<s>[INST] Linear Algebra Question My question is; How can I find an equation relating $a,b$, and $c$ so that the linear system
$$\begin{cases}2x+y-z=a\\ x-2y-3z=b\\ -3x-y+2z=c\end{cases}$$
is consistent for any values of $a,b$, and $c$ that satisfy that equation.
Thanks,
 [/INST] Perform Gaussian elimination on the system. For each row with 3 zeros on the left, the corresponding equation on the right is a restriction on $a$, $b$, $c$.
Here are the details:
$$
\begin{array}{rrrl}
2 & 1 & -1 & a \\
1 & -2 & -3 & b \\
-3 & -1 & -2 & c
\end{array}
\quad\to\quad
\begin{array}{rrrl}
1 & -2 & -3 & b \\
2 & 1 & -1 & a \\
-3 & -1 & -2 & c
\end{array}
\quad\to\quad
\begin{array}{rrrl}
1 & -2 & -3 & b \\
0 & 5 & 5 & a-2b \\
0 & -7 & -7 & c+3b
\end{array}
\quad\to\quad
$$
$$
\begin{array}{rrrl}
1 & -2 & -3 & b \\
0 & 35 & 35 & 7a-14b \\
0 & -35 & -35 & 5c+15b
\end{array}
\quad\to\quad
\begin{array}{rrrl}
1 & -2 & -3 & b \\
0 & 35 & 35 & 7a-14b \\
0 & 0& 0 &7a+b+5c
\end{array}
$$
Hence we must have $7a+b+5c=0$ if the original system has a solution.
</s>