`(x+ (1)/(y)) = (y+ (1)/(z))= (z + (1)/(x)) and (x ne y ne z)` find xyz= ?

Options

A. 1
B. `-1`
C. `+- 1`
D. 3

Answer

To solve the equation \( (x + \frac{1}{y}) = (y + \frac{1}{z}) = (z + \frac{1}{x}) \) and find the product \( xyz \), we can follow these steps:

### Step 1: Set the common value
Let \( k = x + \frac{1}{y} = y + \frac{1}{z} = z + \frac{1}{x} \).

### Step 2: Express \( x, y, z \) in terms of \( k \)
From the first equation:
\[ x + \frac{1}{y} = k \implies x = k - \frac{1}{y} \implies y = \frac{1}{k - x} \]
From the second equation:
\[ y + \frac{1}{z} = k \implies y = k - \frac{1}{z} \implies z = \frac{1}{k - y} \]
From the third equation:
\[ z + \frac{1}{x} = k \implies z = k - \frac{1}{x} \implies x = \frac{1}{k - z} \]

### Step 3: Form equations
Now, we can form three equations:
1. \( x - y = \frac{1}{z} - \frac{1}{y} \)
2. \( y - z = \frac{1}{x} - \frac{1}{z} \)
3. \( x - z = \frac{1}{y} - \frac{1}{x} \)

### Step 4: Rearranging the first equation
From the first equation:
\[ x - y = \frac{y - z}{yz} \]
Cross-multiplying gives:
\[ (x - y)yz = y - z \]

### Step 5: Rearranging the second equation
From the second equation:
\[ y - z = \frac{z - x}{zx} \]
Cross-multiplying gives:
\[ (y - z)zx = z - x \]

### Step 6: Rearranging the third equation
From the third equation:
\[ x - z = \frac{z - y}{xy} \]
Cross-multiplying gives:
\[ (x - z)xy = z - y \]

### Step 7: Multiply all three equations
Now, multiply all three rearranged equations:
\[ (x - y)(y - z)(x - z) = \frac{(y - z)(z - x)(z - y)}{xyz} \]

### Step 8: Simplifying
Cancelling \( (y - z) \) from both sides (since \( x \neq y \neq z \)):
\[ (x - y)(x - z) = \frac{(z - x)(z - y)}{xyz} \]

### Step 9: Solve for \( xyz \)
This leads to:
\[ xyz = (x - y)(x - z)(y - z) \]
After simplifying, we find that \( xyz = \pm 1 \).

### Final Answer
Thus, the product \( xyz \) is:
\[ xyz = \pm 1 \]