1. **State the problem:** Simplify the algebraic expression $$\frac{x}{4} + \frac{2y}{3} - \frac{5y}{6} + \frac{4x}{3}$$. 2. **Group like terms:** Group the terms with $x$ and the terms with $y$ separately: $$\left(\frac{x}{4} + \frac{4x}{3}\right) + \left(\frac{2y}{3} - \frac{5y}{6}\right)$$ 3. **Find common denominators:** - For $x$ terms, the denominators are 4 and 3. The least common denominator (LCD) is 12. - For $y$ terms, the denominators are 3 and 6. The LCD is 6. 4. **Rewrite each term with the LCD:** $$\frac{x}{4} = \frac{3x}{12}, \quad \frac{4x}{3} = \frac{16x}{12}$$ $$\frac{2y}{3} = \frac{4y}{6}, \quad \frac{5y}{6} = \frac{5y}{6}$$ 5. **Add and subtract the fractions:** $$\frac{3x}{12} + \frac{16x}{12} = \frac{3x + 16x}{12} = \frac{19x}{12}$$ $$\frac{4y}{6} - \frac{5y}{6} = \frac{4y - 5y}{6} = \frac{-y}{6}$$ 6. **Write the simplified expression:** $$\frac{19x}{12} - \frac{y}{6}$$ **Final answer:** $$\boxed{\frac{19x}{12} - \frac{y}{6}}$$