1. **State the problem:** Solve the system of linear equations: $$3y = \frac{3}{2}x + 6$$ $$\frac{1}{2}y - \frac{1}{4}x = 3$$ 2. **Rewrite the first equation in standard form:** Divide both sides by 3: $$y = \frac{1}{2}x + 2$$ 3. **Rewrite the second equation in standard form:** Multiply both sides by 4 to clear denominators: $$4 \times \left(\frac{1}{2}y - \frac{1}{4}x\right) = 4 \times 3$$ $$2y - x = 12$$ 4. **Express $y$ from the second equation:** $$2y = x + 12$$ $$y = \frac{x}{2} + 6$$ 5. **Compare the two expressions for $y$:** From equation 1: $$y = \frac{1}{2}x + 2$$ From equation 2: $$y = \frac{1}{2}x + 6$$ 6. **Analyze the system:** Both lines have the same slope $\frac{1}{2}$ but different intercepts (2 and 6), so they are parallel and do not intersect. 7. **Conclusion:** The system has **no solution** because the lines are parallel and never meet. **Final answer:** no solution