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Difficulty: HardAlgebraic Equations and Systems in Data Sufficiency

If xx and yy are real numbers such that x≠yx \neq y, what is the value of x+yx + y?

(1) x3−y3=7(x−y)x^3 - y^3 = 7(x - y)
(2) x2−y2=3(x−y)x^2 - y^2 = 3(x - y)

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

Answer

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
The question stem explicitly states that x≠yx \neq y, establishing that x−y≠0x - y \neq 0. In Statement (2), factoring x2−y2x^2 - y^2 into (x−y)(x+y)=3(x−y)(x - y)(x + y) = 3(x - y) and dividing by non-zero (x−y)(x - y) directly yields the unique value x+y=3x + y = 3. In Statement (1), factoring gives x2+xy+y2=7x^2 + xy + y^2 = 7, which leaves x+yx + y undetermined because xyxy can vary. Consequently, Statement (2) alone is sufficient, while Statement (1) alone is not.

Step-by-Step Solution

1
Analyze the question stem constraints and target expression.
We are given that xx and yy are real numbers with x≠yx \neq y, which implies x−y≠0x - y \neq 0. The goal is to determine a single unique value for x+yx + y.
Establishing x−y≠0x - y \neq 0 allows valid algebraic division by (x−y)(x - y) in the given statements.
2
Evaluate Statement (1) independently: x3−y3=7(x−y)x^3 - y^3 = 7(x - y).
Factor the left side using the difference of cubes identity: (x−y)(x2+xy+y2)=7(x−y)(x - y)(x^2 + xy + y^2) = 7(x - y). Since x−y≠0x - y \neq 0, divide both sides by (x−y)(x - y) to get x2+xy+y2=7x^2 + xy + y^2 = 7, which rewrites to (x+y)2−xy=7(x + y)^2 - xy = 7.
Because xyxy is unknown and variable, (x+y)(x + y) can take multiple values. For instance, (x,y)=(2,1)(x, y) = (2, 1) gives x+y=3x+y=3 and 4+2+1=74+2+1=7, whereas (x,y)=(7,−7)(x, y) = (\sqrt{7}, -\sqrt{7}) gives x+y=0x+y=0 and 7−7+7=77-7+7=7. Thus, Statement (1) alone is NOT sufficient.
3
Evaluate Statement (2) independently: x2−y2=3(x−y)x^2 - y^2 = 3(x - y).
Factor the left side using the difference of squares identity: (x−y)(x+y)=3(x−y)(x - y)(x + y) = 3(x - y). Since x−y≠0x - y \neq 0, divide both sides by (x−y)(x - y) to yield x+y=3x + y = 3.
This provides a single, unambiguous numerical value for x+yx + y. Therefore, Statement (2) alone IS sufficient.

Key Concept

Algebraic expression simplification using stem constraints in Data Sufficiency
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Topics

Algebraic Equations and Systems in Data Sufficiency

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