1. **State the problem:** Solve the linear equation $$\frac{4}{3}x - \frac{3}{2}y = -\frac{5}{3}y + \frac{1}{2}$$ for $x$ in terms of $y$. 2. **Rewrite the equation:** Move all terms involving $y$ to one side and terms involving $x$ to the other side: $$\frac{4}{3}x - \frac{3}{2}y + \frac{5}{3}y = \frac{1}{2}$$ 3. **Combine like terms for $y$:** Find a common denominator for $-\frac{3}{2}y$ and $\frac{5}{3}y$ which is 6: $$-\frac{3}{2}y = -\frac{9}{6}y, \quad \frac{5}{3}y = \frac{10}{6}y$$ So, $$-\frac{9}{6}y + \frac{10}{6}y = \frac{1}{6}y$$ 4. **Rewrite the equation:** $$\frac{4}{3}x + \frac{1}{6}y = \frac{1}{2}$$ 5. **Isolate $x$:** Subtract $\frac{1}{6}y$ from both sides: $$\frac{4}{3}x = \frac{1}{2} - \frac{1}{6}y$$ 6. **Divide both sides by $\frac{4}{3}$:** $$x = \frac{\frac{1}{2} - \frac{1}{6}y}{\frac{4}{3}}$$ 7. **Simplify the division:** Dividing by $\frac{4}{3}$ is multiplying by its reciprocal $\frac{3}{4}$: $$x = \left(\frac{1}{2} - \frac{1}{6}y\right) \times \frac{3}{4}$$ 8. **Distribute $\frac{3}{4}$:** $$x = \frac{1}{2} \times \frac{3}{4} - \frac{1}{6}y \times \frac{3}{4} = \frac{3}{8} - \frac{3}{24}y$$ 9. **Simplify $\frac{3}{24}y$:** $$\frac{3}{24}y = \frac{1}{8}y$$ 10. **Final expression:** $$x = \frac{3}{8} - \frac{1}{8}y$$ This expresses $x$ in terms of $y$.