Đáp án:
\[\left( {x;y} \right) = \left\{ {\left( {6;9} \right);\left( {9;6} \right)} \right\}\]
Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
\left\{ \begin{array}{l}
{x^2} + {y^2} - x - y = 102\\
xy + x + y = 69
\end{array} \right.\\
\Leftrightarrow \left( {{x^2} + {y^2} - x - y} \right) + 2\left( {xy + x + y} \right) = 102 + 69.2\\
\Leftrightarrow \left( {{x^2} + {y^2} + 2xy} \right) + \left( {x + y} \right) = 240\\
\Leftrightarrow {\left( {x + y} \right)^2} + \left( {x + y} \right) - 240 = 0\\
\Leftrightarrow \left[ \begin{array}{l}
x + y = 15\\
x + y = - 16
\end{array} \right.\\
TH1:\,\,\,x + y = 15 \Rightarrow \left\{ \begin{array}{l}
y = 15 - x\\
xy = 54
\end{array} \right.\\
\Rightarrow x\left( {15 - x} \right) = 54\\
\Leftrightarrow {x^2} - 15x + 54 = 0\\
\Leftrightarrow \left[ \begin{array}{l}
x = 6 \Rightarrow y = 9\\
x = 9 \Rightarrow y = 6
\end{array} \right.\\
TH2:\,\,\,x + y = - 16 \Rightarrow \left\{ \begin{array}{l}
y = - 16 - x\\
xy = 85
\end{array} \right.\\
\Rightarrow x.\left( { - 15 - x} \right) = 85\\
\Leftrightarrow {x^2} + 15x + 85 = 0\,\,\,\left( {vn} \right)
\end{array}\)
Vậy \(\left( {x;y} \right) = \left\{ {\left( {6;9} \right);\left( {9;6} \right)} \right\}\)
