Найдите все значения параметра a, при каждом из которых система уравнений (a-1)x-2y=3; 2x-(a+2)y=a+4 не имеет решений.

\[\left\{ \begin{matrix} (a - 1)x - 2y = 3\ \ \ \ \ \ \ \ \\ 2x - (a + 2)y = a + 4 \\ \end{matrix} \right.\ \]

\[\frac{a - 1}{2} = \frac{2}{a + 2} \neq \frac{3}{a + 4}\]

\[\frac{a - 1}{2} = \frac{2}{a + 2}\]

\[(a - 1)(a + 2) = 4\]

\[a^{2} + a - 2 = 4\]

\[a^{2} + a - 6 = 0\]

\[D = 1^{2} - 4 \cdot 1 \cdot ( - 6) = 1 + 24 =\]

\[= 25\]

\[a_{1} = \frac{- 1 + \sqrt{25}}{2} = \frac{- 1 + 5}{2} = \frac{4}{2} =\]

\[= 2\]

\[a_{2} = \frac{- 1 - \sqrt{25}}{2} = \frac{- 1 - 5}{2} =\]

\[= \frac{- 6}{2} = - 3\]

\[\frac{a - 1}{2} \neq \frac{3}{a + 4}\]

\[a_{1} = 2:\]

\[\frac{2 - 1}{2} \neq \frac{3}{2 + 4}\]

\[\frac{1}{2} \neq \frac{3}{6}\]

\[\frac{1}{2} \neq \frac{1}{2}.\]

\[a_{2} = - 3:\]

\[\frac{- 3 - 1}{2} \neq \frac{3}{- 3 + 4}\]

\[\frac{- 4}{2} \neq \frac{3}{1}\]

\[- 2 \neq 3\]

\[Нет\ решения\ при\ a = - 3.\]

\[2)\ \left\{ \begin{matrix} - x - 2y = 3\ \ \ \ (1) \\ 2x - 2y = 4\ \ \ \ \ (2) \\ \end{matrix} \right.\ \]

\[(2) - (1):\ \ \ \ 3x = 1\]

\[\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x = \frac{1}{3}\]

\[\left\{ \begin{matrix} x = \frac{1}{3}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ } \\ - x - 2y = 3 \\ \end{matrix} \right.\ \]

\[- \frac{1}{3} - 2y = 3\]

\[2y = - \frac{1}{3} - 3\]

\[2y = - 3\frac{1}{3}\]

\[2y = - \frac{10}{3}\]

\[y = - \frac{5}{3} = - 1\frac{2}{3}\]

\[\left( \frac{1}{3};\ - 1\frac{2}{3} \right) - решение.\]

\[Ответ:a = - 3.\]