Find the values of the parameter a for which the system has (1) one solution, (2) no solutions, and (3) infinitely many solutions. In case (3), find the solution.
I'm so sorry about the picture, but I need extra help with it. Hope you understand me.
Probably my attempt is not true, can someone help me how to solve it?
If I'm reading your handwriting correctly, it looks like you are studying the system $A\vec{x}=\vec{b}$ where \begin{align*} A &= \left[\begin{array}{rrr} 2 & 2 & a + 3 \\ 2 & 3 & a + 4 \\ 3 & 6 \, a + 5 & 7 \end{array}\right] & \vec{b} &= \left[\begin{array}{r} 8 \\ 12 \\ 20 \end{array}\right] \end{align*} The determinant of $A$ is $$ \det(A)=-15 \, a + 1 $$ So, our system is guaranteed a unique solution if and only if $a\neq\frac{1}{15}$.
If $a=\frac{1}{15}$, then our system can be reduced to $$ \operatorname{rref} \left[\begin{array}{rrr|r} 2 & 2 & \frac{46}{15} & 8 \\ 2 & 3 & \frac{61}{15} & 12 \\ 3 & \frac{27}{5} & 7 & 20 \end{array}\right] =\left[\begin{array}{rrr|r} 1 & 0 & \frac{8}{15} & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] $$ What does this reduction tell us about the solvability of the system in this case?