1. **State the problem:** Solve the system of linear equations: $$\begin{cases} x + 2y = 5 \\ 3x - y = 4 \end{cases}$$ 2. **Formula and rules:** To solve a system of linear equations, we can use substitution or elimination. Here, we'll use elimination. 3. **Multiply the first equation by 1 and the second by 2 to align coefficients of $y$:** $$\begin{cases} x + 2y = 5 \\ 6x - 2y = 8 \end{cases}$$ 4. **Add the two equations to eliminate $y$:** $$ (x + 2y) + (6x - 2y) = 5 + 8 $$ $$ 7x + \cancel{2y} - \cancel{2y} = 13 $$ $$ 7x = 13 $$ 5. **Solve for $x$:** $$ x = \frac{13}{7} $$ 6. **Substitute $x$ back into the first equation to find $y$:** $$ \frac{13}{7} + 2y = 5 $$ 7. **Isolate $y$:** $$ 2y = 5 - \frac{13}{7} $$ $$ 2y = \frac{35}{7} - \frac{13}{7} = \frac{22}{7} $$ 8. **Divide both sides by 2:** $$ y = \frac{\cancel{2} \times 11}{7 \times \cancel{2}} = \frac{11}{7} $$ **Final answer:** $$ \boxed{\left( \frac{13}{7}, \frac{11}{7} \right)} $$