Sure! Let's go through Fatima's steps and label them with the appropriate properties of equality:
1. **4 (¾ (y + 8)) = 4 (½ (2y + 4))** (Multiplication Property of Equality)
- This step shows that she multiplied both sides of the equation by 4 to eliminate the fractions.
2. **3 (y + 8) = 2y + 4** (Simplify)
- Here, she simplified the expressions on both sides.
3. **3(y) + 3(8) = 2y + 4** (Distributive Property)
- In this step, she applied the distributive property to distribute 3 across the terms in the parentheses.
4. **3y + 24 = 2y + 4** (Simplify)
- This is the simplification step, combining the products from the previous step.
5. **3y – 2y + 24 = 2y – 2y + 4** (Subtraction Property of Equality)
- In this step, she subtracted \(2y\) from both sides, maintaining equality.
6. **y + 24 = 4** (Combine like terms)
- At this point, she combined like terms to simplify the equation.
7. **y + 24 – 24 = 4 – 24** (Subtraction Property of Equality)
- Here, she subtracted 24 from both sides to isolate \(y\).
8. **y = -20** (Final Result)
- This is the solution for \(y\).
So, the labeled steps would look like this:
1. **4 (¾ (y + 8)) = 4 (½ (2y + 4))** (Multiplication Property of Equality)
2. **3 (y + 8) = 2y + 4** (Simplify)
3. **3(y) + 3(8) = 2y + 4** (Distributive Property)
4. **3y + 24 = 2y + 4** (Simplify)
5. **3y – 2y + 24 = 2y – 2y + 4** (Subtraction Property of Equality)
6. **y + 24 = 4** (Combine like terms)
7. **y + 24 – 24 = 4 – 24** (Subtraction Property of Equality)
8. **y = -20** (Final Result)
Fatima correctly used properties of equality to solve for \(y\)!