<s>[INST] Gauss-Jordan Elimination Trying to solve this using Gauss-Jordan Elimination.
$x_1 + x_2 - x_3 = -4$
$-x_1 - 2x_2 + x_3 = 3+t$
$2x_1 + x_2 + (s-3)x_3 = st-9$
$2x_1         + (s-3)x_3 = st+2t-10$
I came across:
$t=0$
And
$s\neq 1$
Need to find the values of s and t that the equations will have: no solution, 1 solution, unlimited solutions.
$\left[\begin{array}{ccc|c}1&1&-1&-4\\-1&-2&1&3+t\\2&1&s-3&st-9\\2&0&s-3&st+2t-10\end{array}\right]$
$\left[\begin{array}{ccc|c}1&1&-1&-4\\0&1&0&1-t\\0&0&s-1&t(s-1)\\0&1&0&1-2t\end{array}\right]$
$\left[\begin{array}{ccc|c}1&1&-1&-4\\0&1&0&1-t\\0&0&1&t\\0&0&0&-t\end{array}\right]$
$\left[\begin{array}{ccc|c}1&0&0&2t-5\\0&1&0&1-t\\0&0&1&t\\0&0&0&-t\end{array}\right]$
 [/INST] From your final system, $0=-t$, hence if $t \neq 0$, there is no solution.
If $t=0$, we can drop the very last equation as it is just $0=0$. 
The $3 \times 3 $ matrix on the LHS is non-singular, hence the system has a unique solution.
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