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Difficulty: Very hardAlgebraic Equations and Systems in Data Sufficiency

If xx and yy are real numbers such that x+y≠0x + y \neq 0, what is the value of x3+y3x+y\frac{x^3 + y^3}{x + y}?

(1) x2−xy+y2=12x^2 - xy + y^2 = 12
(2) x2+y2=20x^2 + y^2 = 20 and xy=8xy = 8

  1. A
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
  2. B
    Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
  3. C
    BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
  4. EACH statement ALONE is sufficient.Answer
  5. E
    Statements (1) and (2) TOGETHER are NOT sufficient.

Answer

EACH statement ALONE is sufficient.
Factoring the numerator using the sum of cubes identity x3+y3=(x+y)(x2−xy+y2)x^3 + y^3 = (x + y)(x^2 - xy + y^2) allows canceling (x+y)(x + y), since x+y≠0x + y \neq 0. Thus, the question asks for the value of x2−xy+y2x^2 - xy + y^2. Statement (1) directly states that x2−xy+y2=12x^2 - xy + y^2 = 12, which is sufficient. Statement (2) gives x2+y2=20x^2 + y^2 = 20 and xy=8xy = 8, so x2−xy+y2=(x2+y2)−xy=20−8=12x^2 - xy + y^2 = (x^2 + y^2) - xy = 20 - 8 = 12, which is also sufficient. Since each statement independently yields a unique value, the correct choice states that each statement alone is sufficient.

Step-by-Step Solution

1
Simplify and rephrase the target expression in the question stem.
Since x+y≠0x + y \neq 0, factor the numerator using the sum of cubes formula x3+y3=(x+y)(x2−xy+y2)x^3 + y^3 = (x + y)(x^2 - xy + y^2). The expression simplifies to (x+y)(x2−xy+y2)x+y=x2−xy+y2\frac{(x + y)(x^2 - xy + y^2)}{x + y} = x^2 - xy + y^2. The question target is equivalent to finding the value of x2−xy+y2x^2 - xy + y^2.
Simplifying the target expression before analyzing the statements eliminates unnecessary variables and reveals the exact algebraic value needed.
2
Evaluate Statement (1) independently.
Statement (1) gives x2−xy+y2=12x^2 - xy + y^2 = 12. Since this matches the simplified question target directly, the value is uniquely determined as 12.
Statement (1) provides the exact value of the rephrased target expression.
3
Evaluate Statement (2) independently.
Statement (2) provides x2+y2=20x^2 + y^2 = 20 and xy=8xy = 8. Substituting these into the target expression x2−xy+y2=(x2+y2)−xyx^2 - xy + y^2 = (x^2 + y^2) - xy gives 20−8=1220 - 8 = 12. The value is uniquely determined as 12.
Statement (2) supplies component values that combine to form the target expression uniquely.
4
Determine the final Data Sufficiency decision.
Because Statement (1) alone is sufficient and Statement (2) alone is sufficient, the correct choice is that EACH statement ALONE is sufficient.
Both statements independently yield a single, consistent answer to the rephrased question.

Key Concept

Question Stem Rephrasing and Algebraic Identity Simplification in Data Sufficiency
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Topics

Algebraic Equations and Systems in Data Sufficiency

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