State the axiom or theorem that justifies each step of the following proofs. Prove for all real numbers a, b, c, c does not equal 0, if ac = bc, then a = b.   1. ac=bc   2. 1/c is a real number  3. (ac)(1/c) = (bc)(1/c)  4. a(c times 1/c) = b(c times 1/c)   5. a times 1 = b times 1   6. a=b

Let's go through the proof step by step and identify the axioms or theorems that justify each step:

1. **Statement:** \( ac = bc \)  
   **Justification:** This is the given hypothesis or assumption.

2. **Statement:** \( \frac{1}{c} \) is a real number  
   **Justification:** The multiplicative inverse property, which states that for any nonzero real number \( c \), there exists a unique real number \( \frac{1}{c} \) such that \( c \cdot \frac{1}{c} = 1 \).

3. **Statement:** \( (ac) \cdot \frac{1}{c} = (bc) \cdot \frac{1}{c} \)  
   **Justification:** The multiplication property of equality, which states that if two expressions are equal, multiplying both sides by the same nonzero number does not change the equality.

4. **Statement:** \( a \cdot (c \cdot \frac{1}{c}) = b \cdot (c \cdot \frac{1}{c}) \)  
   **Justification:** The associative property of multiplication, which states that the way in which numbers are grouped in a product does not change the result, i.e., \( (ac) \cdot \frac{1}{c} = a \cdot (c \cdot \frac{1}{c}) \).

5. **Statement:** \( a \cdot 1 = b \cdot 1 \)  
   **Justification:** The multiplicative inverse property, which states that \( c \cdot \frac{1}{c} = 1 \) for any nonzero \( c \).

6. **Statement:** \( a = b \)  
   **Justification:** The identity property of multiplication, which states that multiplying any number by 1 gives the original number, i.e., \( a \cdot 1 = a \) and \( b \cdot 1 = b \).

This proof shows that if \( ac = bc \) and \( c \neq 0 \), then \( a = b \).

Would you like any details on this proof? Here are five related questions:

1. How does the proof change if \( c = 0 \)?
2. Can this proof be extended to complex numbers?
3. What is the significance of the nonzero condition on \( c \)?
4. How does this proof relate to the concept of division in algebra?
5. What are other situations where the multiplication property of equality is used?

**Tip:** Always verify that the operation you're using (such as division or multiplication by an inverse) is valid under the given conditions (e.g., nonzero divisor).

Explore the proof for the statement: for all real numbers a, b, c ≠ 0, if ac = bc, then a = b. Understand the mathematical axioms and theorems used, including the Multiplication Property of Equality, Associative Property of Multiplication, and Identity Property of Multiplication. Suitable for Grades 7-9.

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