Tìm hai số hữu tỉ a và b, sao cho: a - b = 2( a + b ) = a : b

\[\begin{array}{l}
a - b = 2(a + b) = ab = t\\
 \Rightarrow \frac{a}{b} = t\\
 \Rightarrow a = bt\\
 \Rightarrow \left[ \begin{array}{l}
a - b = bt - b = b(t - 1)\\
2(a + b) = 2(bt + b) = b(2t + 1)
\end{array} \right.\\
 \Rightarrow b(t - 1) = b(2t + 1)\\
 \Leftrightarrow t - 1 = 2t + 1\\
 \Leftrightarrow t =  - 2\\
 \Leftrightarrow \left[ \begin{array}{l}
a - b =  - 2\\
\frac{a}{b} =  - 2
\end{array} \right.\,\,\,\,\, \Leftrightarrow \left[ \begin{array}{l}
a =  - 2b\\
 - 2b - b =  - 2
\end{array} \right.\\
 \Leftrightarrow \left[ \begin{array}{l}
a =  - 2b\\
b = \frac{2}{3}
\end{array} \right.\,\,\,\,\,\,\,\,\,\,\, \Leftrightarrow \left[ \begin{array}{l}
a = \frac{{ - 4}}{3}\\
b = \frac{2}{3}
\end{array} \right.\\
\\

\end{array}\]                 

Đáp án:

Giải thích các bước giải: \(\begin{array}{l} a - b = 2\left( {a + b} \right) = \frac{a}{b} = t\\ \Rightarrow \frac{a}{b} = t \Rightarrow a = bt\\ \Rightarrow \left\{ \begin{array}{l} a - b = bt - b = b\left( {t - 1} \right)\\ 2\left( {a + b} \right) = 2\left( {bt + b} \right) = b\left( {2t + 1} \right) \end{array} \right.\\ \Rightarrow b\left( {t - 1} \right) = b\left( {2t + 1} \right)\\ \Leftrightarrow t - 1 = 2t + 1\\ \Leftrightarrow t = - 2\\ \Leftrightarrow \left\{ \begin{array}{l} a - b = - 2\\ \frac{a}{b} = - 2 \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} a = - 2b\\ - 2b - b = - 2 \end{array} \right.\\ \Leftrightarrow \left\{ \begin{array}{l} a = - 2b\\ b = \frac{2}{3} \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} a = \frac{{ - 4}}{3}\\ b = \frac{2}{3} \end{array} \right. \end{array}\)