1. We are asked to solve the system of linear equations using the elimination method. Given the first system: $$\begin{cases} 3x - 2y = 5 \\ 5x - 3y = 8 \end{cases}$$ 2. The elimination method involves eliminating one variable by making the coefficients of that variable equal in both equations. 3. Multiply the first equation by 3 and the second equation by 2 to align the coefficients of $y$: $$\begin{cases} 3(3x - 2y) = 3(5) \\ 2(5x - 3y) = 2(8) \end{cases}$$ which simplifies to $$\begin{cases} 9x - 6y = 15 \\ 10x - 6y = 16 \end{cases}$$ 4. Subtract the first new equation from the second to eliminate $y$: $$ (10x - 6y) - (9x - 6y) = 16 - 15 $$ which simplifies to $$ 10x - 6y - 9x + 6y = 1 $$ $$ (10x - 9x) + (-6y + 6y) = 1 $$ $$ x + 0 = 1 $$ $$ x = 1 $$ 5. Substitute $x=1$ back into the first original equation: $$ 3(1) - 2y = 5 $$ $$ 3 - 2y = 5 $$ 6. Solve for $y$: $$ -2y = 5 - 3 $$ $$ -2y = 2 $$ $$ y = \frac{\cancel{-2}y}{\cancel{-2}} = \frac{2}{-2} $$ $$ y = -1 $$ 7. Final solution: $$ \boxed{(x, y) = (1, -1)} $$