Consider the following system and find the values of b for which the system has a solution So I have this system:
$$\left\{\begin{array}{c}
x_1 &−  x_2 &+ 2x_3 &= 2 \\
x_1 &+ 2x_2 &− x_3  &= 2 \\
x_1 &+  x_2 &      &= 2 \\
x_1 &       & +x_3 &= α
\end{array}\right.$$
And we are asked to find the values of $\alpha$ for which the system has a solution. When I do the coefficient matrix and a Gauss-Jordan elimination i get two entirely zero rows with one of the augmented matrix solutions being $\alpha-2$ so am I correct in saying that the system will have an infinite number of solutions if $\alpha=2$?
The thing that concerns me is that we are then asked to find the value of $\alpha$ if the degree of freedom is $2$?
Any help would be greatly appreciated!
PS. Sorry for any formatting errors!

I'd argue like the following.
From the first two rows it follows, that whatever value $x_1$ has, $x_2$ must equal $x_3$.
If we insert this knowledge into the last two rows, (replacing $x_3$ by $x_2$) then we see, that $\alpha$ must equal $2$. And if we insert also that, 
$$\left\{\begin{array}{c}
x_1 &−  x_2 &+ 2x_3 &= 2 \\
x_1 &+ 2x_2 &− x_3  &= 2 \\
x_1 &+  x_2 &      &= 2 \\
x_1 & +x_2 &&= 2&=α
\end{array}\right.$$
then it follows $x_1 = 2 - x_2$ and (unfortunately) nothing more - and thus we have an infinitude of solutions.
