The system of equations is given as:

$$\frac{9}{20}by + \frac{7}{15} = -\frac{9}{20}ax$$ $$\frac{x}{8} + \frac{y}{18} = \frac{a}{2} + \frac{5x}{12}$$

We are tasked with finding the value of $a - b$.

Solution:

Rewriting the first equation: $$27by + 28 = -27ax$$ and rearranging: $$27ax + 27by = -28 \tag{1}$$

For the second equation, after multiplying through by the LCM of denominators, we get: $$-21x + 4y = 36a \tag{2}$$

Solving the system of equations, from Equation (1): $$\frac{27a}{-21} = \frac{-28}{36a}$$ Simplifying: $$972a^2 = 588$$ $$a^2 = \frac{49}{81} \implies a = \frac{7}{9}$$

From Equation (1) substituting $a$: $$b = -\frac{4}{27}$$

Finally, the value of $a - b$ is: $$a - b = \frac{7}{9} - \left(-\frac{4}{27}\right)$$ Simplifying: $$a - b = \frac{21}{27} + \frac{4}{27} = \frac{25}{27}$$

Answer: C) $\frac{25}{27}$