If the system of linear equations
$$2x - 3y = \gamma + 5$$
$$\alpha x + 5y = \beta + 1$$,
where $$\alpha, \beta, \gamma \in \mathbf{R}$$ has infinitely many solutions, then the value of $$|9\alpha + 3\beta + 5\gamma|$$ is equal to

The system of linear equations is: $$2x - 3y = \gamma + 5 \quad \cdots (1)$$ and $$\alpha x + 5y = \beta + 1 \quad \cdots (2)$$

A system of 2 equations in 2 unknowns has infinitely many solutions when the two equations are proportional (i.e., they represent the same line), so $$\frac{2}{\alpha} = \frac{-3}{5} = \frac{\gamma + 5}{\beta + 1}$$.

$$\frac{2}{\alpha} = \frac{-3}{5} \implies \alpha = \frac{-10}{3}$$

$$\frac{-3}{5} = \frac{\gamma + 5}{\beta + 1} \implies -3(\beta + 1) = 5(\gamma + 5)$$

$$-3\beta - 3 = 5\gamma + 25$$

$$3\beta + 5\gamma = -28 \quad \cdots (3)$$

$$9\alpha = 9 \cdot \frac{-10}{3} = -30$$ From (3): $$3\beta + 5\gamma = -28$$

$$9\alpha + 3\beta + 5\gamma = -30 + (-28) = -58$$

$$|9\alpha + 3\beta + 5\gamma| = |-58| = 58$$

The answer is $$\boxed{58}$$.