1. **Problem 1:** Solve the equation $5(2b + 1) = 76 + 11$ for $b$. 2. **Step 1:** Expand the left side using distributive property: $$5 \times 2b + 5 \times 1 = 10b + 5$$ 3. **Step 2:** Write the equation: $$10b + 5 = 76 + 11$$ 4. **Step 3:** Simplify the right side: $$10b + 5 = 87$$ 5. **Step 4:** Subtract 5 from both sides: $$10b + \cancel{5} - \cancel{5} = 87 - 5$$ $$10b = 82$$ 6. **Step 5:** Divide both sides by 10: $$\frac{10b}{\cancel{10}} = \frac{82}{10}$$ $$b = \frac{82}{10} = \frac{41}{5}$$ --- 7. **Problem 2:** Solve the equation $7 \frac{21}{8} t - \frac{4}{7} = 5 \frac{3}{4} 6 t - 4$ for $t$. 8. **Step 1:** Convert mixed numbers to improper fractions: $$7 \frac{21}{8} = \frac{7 \times 8 + 21}{8} = \frac{56 + 21}{8} = \frac{77}{8}$$ $$5 \frac{3}{4} = \frac{5 \times 4 + 3}{4} = \frac{20 + 3}{4} = \frac{23}{4}$$ 9. **Step 2:** Rewrite the equation: $$\frac{77}{8} t - \frac{4}{7} = \frac{23}{4} \times 6 t - 4$$ 10. **Step 3:** Multiply $\frac{23}{4} \times 6$: $$\frac{23}{4} \times 6 = \frac{23 \times 6}{4} = \frac{138}{4} = \frac{69}{2}$$ 11. **Step 4:** Rewrite equation: $$\frac{77}{8} t - \frac{4}{7} = \frac{69}{2} t - 4$$ 12. **Step 5:** Add $\frac{4}{7}$ to both sides: $$\frac{77}{8} t = \frac{69}{2} t - 4 + \frac{4}{7}$$ 13. **Step 6:** Find common denominator for $-4 + \frac{4}{7}$: $$-4 = -\frac{28}{7}$$ $$-\frac{28}{7} + \frac{4}{7} = -\frac{24}{7}$$ 14. **Step 7:** Equation becomes: $$\frac{77}{8} t = \frac{69}{2} t - \frac{24}{7}$$ 15. **Step 8:** Subtract $\frac{69}{2} t$ from both sides: $$\frac{77}{8} t - \frac{69}{2} t = - \frac{24}{7}$$ 16. **Step 9:** Find common denominator for left side terms (8): $$\frac{77}{8} t - \frac{69 \times 4}{8} t = \frac{77}{8} t - \frac{276}{8} t = \frac{77 - 276}{8} t = -\frac{199}{8} t$$ 17. **Step 10:** Equation is: $$-\frac{199}{8} t = - \frac{24}{7}$$ 18. **Step 11:** Multiply both sides by $-1$: $$\frac{199}{8} t = \frac{24}{7}$$ 19. **Step 12:** Solve for $t$ by dividing both sides by $\frac{199}{8}$: $$t = \frac{24}{7} \times \frac{8}{199} = \frac{24 \times 8}{7 \times 199} = \frac{192}{1393}$$ --- **Final answers:** $$b = \frac{41}{5}$$ $$t = \frac{192}{1393}$$