Here's a clear, educational, and well-structured solution to the problem.

1. Key Concepts and Formulas

   • Image of a Point with respect to a Plane: The image $P'(x', y', z')$ of a point $P(x_1, y_1, z_1)$ with respect to a plane $Ax + By + Cz + D = 0$ is given by the formula: $\frac{x' - x_1}{A} = \frac{y' - y_1}{B} = \frac{z' - z_1}{C} = \frac{-2(Ax_1 + By_1 + Cz_1 + D)}{A^2 + B^2 + C^2}$ This formula is derived from the fact that the line segment $PP'$ is perpendicular to the plane, and the midpoint of $PP'$ lies on the plane.
   • Solving Systems of Linear Equations: To find unknown variables or expressions, we often need to set up and solve a system of linear equations. Techniques like substitution or elimination are commonly used.

2. Step-by-Step Solution

   Step 1: Identify Given Information and the Goal We are given:

   • Original point $P(x_1, y_1, z_1) = (a, b, c)$.
   • Equation of the plane: $3x - 4y + 12z + 19 = 0$. Comparing this to $Ax + By + Cz + D = 0$, we have $A=3$, $B=-4$, $C=12$, and $D=19$.
   • Mirror image point $P'(x', y', z') = (a - 6, \beta, \gamma)$.
   • An additional condition: $a + b + c = 5$.
   • Our goal is to find the value of the expression $7\beta - 9\gamma$.

   Step 2: Calculate the Denominator for the Image Formula First, we calculate $A^2 + B^2 + C^2$: $A^2 + B^2 + C^2 = (3)^2 + (-4)^2 + (12)^2 = 9 + 16 + 144 = 169$

   Step 3: Apply the Image Formula and Determine the Common Ratio Substitute the coordinates of $P$, $P'$, and the plane coefficients into the image formula: $\frac{(a - 6) - a}{3} = \frac{\beta - b}{-4} = \frac{\gamma - c}{12} = \frac{-2(3a - 4b + 12c + 19)}{169}$ Let's simplify the first part of the equation: $\frac{-6}{3} = -2$ This means the common ratio (let's call it $k$) for all parts of the formula is $-2$. $k = -2$

   Step 4: Express $\beta$ and $\gamma$ in terms of $b$ and $c$ Now, we use the common ratio $k=-2$ to find expressions for $\beta$ and $\gamma$:

   • For $\beta$: $\frac{\beta - b}{-4} = -2$ $\beta - b = (-2) \times (-4)$ $\beta - b = 8$ $\beta = b + 8 \quad \text{(Equation 1)}$
   • For $\gamma$: $\frac{\gamma - c}{12} = -2$ $\gamma - c = (-2) \times 12$ $\gamma - c = -24$ $\gamma = c - 24 \quad \text{(Equation 2)}$

   Step 5: Form an Equation involving $a, b, c$ using the Common Ratio Equate the full expression of the common ratio to $-2$: $\frac{-2(3a - 4b + 12c + 19)}{169} = -2$ Divide both sides by $-2$: $\frac{3a - 4b + 12c + 19}{169} = 1$ Multiply both sides by $169$: $3a - 4b + 12c + 19 = 169$ Subtract $19$ from both sides: $3a - 4b + 12c = 150 \quad \text{(Equation 3)}$

   Step 6: Utilize the Given Condition to Find $7b - 9c$ We are given the condition $a + b + c = 5 \quad \text{(Equation 4)}$. Our target expression $7\beta - 9\gamma$ involves $7b - 9c$ (from Equations 1 and 2). To find $7b - 9c$, we can eliminate $a$ from Equations 3 and 4. Multiply Equation 4 by 3: $3(a + b + c) = 3 \times 5$ $3a + 3b + 3c = 15 \quad \text{(Equation 5)}$ Now, subtract Equation 5 from Equation 3: $(3a - 4b + 12c) - (3a + 3b + 3c) = 150 - 15$ $3a - 4b + 12c - 3a - 3b - 3c = 135$ $-7b + 9c = 135$ Multiply by $-1$ to get the desired form $7b - 9c$: $7b - 9c = -135 \quad \text{(Equation 6)}$

   Step 7: Calculate the Final Expression Substitute the expressions for $\beta$ and $\gamma$ from Equations 1 and 2 into $7\beta - 9\gamma$: $7\beta - 9\gamma = 7(b + 8) - 9(c - 24)$ $7\beta - 9\gamma = 7b + 56 - 9c + 216$ Group the terms: $7\beta - 9\gamma = (7b - 9c) + (56 + 216)$ $7\beta - 9\gamma = (7b - 9c) + 272$ Now substitute the value of $7b - 9c$ from Equation 6: $7\beta - 9\gamma = -135 + 272$ $7\beta - 9\gamma = 137$

3. Common Mistakes & Tips

   • Sign Errors: Be meticulous with negative signs, especially in the formula for the image of a point and when combining equations. A common error is mixing up the sign of $D$ or forgetting the $-2$ in the numerator.
   • Formula Accuracy: Ensure you recall the exact formula for the image of a point. The -2 factor is crucial for the image, while for the foot of the perpendicular, it would be -1.
   • Systematic Elimination: When solving a system of equations, choose a variable to eliminate and multiply equations by appropriate constants to make the coefficients of that variable equal (and opposite if adding, or same if subtracting).

4. Summary

   We began by applying the standard formula for the image of a point with respect to a plane. By using the given x-coordinate of the image point, we efficiently determined the common ratio of the formula. This common ratio allowed us to express $\beta$ and $\gamma$ in terms of $b$ and $c$, and also to derive a linear equation relating $a, b, c$. Combining this derived equation with the given condition $a+b+c=5$, we solved for the expression $7b-9c$. Finally, we substituted this value back into the expression for $7\beta-9\gamma$ to find the result.

The final answer is $\boxed{137}$.