Property Identification 1639C3

1. **State the problem:** Identify the property used in each step of solving the equation $2(5x - 7) = 2x + 10$. 2. **Step 1:** From $2(5x - 7) = 2x + 10$ to $10x - 14 = 2x + 10$ - The property used is the **Distributive Property**, which states $a(b + c) = ab + ac$. - Here, $2$ is distributed over $(5x - 7)$: $$2 \times 5x = 10x, \quad 2 \times (-7) = -14$$ 3. **Step 2:** From $10x - 14 = 2x + 10$ to $10x - 14 - 2x = 2x + 10 - 2x$ - The property used is the **Subtraction Property of Equality**, which allows subtracting the same quantity from both sides. - Simplifying: $$10x - 14 - 2x = 8x - 14, \quad 2x + 10 - 2x = 10$$ 4. **Step 3:** From $8x - 14 = 10$ to $8x - 14 + 14 = 10 + 14$ - The property used is the **Addition Property of Equality**, which allows adding the same quantity to both sides. - Simplifying: $$8x - 14 + 14 = 8x, \quad 10 + 14 = 24$$ 5. **Step 4:** From $8x = 24$ to $\frac{8x}{8} = \frac{24}{8}$ - The property used is the **Division Property of Equality**, which allows dividing both sides by the same nonzero number. - Simplifying with cancellation: $$\frac{\cancel{8}x}{\cancel{8}} = \frac{24}{8} \implies x = 3$$ **Final answer:** The property used in the first step (from $2(5x - 7)$ to $10x - 14$) is the **Distributive Property** (Option D).