1. **State the problem:** Solve the linear equation $$\frac{1}{3}x - \frac{3}{4} = \frac{1}{2}x - \frac{2}{3}$$ for $x$. 2. **Write down the equation:** $$\frac{1}{3}x - \frac{3}{4} = \frac{1}{2}x - \frac{2}{3}$$ 3. **Goal:** Isolate $x$ on one side. 4. **Subtract $\frac{1}{2}x$ from both sides:** $$\frac{1}{3}x - \frac{1}{2}x - \frac{3}{4} = - \frac{2}{3}$$ 5. **Find common denominator for $\frac{1}{3}x - \frac{1}{2}x$:** $$\frac{2}{6}x - \frac{3}{6}x = -\frac{1}{6}x$$ So the equation becomes: $$-\frac{1}{6}x - \frac{3}{4} = - \frac{2}{3}$$ 6. **Add $\frac{3}{4}$ to both sides:** $$-\frac{1}{6}x = - \frac{2}{3} + \frac{3}{4}$$ 7. **Find common denominator for the right side:** $$- \frac{2}{3} + \frac{3}{4} = - \frac{8}{12} + \frac{9}{12} = \frac{1}{12}$$ So: $$-\frac{1}{6}x = \frac{1}{12}$$ 8. **Divide both sides by $-\frac{1}{6}$:** $$x = \frac{\frac{1}{12}}{-\frac{1}{6}} = \frac{1}{12} \times \frac{-6}{1} = -\frac{6}{12}$$ 9. **Simplify the fraction:** $$x = -\frac{1}{2}$$ **Final answer:** $$x = -\frac{1}{2}$$