$$2x-y+z=0, 3x+2y-z=0,x+4y+3z=0$$
$$
\begin{vmatrix}
2 & -1 & 1 \\
3 & 2 & -1 \\
1 & 4 & 3 \\
\end{vmatrix}
$$
By reducing row $$R_1=R_1-2R_3\\R_2=R_3-3R_3$$
we get
\begin{vmatrix}
0 & -9 & -5 \\
0 & -10 & -10 \\
1 & 4 & 3 \\
\end{vmatrix}
Expanding along C3 and solving the determinant we get 40
But how to solve further ?
Best Answer
Once all equations are equal to zero then:
$$\begin{vmatrix} 0 & -9 & -5 \\ 0 & -10 & -10 \\ 1 & 4 & 3 \\ \end{vmatrix}=\begin{vmatrix} 0 & -9 & -5 \\ 0 & 1 & 1 \\ 1 & 4 & 3 \\ \end{vmatrix}$$
And now $R_1=R_1+9R_2$ and get
$$\begin{vmatrix} 0 & 0 & 4 \\ 0 & 1 & 1 \\ 1 & 4 & 3 \\ \end{vmatrix}$$
The above system is equivalent to
$$4z=0\to z=0\\ y+z=0\to y=0\\ x+4y+3z=0\to x=0$$