Найдите все значения параметра a, при каждом из которых система уравнений (a-1)x-4y=a+5; x-(a+1)y=3 не имеет решений.

\[\left\{ \begin{matrix} (a + 1)x - 4y = a + 5\ \ \ \ \\ x - (a + 1)y = 3\ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]

\[1)\ a \neq 0:\ \ \ \]

\[\ \frac{a + 1}{1} = \frac{4}{a + 1} \neq \frac{a + 5}{3}\]

\[\frac{a + 1}{1} = \frac{4}{a + 1}\]

\[(a + 1)^{2} = 4\]

\[(a + 1)^{2} - 4 = 0\]

\[(a + 1 - 2)(a + 1 + 2) = 0\]

\[(a - 1)(a + 3) = 0\]

\[a_{1} = 1;\ \ \ \ a_{2} = - 3.\]

\[\frac{a + 1}{1} \neq \frac{a + 5}{3}\text{\ \ \ \ \ }\]

\[a = 1:\ \ \ \ \]

\[\frac{1 + 1}{1} \neq \frac{1 + 5}{3}\text{\ \ \ \ }\]

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

\[2 \neq 2.\]

\[a = - 3:\ \ \]

\[\frac{- 3 + 1}{1} \neq \frac{- 3 + 5}{3}\text{\ \ \ \ }\]

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

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

\[Нет\ решений\ при\ a = - 3.\]

\[y + 3 - 4y = 5\]

\[- 3y = 2\]

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

\[x = - \frac{2}{3} + 3 = 2\frac{1}{3}\]

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

\[Ответ:\ \ \ a = - 3.\]