1. **State the problem:** Solve the system of equations using elimination: $$\frac{x}{6} + \frac{y}{4} = 6$$ $$\frac{5x}{6} - \frac{y}{3} = 11$$ 2. **Rewrite the equations to clear denominators:** Multiply the first equation by 12 (LCM of 6 and 4): $$12 \times \left(\frac{x}{6} + \frac{y}{4}\right) = 12 \times 6$$ $$2x + 3y = 72$$ Multiply the second equation by 6 (LCM of 6 and 3): $$6 \times \left(\frac{5x}{6} - \frac{y}{3}\right) = 6 \times 11$$ $$5x - 2y = 66$$ 3. **Set up the system without fractions:** $$2x + 3y = 72$$ $$5x - 2y = 66$$ 4. **Use elimination to eliminate one variable:** Multiply the first equation by 2 and the second by 3 to align coefficients of $y$: $$2 \times (2x + 3y) = 2 \times 72 \Rightarrow 4x + 6y = 144$$ $$3 \times (5x - 2y) = 3 \times 66 \Rightarrow 15x - 6y = 198$$ 5. **Add the two equations to eliminate $y$:** $$4x + 6y + 15x - 6y = 144 + 198$$ $$19x + \cancel{6y} - \cancel{6y} = 342$$ $$19x = 342$$ 6. **Solve for $x$:** $$x = \frac{342}{19}$$ 7. **Substitute $x$ back into one of the original equations to find $y$:** Use $2x + 3y = 72$: $$2 \times \frac{342}{19} + 3y = 72$$ $$\frac{684}{19} + 3y = 72$$ 8. **Isolate $y$:** $$3y = 72 - \frac{684}{19}$$ $$3y = \frac{72 \times 19}{19} - \frac{684}{19} = \frac{1368 - 684}{19} = \frac{684}{19}$$ 9. **Solve for $y$:** $$y = \frac{684}{19 \times 3} = \frac{684}{57} = 12$$ **Final answer:** $$x = \frac{342}{19}, \quad y = 12$$