For the system of linear equations \(\begin{aligned}& x+y+z=6 \\& \alpha x+\beta y+7 z=3 \\& x+2 y+3 z=14\en…

If \(\alpha=\beta=7\), then the system has no solution

If \(\alpha=\beta\) and \(\alpha \neq 7\) then the system has a unique solution.

There is a unique point \((\alpha, \beta)\) on the line \(x+2 y+18=0\) for which the system has infinitely many solutions

For every point \((\alpha, \beta) \neq(7,7)\) on the line \(x-2 y+7=0\), the system has infinitely many solutions.