The number of solutions of the following equations $x_2-x_3=1$, $-x_1+2x_3=-2$, $x_1-2x_2=3$ is:

Write the system in matrix form $AX=B$:$0x_1+1x_2-1x_3=1$$-1x_1+0x_2+2x_3=-2$$1x_1-2x_2+0x_3=3$$\Delta =\left|\begin{array}{ccc}0& 1& -1\\ -1& 0& 2\\ 1& -2& 0\end{array}\right|=0(0+4)-1(0-2)-1(2-0)=2-2=0$.Since $\Delta =0$, we check ${\Delta }_x$:${\Delta }_x=\left|\begin{array}{ccc}1& 1& -1\\ -2& 0& 2\\ 3& -2& 0\end{array}\right|=1(0+4)-1(0-6)-1(4-0)=4+6-4=6$.Since $\Delta =0$ and ${\Delta }_x\ne 0$, the system is inconsistent and has no solution (Zero solutions).