Question

The Commutative Property The Associative Property a + b = b + ____ a + (b + c) = (a + ____ ) + c 7 + (5 − 7) = 7 + 5 + (−7) 5 − 4 ? 4 − 5 Use the Commutative Property: 1 ≠ −1 7 + ( ____ ) + 5 5 + (−4) = ( ____ ) + 5 Use the Associative Property: 1 = 1 [7 + (−7)] + 5 = ____

Answer 1

Let's fill in the blanks based on the Commutative and Associative Properties:

**Commutative Property:** This property states that the order in which you add two numbers does not affect the sum. Therefore:
- \( a + b = b + a \)

For the first blank:
- \( a + b = b + \text{(blank)} \)
- The blank will be \( a \) (the first number) so the equation becomes:
- \( a + b = b + a \)

For the second blank:
- \( a + (b + c) = (a + \text{(blank)}) + c \)
- The blank will be \( b \) (the second number), so the equation becomes:
- \( a + (b + c) = (a + b) + c \)

Now on to the specific equations:
1. \( 7 + (5 - 7) = 7 + 5 + (-7) \) is valid as it shows the addition of the terms properly by simplifying \( 5 - 7 \) into \( -2 \), therefore equal:
- \( 7 + (-2) = 7 + 5 - 7 \) confirming the equality.

2. \( 5 - 4 \) is equal to \( 1 \).

Next for the Commutative Property:
- \( 7 + ( \text{blank} ) + 5 \)
- To keep the sum correct, the blank should be \( 0 \) because adding zero does not change the sum.

So, we have \( 7 + 0 + 5 \) which can also be rearranged to \( 0 + 7 + 5 \) using the Commutative property.

Next, \( 5 + (-4) = ( \text{blank} ) + 5 \) can be solved by recognizing that:
- \( 5 - 4 = 1 \), hence the blank will be \( 1 \).

For the Associative Property:
- \( [7 + (-7)] + 5 = \text{(blank)} \)
- Since \( 7 + (-7) = 0 \), it follows that \( 0 + 5 = 5 \), so the blank should be filled with \( 5 \).

To summarize:
- The blanks filled:
- \( a + b = b + a \)
- \( a + (b + c) = (a + b) + c \)
- \( 7 + (5 - 7) = 7 + 5 + (-7) \)
- \( 5 - 4 = 1 \)
- Use the Commutative Property: \( 7 + 0 + 5 \)
- \( 5 + (-4) = 1 + 5 \)
- Use the Associative Property: \( [7 + (-7)] + 5 = 5 \)

This confirms proper completion of the properties and summation logic.