1. **Stating the problem:** Add and subtract the given algebraic fractions: $$\frac{4x + 3y}{4x} + \frac{4x - 3y}{6y}$$ and $$9 - \frac{7 + 4x}{3y}$$ 2. **Formula and rules:** To add or subtract fractions, find a common denominator, then combine the numerators. 3. **First expression:** Find the least common denominator (LCD) of $4x$ and $6y$. $$\text{LCD} = \text{lcm}(4x, 6y) = 12xy$$ Rewrite each fraction with denominator $12xy$: $$\frac{4x + 3y}{4x} = \frac{(4x + 3y) \cdot \cancel{3y}}{\cancel{4x} \cdot 3y} = \frac{3y(4x + 3y)}{12xy}$$ $$\frac{4x - 3y}{6y} = \frac{(4x - 3y) \cdot \cancel{2x}}{\cancel{6y} \cdot 2x} = \frac{2x(4x - 3y)}{12xy}$$ 4. **Add the fractions:** $$\frac{3y(4x + 3y)}{12xy} + \frac{2x(4x - 3y)}{12xy} = \frac{3y(4x + 3y) + 2x(4x - 3y)}{12xy}$$ Expand the numerators: $$3y(4x + 3y) = 12xy + 9y^2$$ $$2x(4x - 3y) = 8x^2 - 6xy$$ Sum: $$12xy + 9y^2 + 8x^2 - 6xy = 8x^2 + (12xy - 6xy) + 9y^2 = 8x^2 + 6xy + 9y^2$$ So the sum is: $$\frac{8x^2 + 6xy + 9y^2}{12xy}$$ 5. **Second expression:** $$9 - \frac{7 + 4x}{3y}$$ Rewrite 9 as a fraction with denominator $3y$: $$9 = \frac{9 \cdot 3y}{3y} = \frac{27y}{3y}$$ Subtract: $$\frac{27y}{3y} - \frac{7 + 4x}{3y} = \frac{27y - (7 + 4x)}{3y} = \frac{27y - 7 - 4x}{3y}$$ 6. **Final answers:** $$\frac{8x^2 + 6xy + 9y^2}{12xy}$$ and $$\frac{27y - 7 - 4x}{3y}$$