Based on the uploaded image, I can identify the mathematical properties of addition being applied in each case. Here's the breakdown of the equations and the corresponding properties:

1. Equation: $-b + b = 0$
   Property: This is the Inverse Property of Addition, which states that the sum of a number and its additive inverse (negative) is zero.

2. Equation: $3 + (9 + a) = (3 + 9) + a$
   Property: This is the Associative Property of Addition, which states that the way numbers are grouped in an addition problem does not affect the sum.

3. Equation: $6 = 0 + 6$
   Property: This is the Identity Property of Addition, which states that adding zero to any number does not change the value of that number.

If you have any follow-up questions or need further explanations, let me know!

Follow-Up Questions:

1. Can you explain how the Inverse Property of Addition works with real numbers?
2. Why does the Associative Property only apply to addition and multiplication?
3. What is the significance of the Identity Property in basic arithmetic?
4. Are there other properties of addition that relate to these examples?
5. How can understanding these properties help in solving algebraic equations?

Tip:

Remember that identifying these properties can simplify problem-solving and proofs in algebra and arithmetic!