Given,

$\begin{array}{c}2\text{x}-\text{y}-\text{z}=-\text{a}\\ \text{x}-2\text{y}+\text{z}=\text{b}\\ \text{x}+\text{y}-2\text{z}=\text{c}\end{array}$

For consistent system of equations $∆\ne 0$ or $∆={∆}_1={∆}_2={∆}_3=0$
$∆=\left|\begin{array}{ccc}2& -1& -1\\ 1& -2& 1\\ 1& 1& -2\end{array}\right|=0$
${∆}_1=\left|\begin{array}{ccc}-\text{a}& -1& -1\\ \text{b}& -2& 1\\ \text{c}& 1& -2\end{array}\right|=0$
$\Rightarrow \mathrm{\text{a}}+\mathrm{\text{b}}+\mathrm{\text{c}}=0$
${∆}_2=\left|\begin{array}{ccc}2& -\text{a}& -1\\ 1& \text{b}& 1\\ 1& \text{c}& -2\end{array}\right|=0$
$\Rightarrow \mathrm{\text{a}}+\mathrm{\text{b}}+\mathrm{\text{c}}=0$
${∆}_3=\left|\begin{array}{ccc}2& -1& -\text{a}\\ 1& -2& \text{b}\\ 1& 1& \text{c}\end{array}\right|=0$
$\Rightarrow \mathrm{\text{a}}+\mathrm{\text{b}}+\mathrm{\text{c}}=0$
$\left.\begin{array}{c}\text{a}+\text{b}+\text{c}=0\\ 16\text{a}-4\text{b}+\text{c}=0\end{array}\right\}\Rightarrow \frac{\mathrm{\text{a}}}{5}=\frac{\mathrm{\text{b}}}{15}=\frac{\mathrm{\text{c}}}{-20}$
$\Rightarrow \mathrm{\text{a}}=\mathrm{\text{λ}}$
$\mathrm{\text{b}}=3\mathrm{\text{λ}}$
$\mathrm{\text{c}}=-4\mathrm{\text{λ}}$
$\Rightarrow$ Sum of roots $=\frac{-\mathrm{\text{b}}}{\mathrm{\text{a}}}=-3$