1. The problem is to simplify each given fraction expression step-by-step. 2. For expression 1: $\frac{a}{b} + \frac{c}{d}$, find common denominator $bd$, then write $$\frac{ad}{bd} + \frac{cb}{bd} = \frac{ad+cb}{bd}.$$ This is the simplified form. 3. For expression 2: $\frac{a}{b} - \frac{c}{d}$, similarly use common denominator $bd$, $$\frac{ad}{bd} - \frac{cb}{bd} = \frac{ad-cb}{bd}.$$ 4. For expression 3: $\frac{2}{2} + \frac{3}{y}$, simplify $\frac{2}{2}=1$, then sum $$1 + \frac{3}{y} = \frac{y}{y} + \frac{3}{y} = \frac{y+3}{y}.$$ 5. For expression 4: $\frac{2}{3} - \frac{1}{24}$, common denominator is $24$, rewrite $$\frac{16}{24} - \frac{1}{24} = \frac{15}{24} = \frac{5}{8}.$$ 6. For expression 5: $\frac{x}{x+y} + \frac{y}{y+z}$, this generally cannot be simplified further without knowing relationships between variables, so leave it as is. 7. For expression 6: $\frac{2}{5} - \frac{1}{7}$, common denominator $35$, write $$\frac{14}{35} - \frac{5}{35} = \frac{9}{35}.$$ 8. For expression 7: $\frac{2}{3} + \frac{1}{7} - \frac{1}{5}$, common denominator $105$, write $$\frac{70}{105} + \frac{15}{105} - \frac{21}{105} = \frac{70+15-21}{105} = \frac{64}{105}.$$ 9. For expression 8: simplify the compound fraction $$\frac{\frac{1}{3} - \frac{1}{4}}{\frac{1}{4} + \frac{1}{3}}.$$ Calculate numerator: $$\frac{1}{3} - \frac{1}{4} = \frac{4}{12} - \frac{3}{12} = \frac{1}{12}.$$ Calculate denominator: $$\frac{1}{4} + \frac{1}{3} = \frac{3}{12} + \frac{4}{12} = \frac{7}{12}.$$ The entire expression is $$\frac{\frac{1}{12}}{\frac{7}{12}} = \frac{1}{12} \times \frac{12}{7} = \frac{1}{7}.$$ Final answers: 1) $\frac{ad+cb}{bd}$ 2) $\frac{ad-cb}{bd}$ 3) $\frac{y+3}{y}$ 4) $\frac{5}{8}$ 5) $\frac{x}{x+y} + \frac{y}{y+z}$ (no simpler form) 6) $\frac{9}{35}$ 7) $\frac{64}{105}$ 8) $\frac{1}{7}$