1. **State the problem:** Solve the system of linear equations: $$\frac{2x-7}{2} - \frac{2(x-y)}{3} = 0$$ $$\frac{x - y}{2} - \frac{7x - y}{6} = -2$$ 2. **Rewrite each equation to clear denominators:** Multiply the first equation by 6 (the least common multiple of 2 and 3): $$6 \times \left(\frac{2x-7}{2} - \frac{2(x-y)}{3}\right) = 6 \times 0$$ $$3(2x-7) - 2 \times 2(x-y) = 0$$ $$3(2x-7) - 4(x-y) = 0$$ Multiply the second equation by 6 (LCM of 2 and 6): $$6 \times \left(\frac{x - y}{2} - \frac{7x - y}{6}\right) = 6 \times (-2)$$ $$3(x - y) - (7x - y) = -12$$ 3. **Simplify both equations:** First equation: $$3(2x-7) - 4(x-y) = 0$$ $$6x - 21 - 4x + 4y = 0$$ $$\cancel{6x} - 21 - \cancel{4x} + 4y = 0$$ $$2x + 4y - 21 = 0$$ Second equation: $$3(x - y) - (7x - y) = -12$$ $$3x - 3y - 7x + y = -12$$ $$\cancel{3x} - 3y - \cancel{7x} + y = -12$$ $$-4x - 2y = -12$$ 4. **Rewrite the system:** $$2x + 4y = 21$$ $$-4x - 2y = -12$$ 5. **Solve the system by elimination:** Multiply the first equation by 1 and the second by 1 to align coefficients: $$2x + 4y = 21$$ $$-4x - 2y = -12$$ Multiply the first equation by 2: $$4x + 8y = 42$$ Add to the second equation: $$(4x + 8y) + (-4x - 2y) = 42 + (-12)$$ $$\cancel{4x} + 8y - \cancel{4x} - 2y = 30$$ $$6y = 30$$ 6. **Solve for $y$:** $$y = \frac{30}{6} = 5$$ 7. **Substitute $y=5$ into one of the original simplified equations:** Using $$2x + 4y = 21$$: $$2x + 4(5) = 21$$ $$2x + 20 = 21$$ $$2x = 21 - 20 = 1$$ $$x = \frac{1}{2}$$ **Final answer:** $$x = \frac{1}{2}, \quad y = 5$$