If a point $P(\alpha ,\beta ,\gamma )$ satisfying $\left[\alpha \beta \gamma \right]\left[\begin{array}{ccc}2& 10& 8\\ 9& 3& 8\\ 8& 4& 8\end{array}\right]=\left[\begin{array}{ccc}0& 0& 0\end{array}\right]$ lies on the plane $2x+4y+3z=5$ then $6\alpha +9\beta +7\gamma$ is equal to :

$$\begin{gathered} 2\alpha +4\beta +3\gamma =5 \dots \dots \dots .\left(1\right) \\ 2\alpha +9\beta +8\gamma =0 \dots \dots \dots .\left(2\right) \\ 10\alpha +3\beta +4\gamma =0 \dots \dots \dots .\left(3\right) \\ 8\alpha +8\beta +8\gamma =0 \dots \dots \dots .\left(4\right) \\ -6\alpha +\beta =0 \\ \beta =6\alpha \dots \dots \dots .\left(5\right) \end{gathered}$$

From equation (4)

$$\begin{gathered} 8\alpha +48\alpha +8\gamma =0 \\ \gamma =-7\alpha \dots \dots \dots .\left(6\right) \end{gathered}$$

From equation(1)

$$\begin{gathered} 8\alpha +24\alpha -21\alpha =5 \\ 5\alpha =5 \\ \alpha =1 \\ \beta =+6, \gamma =-7 \\ \therefore 6\alpha +9\beta +7\gamma \\ =6+54-49 \\ =11 \end{gathered}$$