<s>[INST] Reducing the System of linear equations \begin{align*}
x+2y-3z&=4 \\
3x-y+5z&=2 \\
4x+y+(k^2-14)z&=k+2
\end{align*}
I started doing the matrix of the system:
$$ \begin{pmatrix} 1 & 2 & -3 & 4 \\ 3 & -1 & 5 & 2 \\ 4 & 1 & k^2-14 & k+2  \end{pmatrix} $$
But I still don't know how can I reduce this, could you explain me what steps I should follow?
Which values of $k$ make the system inconsistent, determined, undetermined?
 [/INST] Let our augmented matrix be $$A=\begin{pmatrix} 1 & 2 & -3 & 4 \\ 3 & -1 & 5 & 2 \\ 4 & 1 & k^2-14 & k+2  \end{pmatrix}$$ as you pointed out. So, if we do $$-3R_1+R_2\to R_2~(I),~~~ -4R_1+R_3\to R_3~(II),~~~R_2-R_3\to R_3~(III)$$ then we get $$A\to B=\begin{pmatrix} 1 & 2 & -3 & 4 \\ 0 & -7 & 14 & -10 \\ 0 & 0 & 16-k^2 & -k+4  \end{pmatrix}$$ Now think about 


*

*$k=4$ which makes your system consistent with infinitely many solutions,

*$k=-4$ which makes your system inconsistent,
and...
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