To solve for \( x \) in the equation \( xy = yx \), we need to analyze the equation. This equation is symmetric and holds true for all real numbers \( x \) and \( y \), meaning \( x \) can be any real number.

Let's explore a few specific cases to understand why:

1. **If \( y = 0 \):**  
   The equation becomes \( 0 = 0 \), which is true for any \( x \).

2. **If \( y = 1 \):**  
   The equation becomes \( x = x \), which is obviously true for any \( x \).

3. **If \( y = 2 \):**  
   The equation becomes \( 2x = 2x \), which is also true for any \( x \).

4. **General Case:**  
   The equation \( xy = yx \) holds true for all real numbers \( x \) and \( y \) because multiplication is commutative in the set of real numbers.

Thus, \( x \) can be any real number. There is no restriction on \( x \) based on the equation \( xy = yx \).

### Questions for Further Exploration:
1. What happens if \( x \) and \( y \) are matrices? Does \( xy = yx \) still hold?
2. How does the equation \( xy = yx \) change if \( x \) and \( y \) are complex numbers?
3. What are the implications of commutative multiplication in other algebraic structures, such as rings or fields?
4. Can you find examples where \( xy \neq yx \)? Under what conditions does this occur?
5. How does this equation relate to the concept of commutative groups?

### Tip:
When solving equations involving variables, consider different types of numbers (integers, real numbers, complex numbers) and algebraic structures to fully explore the possible solutions.

Explore the commutative property of multiplication in equations and understand why the equation xy = yx holds true for all real numbers x and y. This detailed explanation covers algebraic concepts and real number properties, suitable for advanced high school students.

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