If (x + y)^2 = xy + 1 and x^3 − y^3 = 1 for real numbers x and y, then what is the value of the difference x − y?

Introduction / Context:
This algebra question combines symmetric expressions and identities involving x + y, xy, and x^3 − y^3. It is designed to test the ability to use algebraic identities cleverly instead of expanding everything directly. Such problems appear frequently in aptitude exams and algebra sections of school level mathematics.

Given Data / Assumptions:

(x + y)^2 = xy + 1.
x^3 − y^3 = 1.
x and y are real numbers.
We need to find x − y.

Concept / Approach:
The identity for the difference of cubes, x^3 − y^3 = (x − y)(x^2 + xy + y^2), is central here. The expression x^2 + xy + y^2 can be written in terms of x + y and xy. The equation (x + y)^2 = xy + 1 gives a direct relationship between these symmetric parts. The trick is to express x^2 + xy + y^2 in a form that can be evaluated using the given relation without solving for x and y individually.

Step-by-Step Solution:
Start with the identity: x^3 − y^3 = (x − y)(x^2 + xy + y^2).We are given x^3 − y^3 = 1, so 1 = (x − y)(x^2 + xy + y^2).Now use (x + y)^2 = x^2 + 2xy + y^2 = xy + 1.From this, x^2 + 2xy + y^2 = xy + 1.Rearrange: x^2 + y^2 = (xy + 1) − 2xy = 1 − xy.Now compute x^2 + xy + y^2 = (x^2 + y^2) + xy = (1 − xy) + xy = 1.Thus x^2 + xy + y^2 equals 1.Substitute back into 1 = (x − y)(x^2 + xy + y^2): 1 = (x − y) * 1.Therefore x − y = 1.

Verification / Alternative check:
We can check the result by constructing a concrete pair (x, y) that satisfies both given conditions. If x − y = 1 and x^2 + xy + y^2 = 1, then x and y will satisfy x^3 − y^3 = 1 by the cube identity. Solving the system to find actual values is possible but not required, because the algebraic manipulation already shows that the only value consistent with x^3 − y^3 = 1 and x^2 + xy + y^2 = 1 is x − y = 1.

Why Other Options Are Wrong:
Values such as 0, −1, 2, or 3 for x − y would change x^3 − y^3 to some number other than 1 when multiplied by x^2 + xy + y^2, which we have shown equals 1. Therefore they are inconsistent with the given condition x^3 − y^3 = 1 and must be rejected.

Common Pitfalls:
Students sometimes expand everything and attempt to solve for x and y directly, which leads to unnecessary complexity. Others misapply the square identity and incorrectly compute x^2 + xy + y^2. Keeping track of the symmetric forms and using identities systematically is the fastest and safest route. Remember that x^2 + xy + y^2 = (x + y)^2 − xy, and that can be easily evaluated from the given relation.

Final Answer:
The required value of x − y is 1.