29А. При каких значениях параметра a система уравнений \(\left\{ {\begin{array}{*{20}{c}} {3x-6y = 1,} \\ {5x-a\,y = 2} \end{array}} \right.\) имеет решения \(x < 0,\,\,\,y < 0\)?
ОТВЕТ: \(a \in \left( {10;12} \right).\)
\(\left\{ {\begin{array}{*{20}{c}}{3x-6y = 1,}\\{5x-a\,y = 2}\end{array}} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{x = \frac{{6y + 1}}{3},\,\,\,\,\,\,\,\,\,\,\,\,}\\{\frac{{30y + 5}}{3}-ay = 2}\end{array}} \right.\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{x = \frac{{6y + 1}}{3},\,\,}\\{y = \frac{1}{{30-3a}}}\end{array}} \right.\,\,\,\, \Leftrightarrow \) \( \Leftrightarrow \,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{x = \frac{1}{3}\left( {\frac{6}{{30-3a}} + 1} \right),}\\{y = \frac{1}{{30-3a}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\end{array}} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{x = \frac{{12-a}}{{30-3a}},}\\{y = \frac{1}{{30-3a}}.\,}\end{array}} \right.\) По условию \(x < 0,\,\,\,\,y < 0,\) поэтому: \(\left\{ {\begin{array}{*{20}{c}}{\frac{{12-a}}{{30-3a}} < 0,}\\{\frac{1}{{30-3a}} < 0}\end{array}\,} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{a \in \left( {10;12} \right),}\\{a \in \left( {10;\infty } \right)\,\,\,}\end{array}} \right.\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,a \in \left( {10;12} \right).\) Ответ: \(\left( {10;12} \right)\).