= Solution
<Gaussian elimination> of the first two equations gives $x=2y-1$ and $z=3-4y$. The third then reduces to $(\alpha-2)y=\beta-3$. Hence
$$
\boxed{\begin{array}{ll}
\alpha=2,\ \beta\ne3:&\text{no solution},\\
\alpha=2,\ \beta=3:&\text{infinitely many solutions},\\
\alpha\ne2:&\text{one solution}.
\end{array}}
$$
In the infinite family $y=t$, $x=2t-1$ and $z=3-4t$, for $t\in\mathbb R$. This makes the distinction between an inconsistent <linear system> and a free parameter explicit; its <coefficient matrix> has <determinant> $4-2\alpha$.
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