2};  \dfrac{-1}{7}\right); \left(-2;  \dfrac{-16}{7}\right); \left( \dfrac{-9+ 3\sqrt{33}}{4}; 3\right) ; \left( \dfrac{-9- 3\sqrt{33}}{4}; 3\right)\right\rbrace\]