# For how many values of &#x03B1;<!-- α --> does this system of equations have infinitely many s

For how many values of $\alpha$ does this system of equations have infinitely many solutions ?
$\left(\begin{array}{ccc}2& 1& -4\\ 4& 3& -12\\ 1& 2& -8\end{array}\right)\left(\begin{array}{c}x\\ y\\ z\end{array}\right)=\left(\begin{array}{c}\alpha \\ 5\\ 7\end{array}\right)$
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fongama33
The point is that the system has a kernel; since the column space is not 3-dimensional there are nontrivial linear combinations of the columns that vanish, namely multiples of $x=0,y=4,z=1$. So given one solution, you find another by adding any element of this kernel. Do be sure to check your claim about there being exactly one $\alpha ,$ though: not every line intersects every plane, and a given line intersects some planes infinitely often.