1. **Problem statement:** Simplify the expressions in exercise 2.222 parts a), b), and c). 2. **Recall:** To simplify expressions with fractions, find a common denominator, combine terms, and simplify. --- ### a) Simplify $$\frac{3z + 1}{5z} - \frac{4 - 2x}{3x} + 3$$ 3. Find a common denominator for the fractions: $$5z$$ and $$3x$$. The common denominator is $$15xz$$. 4. Rewrite each fraction with denominator $$15xz$$: $$\frac{3z + 1}{5z} = \frac{(3z + 1) \cdot 3x}{15xz} = \frac{3x(3z + 1)}{15xz}$$ $$\frac{4 - 2x}{3x} = \frac{(4 - 2x) \cdot 5z}{15xz} = \frac{5z(4 - 2x)}{15xz}$$ 5. Write 3 as $$\frac{3 \cdot 15xz}{15xz} = \frac{45xz}{15xz}$$ 6. Combine all terms over the common denominator: $$\frac{3x(3z + 1) - 5z(4 - 2x) + 45xz}{15xz}$$ 7. Expand numerators: $$3x(3z + 1) = 9xz + 3x$$ $$-5z(4 - 2x) = -20z + 10xz$$ 8. Sum numerator terms: $$9xz + 3x - 20z + 10xz + 45xz = (9xz + 10xz + 45xz) + 3x - 20z = 64xz + 3x - 20z$$ 9. Final simplified form: $$\frac{64xz + 3x - 20z}{15xz}$$ --- ### b) Simplify $$\frac{2y - z}{2z} - \frac{32z^2 + y}{18y} + 2$$ 10. Common denominator for $$2z$$ and $$18y$$ is $$18yz$$. 11. Rewrite fractions: $$\frac{2y - z}{2z} = \frac{(2y - z) \cdot 9y}{18yz} = \frac{9y(2y - z)}{18yz}$$ $$\frac{32z^2 + y}{18y} = \frac{32z^2 + y}{18y} = \frac{(32z^2 + y) \cdot z}{18yz} = \frac{z(32z^2 + y)}{18yz}$$ 12. Write 2 as $$\frac{2 \cdot 18yz}{18yz} = \frac{36yz}{18yz}$$ 13. Combine terms: $$\frac{9y(2y - z) - z(32z^2 + y) + 36yz}{18yz}$$ 14. Expand numerator: $$9y(2y - z) = 18y^2 - 9yz$$ $$-z(32z^2 + y) = -32z^3 - yz$$ 15. Sum numerator terms: $$18y^2 - 9yz - 32z^3 - yz + 36yz = 18y^2 + ( -9yz - yz + 36yz ) - 32z^3 = 18y^2 + 26yz - 32z^3$$ 16. Final simplified form: $$\frac{18y^2 + 26yz - 32z^3}{18yz}$$ --- ### c) Simplify $$\frac{x - 2y}{4y} - \frac{y - 4x}{6x} - 5$$ 17. Common denominator for $$4y$$ and $$6x$$ is $$12xy$$. 18. Rewrite fractions: $$\frac{x - 2y}{4y} = \frac{(x - 2y) \cdot 3x}{12xy} = \frac{3x(x - 2y)}{12xy}$$ $$\frac{y - 4x}{6x} = \frac{(y - 4x) \cdot 2y}{12xy} = \frac{2y(y - 4x)}{12xy}$$ 19. Write 5 as $$\frac{5 \cdot 12xy}{12xy} = \frac{60xy}{12xy}$$ 20. Combine terms: $$\frac{3x(x - 2y) - 2y(y - 4x) - 60xy}{12xy}$$ 21. Expand numerator: $$3x(x - 2y) = 3x^2 - 6xy$$ $$-2y(y - 4x) = -2y^2 + 8xy$$ 22. Sum numerator terms: $$3x^2 - 6xy - 2y^2 + 8xy - 60xy = 3x^2 + ( -6xy + 8xy - 60xy ) - 2y^2 = 3x^2 - 58xy - 2y^2$$ 23. Final simplified form: $$\frac{3x^2 - 58xy - 2y^2}{12xy}$$ --- **Final answers:** - a) $$\frac{64xz + 3x - 20z}{15xz}$$ - b) $$\frac{18y^2 + 26yz - 32z^3}{18yz}$$ - c) $$\frac{3x^2 - 58xy - 2y^2}{12xy}$$