The system of equations $-2x+y+z=a,x-2y+z=b,x+y-2z=c$ is consistent if

$$\begin{gathered} We have \Delta = \left|\begin{array}{ccc}-2& 1& 1\\ 1& -2& 1\\ 1& 1& -2\end{array}\right|=-2\left(4-1\right)-1\left(-2-1\right)+1\left(1+2\right)=0 \\ \therefore For consistency we must have {\Delta }_1={\Delta }_2={\Delta }_3=0 \\ \Rightarrow \left|\begin{array}{ccc}a& 1& 1\\ b& -2& 1\\ c& 1& -2\end{array}\right|=0 \\ \Rightarrow a\left(4-1\right)-1\left(-2b-c\right)+1\left(b+2c\right)=0 \\ \Rightarrow 3\left(a+b+c\right)=0 \\ \Rightarrow a+b+c=0 \end{gathered}$$