I have a matrix $A = $$ \left[ \begin{array}{ccc|c} 1&2&3&-17\\ -2&-4&-k&34 \end{array} \right] $$ $ and want to check for consistency of the system for the possible values of $k$.
I am, however unsure if my answer is correct:
$$ \left[ \begin{array}{ccc|c} 1&2&3&-17\\ -2&-4&-k&34 \end{array} \right] $$
$$ \left[ \begin{array}{ccc|c} 1&2&3&-17\\ 0&0&6-k&0 \end{array} \right] $$
$$ \left[ \begin{array}{ccc|c} 1&2&3&-17\\ 0&0&1&0 \end{array} \right] $$
$$ \left[ \begin{array}{ccc|c} 1&2&0&-17\\ 0&0&1&0 \end{array} \right] $$
Am I correct in assuming, that as long as variable $x_3 = 0$, every value of $k$ makes the system consistent?
Yes, you're correct.
The only trouble that might arise is dividing by $6-k$ during Gaussian elimination, because in the case that $k=6$, that's not justifiable. By examining the case $k=6$ separately though, you can see that the system is still consistent.