In algebra, the real numbers x and y satisfy x − y = 7. Using this relationship, evaluate the cubic difference (x − 15)^3 − (y − 8)^3 exactly, without expanding both cubes fully from scratch.

Introduction / Context:
This question tests your ability to recognise structure in algebraic expressions rather than performing long and unnecessary expansions. It involves a known difference between two cubic expressions, but with shifted arguments x − 15 and y − 8. When such shifts are present, the key is to see whether these shifted quantities might actually be equal, which would make their difference very simple to evaluate.

Given Data / Assumptions:
- x and y are real numbers related by x − y = 7.
- We must compute (x − 15)^3 − (y − 8)^3 exactly.
- No specific numerical values of x and y are given, so the answer must be independent of the individual values and depend only on the relation between them.

Concept / Approach:
The expression (x − 15)^3 − (y − 8)^3 is a difference of cubes. Normally, one might use the identity A^3 − B^3 = (A − B)(A^2 + AB + B^2). However, before applying this identity, we should check whether A and B themselves might be equal. If the inner expressions are equal, then A^3 − B^3 is automatically zero, and the problem becomes trivial.

Step-by-Step Solution:
1) Let A = x − 15 and B = y − 8. Then the expression is A^3 − B^3. 2) Compute A − B = (x − 15) − (y − 8) = x − y − 7. 3) Substitute the known relation x − y = 7, giving A − B = 7 − 7 = 0. 4) Hence A = B, which means x − 15 = y − 8. 5) When A = B, the expression A^3 − B^3 equals zero for any real values of A and B. 6) Therefore (x − 15)^3 − (y − 8)^3 = 0.

Verification / Alternative check:
Choose a simple numerical example that satisfies x − y = 7, for instance x = 10 and y = 3. Then x − 15 = −5 and y − 8 = −5, so both inner terms are equal. Their cubes are also equal: (−5)^3 = −125. The difference between these two identical cubes is −125 − (−125) = 0. This confirms that the result is zero for at least one example, and the algebraic reasoning shows that it must be zero for all pairs satisfying x − y = 7.

Why Other Options Are Wrong:
Option A (343), Option C (392), Option D (399), and Option E (49) are all non zero values that might appear if someone incorrectly plugged x − y directly into a cube without addressing the shifts, or tried an incomplete expansion. However, once we know that the shifted terms are equal, any non zero answer must be incorrect. Only Option B reflects the correct conclusion that the difference is zero.

Common Pitfalls:
A typical mistake is to expand both cubes fully, which is time consuming and prone to algebraic sign errors, especially under exam pressure. Another pitfall is to try to substitute x − y = 7 directly into the expression without first adjusting for the −15 and −8 shifts, missing the key insight that these shifts make the cubes identical. Recognising patterns and checking for equality of inner expressions can save valuable time and prevent errors.

Final Answer:
Because the relation x − y = 7 makes x − 15 and y − 8 equal, the two cubes are identical, so their difference is 0.