1. **State the problem:** Solve the system of linear equations: $$\frac{x}{6} - \frac{y}{2} = \frac{1}{6}$$ $$\frac{3}{4}x - y = 2$$ 2. **Rewrite the first equation to clear denominators:** Multiply both sides by 6 to eliminate fractions: $$6 \times \left(\frac{x}{6} - \frac{y}{2}\right) = 6 \times \frac{1}{6}$$ $$x - 3y = 1$$ 3. **Rewrite the second equation for clarity:** $$\frac{3}{4}x - y = 2$$ 4. **Express the system now as:** $$\begin{cases} x - 3y = 1 \\ \frac{3}{4}x - y = 2 \end{cases}$$ 5. **Solve the first equation for $x$:** $$x = 1 + 3y$$ 6. **Substitute $x = 1 + 3y$ into the second equation:** $$\frac{3}{4}(1 + 3y) - y = 2$$ 7. **Distribute $\frac{3}{4}$:** $$\frac{3}{4} + \frac{9}{4}y - y = 2$$ 8. **Combine like terms for $y$:** $$\frac{9}{4}y - y = \frac{9}{4}y - \frac{4}{4}y = \frac{5}{4}y$$ So the equation becomes: $$\frac{3}{4} + \frac{5}{4}y = 2$$ 9. **Isolate $y$:** $$\frac{5}{4}y = 2 - \frac{3}{4} = \frac{8}{4} - \frac{3}{4} = \frac{5}{4}$$ 10. **Divide both sides by $\frac{5}{4}$:** $$y = \frac{\frac{5}{4}}{\frac{5}{4}} = \cancel{\frac{5}{4}} \times \frac{4}{\cancel{5}} = 1$$ 11. **Substitute $y=1$ back into $x = 1 + 3y$:** $$x = 1 + 3(1) = 4$$ **Final answer:** $$\boxed{(x, y) = (4, 1)}$$