= Solution
Subtracting the first equation from the third gives $x-z=3/2$. The second equation then becomes $(\alpha-2)x=\beta-3$, and the third fixes $y=(7-7x)/5$. The <linear system> has the following three consistency cases:
$$
\boxed{\begin{aligned}
\alpha\ne2:\quad &(x,y,z)=\left(\frac{\beta-3}{\alpha-2},\frac{7(\alpha-\beta+1)}{5(\alpha-2)},\frac{2\beta-3\alpha}{2(\alpha-2)}\right);\\
\alpha=2,\ \beta=3:\quad &(x,y,z)=\left(t,\frac{7(1-t)}5,t-\frac32\right),\quad t\in\mathbb R;\\
\alpha=2,\ \beta\ne3:\quad &\text{no solution}.
\end{aligned}}
$$
The first case is a unique solution, the second is an <affine line in a vector space> of solutions, and the third is an <inconsistent linear system>.
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