1. **State the problem:** Solve the system of linear equations using the elimination method: $$5x + 4y = -14$$ $$x + 6y = 6$$ 2. **Explain the elimination method:** The goal is to eliminate one variable by adding or subtracting the equations after multiplying them by suitable numbers. 3. **Multiply the second equation by -5 to align coefficients of $x$:** $$-5(x + 6y) = -5(6)$$ $$-5x - 30y = -30$$ 4. **Add this to the first equation:** $$5x + 4y = -14$$ $$-5x - 30y = -30$$ Adding gives: $$0x - 26y = -44$$ 5. **Simplify the resulting equation:** $$-26y = -44$$ 6. **Solve for $y$:** $$y = \frac{-44}{-26} = \frac{22}{13}$$ 7. **Substitute $y = \frac{22}{13}$ into the second original equation to find $x$:** $$x + 6\left(\frac{22}{13}\right) = 6$$ $$x + \frac{132}{13} = 6$$ $$x = 6 - \frac{132}{13} = \frac{78}{13} - \frac{132}{13} = -\frac{54}{13}$$ 8. **Final solution:** $$x = -\frac{54}{13}, \quad y = \frac{22}{13}$$