For which Value of a Variable Gives Solutions of Linear System?

Question

So I have found this question to be very vague and I am struggling to understand it and I believe I am close to the solution but I am not sure if it is correct. The question involves a linear system like so:
$x_1+2x_4=1$
$2x_1+x_3+3x_4=2$
$x_1+x_3+x_4=a$
Which I converted into an augmented matrix:
$$\left[\begin{matrix} 1 & 0 & 0 & 2 & 1 \\ 2 & 0 & 1 & 3 & 2 \\ 1 & 0 & 1 & 1 & a\end{matrix}\right]$$
So I am asked to reduce this to reduced row echelon form, find which values of a the system has solutions for, and to write those solutions in vector form, e.g. $[x_1,x_2,x_3,x_4]$. The part where this becomes tricky for me is I had reduced the system as far as I could down to this:
$$\left[\begin{matrix} 1 & 0 & 0 & 2 & 1 \\ 0 & 0 & 1 & -1 & 0 \\ 0 & 0 & 0 & 0 & a-1\end{matrix}\right]$$
And I suppose the system only has one kind of solution, which is the case of infinite solutions where $a=1$.
My question is: am I blind or is this the only value of a for which this system has a solution, and how am I supposed to write the solution to a system for which there is infinite solutions?

Answer 1

You are right, $a=1$ is the only value of $a$ for which the system has a solution. As for your other question, the matrix you have row-reduced tells you that the system you started with is reduced to the system:

\begin{equation} \begin{split} x_1 + 2x_4 &= 1 \\ x_3 -x_4 &= 0 \end{split} \end{equation}

Now, can you finish the question, and write the solution of this system in vector form? If not, just post a comment, and I'll edit this answer to include the last bit.