1. **State the problem:** Solve the system of linear equations: $$\begin{cases} x + 2y = 5 \\ 5x + 6y = 9 \end{cases}$$ 2. **Use substitution or elimination method.** Here, we use elimination. 3. Multiply the first equation by 3 to align coefficients of $y$: $$3(x + 2y) = 3(5) \Rightarrow 3x + 6y = 15$$ 4. Subtract the second equation from this new equation: $$\cancel{3x} + 6y - (5x + 6y) = 15 - 9$$ $$3x + 6y - 5x - 6y = 6$$ $$-2x = 6$$ 5. Solve for $x$: $$x = \frac{6}{-2} = -3$$ 6. Substitute $x = -3$ into the first original equation: $$-3 + 2y = 5$$ 7. Solve for $y$: $$2y = 5 + 3 = 8$$ $$y = \frac{8}{2} = 4$$ **Final answer:** $$x = -3, \quad y = 4$$