Solve the system by elimination. First, clear the denominators.
\[
\begin{array}{r}
\frac{5 x}{2}+\frac{5 y}{3}=\frac{40}{3} \\
\frac{x}{4}+\frac{y}{3}=\frac{13}{6}
\end{array}
\]

Step 1 ：Given the system of equations: \[\frac{5 x}{2}+\frac{5 y}{3}=\frac{40}{3}\] and \[\frac{x}{4}+\frac{y}{3}=\frac{13}{6}\]

Step 2 ：First, we clear the denominators by multiplying each equation by the least common multiple (LCM) of the denominators. The LCM of 2, 3, 4, and 6 is 12.

Step 3 ：After clearing the denominators, the equations become: \[30x + 20y = 160\] and \[3x + 4y = 26\]

Step 4 ：Next, we use the elimination method to solve the system of equations.

Step 5 ：By solving the system, we find that the solution is \(x = 2\) and \(y = 5\)

Step 6 ：Final Answer: The solution to the system of equations is \(\boxed{x = 2, y = 5}\)