 Solution

The last equation gives $x=1/2-y$, and the first then gives $y+3z=3/2$. Together with the middle equation this reduces the consistency question to $(1-3\alpha )z=\beta -3\alpha /2$.

If $\alpha \ne 1/3$, the linear system has the unique solutionIf $\alpha =1/3$ and $\beta =1/2$, the middle equation repeats one third of $y+3z=3/2$, giving the affine line of solutionsIf $\alpha =1/3$ and $\beta \ne 1/2$, the two equations contradict each other and there is no solution. These are three possible types of solution set, rather than three individual solutions for one fixed choice of parameters.