Let the mirror image of the point (a, b, c) with respect to the plane 3x $-$ 4y + 12z + 19 = 0 be (a $-$ 6, $\beta$, $\gamma$). If a + b + c = 5, then 7$\beta$ $-$ 9$\gamma$ is equal to ______________.

<p>$${{x - a} \over 3} = {{y - b} \over { - 4}} = {{z - c} \over {12}} = {{ - 2(3a - 4b + 12c + 19)} \over {{3^2} + {{( - 4)}^2} + {{12}^2}}}$$</p> <p>$${{x - a} \over 3} = {{y - b} \over { - 4}} = {{z - c} \over {12}} = {{ - 6a + 8b - 24c - 38} \over {169}}$$</p> <p>$(x,y,z) \equiv (a - 6,\,\beta ,\gamma )$</p> <p>$${{(a - 6) - a} \over 3} = {{\beta - b} \over { - 4}} = {{\gamma - c} \over {12}} = {{ - 6a + 8b - 24c - 38} \over {169}}$$</p> <p>${{\beta - b} \over { - 4}} = - 2 \Rightarrow \beta = 8 + b$</p> <p>${{\gamma - c} \over {12}} = - 2 \Rightarrow \gamma = - 24 + c$</p> <p>${{ - 6a + 8b - 24c - 38} \over {169}} = - 2$</p> <p>$\Rightarrow 3a - 4b + 12c = 150$ ..... (1)</p> <p>$a + b + c = 5$</p> <p>$3a + 3b + 3c = 15$ ...... (2)</p> <p>Applying (1) - (2)</p> <p>$- 7b + 9c = 135$</p> <p>$7b - 9c = - 135$</p> <p>$7\beta - 9\gamma = 7(8 + b) - 9( - 24 + c)$</p> <p>$= 56 + 216 + 7b - 9c$</p> <p>$= 56 + 216 - 135 = 137$</p>