Найдите все значения параметра a, при каждом из которых система уравнений (a-1)x+6y=a+7; x+ay=3 не имеет решений.

\[\left\{ \begin{matrix} (a + 1)x + 6y = a + 7\ \\ x + ay = 3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]

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

\[\ \frac{a + 1}{1} = \frac{6}{a} \neq \frac{a + 7}{3}\]

\[\frac{a + 1}{1} = \frac{6}{a}\]

\[a(a + 1) = 6\]

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

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

\[= 25\]

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

\[= 2\]

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

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

\[\frac{a + 1}{1} \neq \frac{a + 7}{3}\text{\ \ \ \ \ }\]

\[a = 2:\ \ \ \]

\[\frac{2 + 1}{1} \neq \frac{2 + 7}{3}\text{\ \ \ }\]

\[\frac{3}{1} \neq \frac{9}{3}\text{\ \ \ }\]

\[3 \neq 3.\]

\[a = - 3:\ \ \ \]

\[\frac{- 3 + 1}{1} \neq \frac{- 3 + 7}{3}\text{\ \ \ \ \ }\]

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

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

\[Нет\ решений\ при\ a = - 3.\]

\[2)\ a = 0:\ \ \ \ \ \left\{ \begin{matrix} x + 6y = 7 \\ x = 3\ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]

\[3 + 6y = 7\]

\[6y = 4\]

\[y = \frac{4}{6} = \frac{2}{3}.\]

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

\[Ответ:\ \ a = - 3.\]