If \(A=\left[\begin{array}{ccc}1 & -1 & 1 \\ 2 & -1 & 0 \\ 3 & 3 & -4\end{array}\right], B=\left[\begin{array}{l}1 \\ 1 \\ 2\end{array}\right]\) and \(X=\left[\begin{array}{l}x_1 \\ x_2 \\ x_3\end{array}\right]\) such that \(A X=B\), then the value of \(x_1+x_2+x_3=\)

\(\begin{aligned} & {\left[\begin{array}{ccc}1 & -1 & 1 \\ 2 & -1 & 0 \\ 3 & 3 & -4\end{array}\right]\left[\begin{array}{l}\mathrm{x}_1 \\ \mathrm{x}_2 \\ \mathrm{x}_3\end{array}\right]=\left[\begin{array}{l}1 \\ 1 \\ 2\end{array}\right]} \\ & \mathrm{R}_3 \rightarrow \mathrm{R}_3-3 \mathrm{R}_1 \text { and } \mathrm{R}_2 \rightarrow \mathrm{R}_2-2 \mathrm{R}_1 \\ & {\left[\begin{array}{ccc}1 & -1 & 1 \\ 0 & 1 & -2 \\ 0 & 6 & -7\end{array}\right]\left[\begin{array}{l}\mathrm{x}_1 \\ \mathrm{x}_2 \\ \mathrm{x}_3\end{array}\right]=\left[\begin{array}{c}1 \\ -1 \\ -1\end{array}\right]} \\ & \mathrm{R}_3 \rightarrow \mathrm{R}_3-6 \mathrm{R}_2 \\ & {\left[\begin{array}{ccc}1 & -1 & 1 \\ 0 & 1 & -2 \\ 0 & 0 & 5\end{array}\right]\left[\begin{array}{l}\mathrm{x}_1 \\ \mathrm{x}_2 \\ \mathrm{x}_3\end{array}\right]=\left[\begin{array}{c}1 \\ -1 \\ 5\end{array}\right]} \\ & \therefore \quad \mathrm{x}_1-\mathrm{x}_2+\mathrm{x}_3=1 \\ & \quad \mathrm{x}_2-2 \mathrm{x}_3=-1 \\ & \quad 5 \mathrm{x}_3=5 \\ & \therefore \mathrm{x}_3=1, \mathrm{x}_2=1, \mathrm{x}_1=1\end{aligned}\)