Property Matching A9Fbbc

1. **State the problem:** We need to identify the property of each equation and each step in the simplification. 2. **Properties for equations 1 to 6:** - Equation 1: $ (5 + 2) + 8 = 5 + (2 + 8) $ is the **Associative Property of Addition** because it changes grouping without changing the sum. - Equation 2: $ 5 + 4 = 4 + 5 $ is the **Commutative Property of Addition** because it changes the order of addends. - Equation 3: $ 5 \cdot 1 = 5 $ is the **Multiplicative Identity Property** because multiplying by 1 leaves the number unchanged. - Equation 4: $ 0 + 2 = 2 $ is the **Additive Identity Property** because adding zero leaves the number unchanged. - Equation 5: $ 8(0) = 0 $ is the **Multiplicative Property of Zero** because any number times zero is zero. - Equation 6: $ 2 (3 + 5) = 2 \cdot 3 + 2 \cdot 5 $ is the **Distributive Property** because multiplication is distributed over addition. 3. **Properties for simplification steps in problem 7:** Given expression: $$7.5x + [2.8 + (-3.2x)]$$ - Step 1 to 2: $$7.5x + [2.8 + (-3.2x)] = 7.5x + [-3.2x + 2.8]$$ This is the **Commutative Property of Addition** inside the brackets, changing order of terms. - Step 2 to 3: $$7.5x + [-3.2x + 2.8] = [7.5x + (-3.2x)] + 2.8$$ This is the **Associative Property of Addition**, regrouping terms. - Step 3 to 4: $$[7.5x + (-3.2x)] + 2.8 = 4.3x + 2.8$$ This is **Combining Like Terms** by adding coefficients: $$7.5x - 3.2x = 4.3x$$. 4. **Summary of properties:** | Equation/Step | Property Name | |--------------|----------------| | 1 | Associative Property of Addition | | 2 | Commutative Property of Addition | | 3 | Multiplicative Identity Property | | 4 | Additive Identity Property | | 5 | Multiplicative Property of Zero | | 6 | Distributive Property | | 7 Step 1-2 | Commutative Property of Addition | | 7 Step 2-3 | Associative Property of Addition | | 7 Step 3-4 | Combining Like Terms | **Final answer:** The properties justifying each step are as above.