c) \( \left\{\begin{array}{l}\frac{1}{2} x+\frac{3}{2} y=5 \\ 3 x-y=5 y\end{array}\right. \)

To solve the system of equations \( \left\{\begin{array}{l}\frac{1}{2} x+\frac{3}{2} y=5 \\ 3 x-y=5 y\end{array}\right. \), let's first simplify the second equation. Rearranging gives \( 3x = 6y \), or \( x = 2y \). Substituting \( x = 2y \) into the first equation \( \frac{1}{2}(2y) + \frac{3}{2}y = 5 \) simplifies to \( y + \frac{3}{2}y = 5 \). Combining like terms gives \( \frac{5}{2}y = 5 \), leading to \( y = 2 \). Now substituting \( y = 2 \) back into \( x = 2y \) gives \( x = 4 \). Thus, the solution is \( (x, y) = (4, 2) \). Always double-check your solutions by substituting back into the original equations to ensure they hold true!