1. **Problem Statement:** Solve the linear equation involving fractions, for example, $$\frac{2x}{3} + \frac{1}{4} = \frac{5x}{6} - \frac{1}{2}$$. 2. **Formula and Rules:** To solve linear equations with fractions, first find a common denominator to clear the fractions by multiplying every term by the least common denominator (LCD). 3. **Find the LCD:** The denominators are 3, 4, 6, and 2. The LCD of 3, 4, 6, and 2 is 12. 4. **Multiply every term by 12:** $$12 \times \frac{2x}{3} + 12 \times \frac{1}{4} = 12 \times \frac{5x}{6} - 12 \times \frac{1}{2}$$ 5. **Simplify each term:** $$4 \times 2x + 3 \times 1 = 2 \times 5x - 6 \times 1$$ $$8x + 3 = 10x - 6$$ 6. **Isolate variable terms on one side:** Subtract $8x$ from both sides: $$3 = 10x - 8x - 6$$ $$3 = 2x - 6$$ 7. **Isolate the constant term:** Add 6 to both sides: $$3 + 6 = 2x$$ $$9 = 2x$$ 8. **Solve for $x$:** Divide both sides by 2: $$x = \frac{9}{2}$$ **Final answer:** $$x = \frac{9}{2}$$ or 4.5.