Let \((\alpha, \beta, \gamma)\)\(\text{ be the image of the point }\)\(P(3, 3, 5) \text{ in the plane } 2x + y - 3z = 6\). \(\text{ Then } \alpha + \beta + \gamma \text{ is equal to:}\)
The equation of the plane is given as:
\[
2x + y - 3z = 6
\]
Let the point \( P(3, 3, 5) \) be the point whose image is \( (\alpha, \beta, \gamma) \). The image of a point in a plane can be found by using the formula for the reflection of a point across a plane.
The reflection formula is:
\[
\alpha - x = \frac{2 \times \left(2x + y - 3z - 6 \right)}{2 + 1 + 1}
\]
Substitute the given values of \( x = 3, y = 3, z = 5 \), and calculate the values of \( \alpha, \beta, \gamma \). After solving for the reflection, we obtain:
\[
\alpha = 6, \quad \beta = 5, \quad \gamma = -1
\]
Now, calculate \( \alpha + \beta + \gamma \):
\[
\alpha + \beta + \gamma = 6 + 5 - 1 = 10
\]
Thus, the value of \( \alpha + \beta + \gamma \) is 10.
