If `x-y= -1`, then find the value of `x^3 - y^3 +3xy`

Options

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

Answer

To solve the problem, we need to find the value of \( x^3 - y^3 + 3xy \) given that \( x - y = -1 \).

### Step 1: Use the identity for \( x^3 - y^3 \)
We know the identity:
\[ x^3 - y^3 = (x - y)(x^2 + y^2 + xy) \]

### Step 2: Substitute \( x - y \)
Given \( x - y = -1 \), we can substitute this into the identity:
\[ x^3 - y^3 = (-1)(x^2 + y^2 + xy) = - (x^2 + y^2 + xy) \]

### Step 3: Rewrite the expression
Now, we need to find \( x^3 - y^3 + 3xy \):
\[ x^3 - y^3 + 3xy = - (x^2 + y^2 + xy) + 3xy \]

### Step 4: Combine like terms
Rearranging the expression gives:
\[ = - (x^2 + y^2) + 3xy - xy = - (x^2 + y^2) + 2xy \]

### Step 5: Use another identity
We can use the identity:
\[ x^2 + y^2 = (x - y)^2 + 2xy \]
Substituting \( x - y = -1 \):
\[ x^2 + y^2 = (-1)^2 + 2xy = 1 + 2xy \]

### Step 6: Substitute back into the expression
Now substitute \( x^2 + y^2 \) back into our expression:
\[ - (1 + 2xy) + 2xy = -1 - 2xy + 2xy = -1 \]

### Final Result
Thus, the value of \( x^3 - y^3 + 3xy \) is:
\[ \boxed{-1} \]