Consider the following system of linear equations \begin{bmatrix}2 & 1 & -4…

Consider the following system of linear equations

$$\begin{bmatrix}2 & 1 & -4 \\4 & 3 & -12 \\1 & 2 & -8\end{bmatrix}\begin{bmatrix}x \\y \\z\end{bmatrix}=\begin{bmatrix}\alpha \\5 \\7\end{bmatrix}$$

Notice that the second and the third columns of the coefficient matrix are linearly dependent. For how many values of α does this system of equations have infinitely many solutions?