If ` x + y = 1 + xy ` then ` x^(3) + y^(3) - x^(3) y^(3)` is equal to :

Options

A. 0
B. 1
C. `-1`
D. 2

Answer

To solve the problem, we need to evaluate the expression \( x^3 + y^3 - x^3 y^3 \) given the equation \( x + y = 1 + xy \).

### Step-by-Step Solution:

1. **Given Equation**: Start with the equation provided:
\[ x + y = 1 + xy \]

2. **Trial Values**: Let's try substituting \( x = 1 \) and \( y = 1 \) to see if they satisfy the equation:
\[ 1 + 1 = 1 + 1 \cdot 1 \]
Simplifying both sides:
\[ 2 = 2 \]
This shows that \( x = 1 \) and \( y = 1 \) satisfy the equation.

3. **Calculate \( x^3 + y^3 - x^3 y^3 \)**: Now, substitute \( x = 1 \) and \( y = 1 \) into the expression we want to evaluate:
\[ x^3 + y^3 - x^3 y^3 = 1^3 + 1^3 - 1^3 \cdot 1^3 \]

4. **Evaluate Each Term**:
- Calculate \( 1^3 + 1^3 \):
\[ 1 + 1 = 2 \]
- Calculate \( 1^3 \cdot 1^3 \):
\[ 1 \cdot 1 = 1 \]

5. **Combine the Results**:
Now substitute these values back into the expression:
\[ 2 - 1 = 1 \]

Thus, the final answer is:
\[ \boxed{1} \]