The number of lines represented by the equation `x^(2)=y^(2)=z^(2)`, is ______________

Answer

To solve the equation \( x^2 = y^2 = z^2 \) and determine the number of lines represented by it, we can break it down into steps.

### Step-by-Step Solution:

1. **Understanding the Equation**:
The equation \( x^2 = y^2 = z^2 \) implies that \( x^2 = y^2 \) and \( y^2 = z^2 \). This means we can analyze each pair of variables separately.

2. **Analyzing \( x^2 = y^2 \)**:
We can rewrite this as:
\[ x^2 - y^2 = 0 \]
This can be factored using the difference of squares:
\[ (x + y)(x - y) = 0 \]
From this factorization, we get two equations:
\[ x + y = 0 \quad \text{(1)} \]
\[ x - y = 0 \quad \text{(2)} \]
From equation (1), we have \( y = -x \), and from equation (2), we have \( y = x \). Thus, we have two lines:
- Line 1: \( y = x \)
- Line 2: \( y = -x \)

3. **Analyzing \( y^2 = z^2 \)**:
Similarly, we can analyze \( y^2 = z^2 \):
\[ y^2 - z^2 = 0 \]
Factoring gives us:
\[ (y + z)(y - z) = 0 \]
This results in:
\[ y + z = 0 \quad \text{(3)} \]
\[ y - z = 0 \quad \text{(4)} \]
From equation (3), we have \( z = -y \), and from equation (4), we have \( z = y \). Thus, we have two more lines:
- Line 3: \( z = y \)
- Line 4: \( z = -y \)

4. **Analyzing \( x^2 = z^2 \)**:
Finally, we analyze \( x^2 = z^2 \):
\[ x^2 - z^2 = 0 \]
Factoring gives us:
\[ (x + z)(x - z) = 0 \]
This results in:
\[ x + z = 0 \quad \text{(5)} \]
\[ x - z = 0 \quad \text{(6)} \]
From equation (5), we have \( z = -x \), and from equation (6), we have \( z = x \). Thus, we have two more lines:
- Line 5: \( z = x \)
- Line 6: \( z = -x \)

5. **Counting the Lines**:
We have derived a total of 6 distinct lines from the equations:
- Line 1: \( y = x \)
- Line 2: \( y = -x \)
- Line 3: \( z = y \)
- Line 4: \( z = -y \)
- Line 5: \( z = x \)
- Line 6: \( z = -x \)

Thus, the total number of lines represented by the equation \( x^2 = y^2 = z^2 \) is **6**.

### Final Answer:
The number of lines represented by the equation \( x^2 = y^2 = z^2 \) is **6**.