1. **State the problem:** Solve the system of equations: $$\frac{1}{2}x + \frac{1}{3}y = 4$$ $$\frac{1}{4}x - \frac{1}{6}y = 1$$ 2. **Rewrite the equations to clear denominators:** Multiply the first equation by 6 (LCM of 2 and 3): $$6 \times \left(\frac{1}{2}x + \frac{1}{3}y\right) = 6 \times 4$$ $$3x + 2y = 24$$ Multiply the second equation by 12 (LCM of 4 and 6): $$12 \times \left(\frac{1}{4}x - \frac{1}{6}y\right) = 12 \times 1$$ $$3x - 2y = 12$$ 3. **Add the two new equations to eliminate $y$:** $$(3x + 2y) + (3x - 2y) = 24 + 12$$ $$6x = 36$$ 4. **Solve for $x$:** $$x = \frac{36}{6} = 6$$ 5. **Substitute $x=6$ into one of the simplified equations to find $y$:** Using $3x + 2y = 24$: $$3(6) + 2y = 24$$ $$18 + 2y = 24$$ $$2y = 24 - 18 = 6$$ $$y = \frac{6}{2} = 3$$ 6. **Final answer:** $$x = 6, \quad y = 3$$