Consider the set of equation `x+y+xy=35,y+z+yz=8,z+x+zx=3`. If `z_(1),z_(2)` are possible values of z, then `|z_(1)-z_(2)|` is equal to

Options

A. 1
B. 3
C. 2
D. none of these

Answer

To solve the given set of equations:

1. **Equations Given:**
\[ x + y + xy = 35 \quad (1) \]
\[ y + z + yz = 8 \quad (2) \]
\[ z + x + zx = 3 \quad (3) \]

2. **Rearranging Each Equation:**
We can rewrite each equation in a more manageable form. We can add 1 to both sides of each equation to factor them.

For equation (1):
\[ x + y + xy + 1 = 36 \implies (1 + x)(1 + y) = 36 \]

For equation (2):
\[ y + z + yz + 1 = 9 \implies (1 + y)(1 + z) = 9 \]

For equation (3):
\[ z + x + zx + 1 = 4 \implies (1 + z)(1 + x) = 4 \]

3. **Letting Variables:**
Let \( a = 1 + x \), \( b = 1 + y \), \( c = 1 + z \). Then we can rewrite the equations as:
\[ ab = 36 \quad (4) \]
\[ bc = 9 \quad (5) \]
\[ ca = 4 \quad (6) \]

4. **Finding Relationships:**
From equations (4), (5), and (6), we can express \( a \), \( b \), and \( c \) in terms of each other.

From (4):
\[ b = \frac{36}{a} \]

Substitute \( b \) into (5):
\[ \frac{36}{a}c = 9 \implies c = \frac{9a}{36} = \frac{a}{4} \]

Substitute \( c \) into (6):
\[ \left(\frac{a}{4}\right)a = 4 \implies \frac{a^2}{4} = 4 \implies a^2 = 16 \implies a = 4 \quad \text{(since } a > 0\text{)} \]

5. **Finding \( b \) and \( c \):**
Now substituting \( a = 4 \) back into our equations for \( b \) and \( c \):
\[ b = \frac{36}{4} = 9 \]
\[ c = \frac{4}{4} = 1 \]

6. **Finding \( z \):**
Recall that \( c = 1 + z \):
\[ 1 + z = 1 \implies z = 0 \]

Now we need to find the second value of \( z \). We can use the relationships we derived earlier.

From \( bc = 9 \):
\[ 9c = 9 \implies c = 1 \quad \text{(already found)} \]

From \( ab = 36 \):
\[ 4b = 36 \implies b = 9 \quad \text{(already found)} \]

From \( ca = 4 \):
\[ 1 \cdot 4 = 4 \quad \text{(already found)} \]

We can also check the quadratic formed by substituting back into the equations.

7. **Finding Possible Values of \( z \):**
From the equations, we can derive:
\[ z^2 + 2z = 0 \implies z(z + 2) = 0 \]
Thus, the possible values of \( z \) are:
\[ z_1 = 0, \quad z_2 = -2 \]

8. **Finding the Absolute Difference:**
Now we need to find \( |z_1 - z_2| \):
\[ |0 - (-2)| = |0 + 2| = 2 \]

Thus, the final answer is:
\[ \boxed{2} \]