1. **Problem 1: Ages of Bianca and Bryan** The age of Bianca is 4 more than 3 times the age of Bryan. The sum of their ages is 32. Find their ages. 2. **Define variables:** Let Bryan's age be $x$. Then Bianca's age is $3x + 4$. 3. **Write the equation for the sum of their ages:** $$x + (3x + 4) = 32$$ 4. **Simplify the equation:** $$x + 3x + 4 = 32$$ $$4x + 4 = 32$$ 5. **Isolate $x$ by subtracting 4 from both sides:** $$4x + \cancel{4} - \cancel{4} = 32 - 4$$ $$4x = 28$$ 6. **Solve for $x$ by dividing both sides by 4:** $$\frac{4x}{\cancel{4}} = \frac{28}{\cancel{4}}$$ $$x = 7$$ 7. **Find Bianca's age:** $$3x + 4 = 3(7) + 4 = 21 + 4 = 25$$ 8. **Answer:** Bryan is 7 years old and Bianca is 25 years old. --- 9. **Problem 2: Ely's age** Ely's mother is 35 years old. Three years ago, she was 4 times as old as Ely was then. Find Ely's current age. 10. **Define variables:** Let Ely's current age be $y$. 11. **Write the equation for ages 3 years ago:** Mother's age 3 years ago: $35 - 3 = 32$ Ely's age 3 years ago: $y - 3$ 12. **Set up the equation:** $$32 = 4(y - 3)$$ 13. **Expand the right side:** $$32 = 4y - 12$$ 14. **Add 12 to both sides:** $$32 + 12 = 4y - 12 + 12$$ $$44 = 4y$$ 15. **Divide both sides by 4:** $$\frac{44}{\cancel{4}} = \frac{4y}{\cancel{4}}$$ $$11 = y$$ 16. **Answer:** Ely is 11 years old now.