Prove the following theorem. Assume that each variable represents a real number. If ca = cb and c does not equal 0, then a = b

To prove the theorem, we can use a logical sequence of algebraic steps justified by well-known axioms and properties of real numbers.

**Theorem:** If \( ca = cb \) and \( c \neq 0 \), then \( a = b \).

### Proof:

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

2. **Statement:** \( c \neq 0 \)  
   **Justification:** This is also a part of the given hypothesis.

3. **Statement:** \( a = \frac{ca}{c} \) and \( b = \frac{cb}{c} \)  
   **Justification:** The division property of equality, which states that if \( x = y \) and \( z \neq 0 \), then \( \frac{x}{z} = \frac{y}{z} \). Here, we divide both sides of the equation \( ca = cb \) by the nonzero real number \( c \).

4. **Statement:** \( \frac{ca}{c} = \frac{cb}{c} \)  
   **Justification:** Since both sides of the equation \( ca = cb \) are divided by the same nonzero number \( c \), their quotients must be equal.

5. **Statement:** \( a = b \)  
   **Justification:** The cancellation law of multiplication, which states that if \( \frac{ca}{c} = \frac{cb}{c} \) and \( c \neq 0 \), then \( a = b \).

This completes the proof that if \( ca = cb \) and \( c \neq 0 \), then \( a = b \).

Do you have any questions or need further clarification? Here are five related questions:

1. How does the proof rely on the condition \( c \neq 0 \)?
2. Can this proof be applied to complex numbers as well?
3. How would the proof change if \( c = 0 \)?
4. Why is the division property of equality crucial in this proof?
5. Can you think of a real-world scenario where this theorem might be useful?

**Tip:** When proving theorems, ensure each step is logically justified by a known property or axiom to maintain the rigor of the proof.

Explore the proof of the theorem: If ca = cb and c ≠ 0, then a = b. This mathematical problem involves real numbers, using the division property of equality and the cancellation law of multiplication to establish the relationship between variables.

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