Calculate Value of Expression Involving Logarithmic Equations

To find the value of $4x + 5y$, we first simplify the logarithmic expressions using the definitions of $a$ and $b$.

Given $a = 3\sqrt{2} = \sqrt{18} = 18^{1/2}$, we rewrite the first equation as: $3x + 2y = \log_a (18)^{5/4} = \log_a (a^2)^{5/4} = \frac{5}{2}$

Given $b = \frac{1}{5^{1/6}\sqrt{6}}$, we note that $b^{-6} = 1080$, so $\sqrt{1080} = b^{-3}$. This simplifies the second equation to: $2x - y = \log_b (b^{-3}) = -3$

Solving the system of linear equations $3x + 2y = \frac{5}{2}$ and $2x - y = -3$ yields $x = -\frac{1}{2}$ and $y = 2$.

Finally, evaluating the required expression: $4x + 5y = 4\left(-\frac{1}{2}\right) + 5(2) = 8$

Thus, $4x + 5y$ is equal to $8$.