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Difficulty: MediumAlgebraic Equations and Systems in Data Sufficiency

If xx and yy are real numbers, what is the value of x−yx - y?

(1) (x−y)2=16(x - y)^2 = 16
(2) x2−y2=24x^2 - y^2 = 24

  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 (1) and (2) TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
  4. D
    EACH statement ALONE is sufficient.
  5. Statements (1) and (2) TOGETHER are NOT sufficient.Answer

Answer

Statements (1) and (2) TOGETHER are NOT sufficient.
The option stating that statements (1) and (2) together are not sufficient is correct because combining both equations results in two valid candidate solution sets for (x,y)(x, y), specifically (5,1)(5, 1) and (−5,−1)(-5, -1). These produce two different values for the target expression x−yx - y (44 and −4-4), preventing a single unique determination.

Step-by-Step Solution

1
Evaluate Statement (1) independently.
(x−y)2=16  ⟹  x−y=4(x - y)^2 = 16 \implies x - y = 4 or x−y=−4x - y = -4.
Taking the square root of both sides yields two possible values for x−yx - y, so Statement (1) alone is not sufficient.
2
Evaluate Statement (2) independently.
x2−y2=(x−y)(x+y)=24x^2 - y^2 = (x - y)(x + y) = 24.
Without knowing x+yx + y, the expression x−yx - y can take infinitely many numerical values, so Statement (2) alone is not sufficient.
3
Evaluate Statements (1) and (2) together.
Case 1: If x−y=4x - y = 4, then 4(x+y)=24  ⟹  x+y=64(x + y) = 24 \implies x + y = 6, giving (x,y)=(5,1)(x, y) = (5, 1) where x−y=4x - y = 4. Case 2: If x−y=−4x - y = -4, then −4(x+y)=24  ⟹  x+y=−6-4(x + y) = 24 \implies x + y = -6, giving (x,y)=(−5,−1)(x, y) = (-5, -1) where x−y=−4x - y = -4.
Both coordinate pairs (5,1)(5, 1) and (−5,−1)(-5, -1) satisfy both given statements, but yield two distinct values (44 and −4-4) for x−yx - y. Therefore, both statements combined remain insufficient.

Key Concept

Degree Ambiguity in Systems of Non-Linear Algebraic Equations
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Topics

Algebraic Equations and Systems in Data Sufficiency

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