CSSS Identify Structure Explain how to use the Multiplication Property of Equality o solve \( \frac{2}{5} y=6 \). Then explain how to solve the equation using the Division Property Equality.

To solve the equation \( \frac{2}{5} y = 6 \) using the Multiplication Property of Equality and the Division Property of Equality, we will follow these steps: ### Step 1: Multiplication Property of Equality The Multiplication Property of Equality states that if you multiply both sides of an equation by the same non-zero number, the two sides remain equal. 1. **Identify the equation**: \[ \frac{2}{5} y = 6 \] 2. **Multiply both sides by the reciprocal of \( \frac{2}{5} \)**, which is \( \frac{5}{2} \): \[ \frac{5}{2} \cdot \frac{2}{5} y = \frac{5}{2} \cdot 6 \] 3. **Simplify the left side**: \[ y = \frac{5}{2} \cdot 6 \] 4. **Calculate \( \frac{5}{2} \cdot 6 \)**: \[ y = \frac{5 \cdot 6}{2} = \frac{30}{2} = 15 \] So, using the Multiplication Property of Equality, we find that: \[ y = 15 \] ### Step 2: Division Property of Equality The Division Property of Equality states that if you divide both sides of an equation by the same non-zero number, the two sides remain equal. 1. **Start with the original equation**: \[ \frac{2}{5} y = 6 \] 2. **Divide both sides by \( \frac{2}{5} \)**: \[ y = 6 \div \frac{2}{5} \] 3. **Dividing by a fraction is the same as multiplying by its reciprocal**: \[ y = 6 \cdot \frac{5}{2} \] 4. **Calculate \( 6 \cdot \frac{5}{2} \)**: \[ y = \frac{6 \cdot 5}{2} = \frac{30}{2} = 15 \] So, using the Division Property of Equality, we also find that: \[ y = 15 \] ### Conclusion In both methods, we arrive at the same solution: \( y = 15 \). The Multiplication Property of Equality allows us to eliminate the fraction by multiplying, while the Division Property of Equality allows us to isolate \( y \) by dividing by the fraction. Both properties are fundamental in solving equations effectively.