$\begin{array}{c}x+8y+7z=0 ...\left(i\right) \\ 9x+2y+3z=0 ...\left(ii\right) \\ x+y+z=0 ...\left(iii\right) \end{array}$.....

Solve the equation $\left(i\right) \&\left(iii\right)$

 $\Rightarrow 7y+6z=0$

Solve the equation $\left(ii\right) \&\left(iii\right)$

$\Rightarrow 7x+z=0$

$x=\lambda \& \left|y=\frac{-6\left(-7\lambda \right)}{7}\right| \& z=-7\lambda$

$\begin{array}{ccc}x=\lambda & \& y=6\lambda & \& z=-7\lambda \end{array}$

  $\begin{array}{c}\Rightarrow \lambda +12\lambda -7\lambda =6\\ \Rightarrow 6\lambda =6\\ \Rightarrow \lambda =1\end{array}|\begin{array}{c}\Rightarrow 2\lambda +6\lambda -7\lambda \\ \Rightarrow \lambda =1\end{array}$

Hence, $a=1, b=6, c=-7$

$\therefore 2a+b+c=1$