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Difficulty: HardAlgebraic Equations and Systems in Data Sufficiency

If xx and yy are real numbers, what is the value of x2−y2x^2 - y^2?

(1) x3−y3=26x^3 - y^3 = 26 and x2+xy+y2=13x^2 + xy + y^2 = 13
(2) x+y=4x + y = 4

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

Answer

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Factoring the target expression gives x2−y2=(x+y)(x−y)x^2 - y^2 = (x + y)(x - y). Statement (1) simplifies via the difference of cubes formula x3−y3=(x−y)(x2+xy+y2)x^3 - y^3 = (x - y)(x^2 + xy + y^2) to x−y=2x - y = 2 and xy=3xy = 3. However, solving for x+yx + y yields (x+y)2=16(x + y)^2 = 16, which gives two solutions (x+y=4x + y = 4 or x+y=−4x + y = -4) and thus two values for x2−y2x^2 - y^2 (88 or −8-8). Thus, Statement (1) alone is insufficient. Statement (2) gives x+y=4x + y = 4 without constraining x−yx - y, making it insufficient alone. Combining both statements provides x+y=4x + y = 4 and x−y=2x - y = 2, giving a unique product of 88. Hence, both statements together are sufficient.

Step-by-Step Solution

1
Rephrase the target expression using algebraic identities
x2−y2=(x+y)(x−y)x^2 - y^2 = (x + y)(x - y)
Determining x2−y2x^2 - y^2 requires knowing either the product (x+y)(x−y)(x + y)(x - y) or the individual values of x+yx + y and x−yx - y.
2
Evaluate Statement (1) independently
Using the difference of cubes identity x3−y3=(x−y)(x2+xy+y2)x^3 - y^3 = (x - y)(x^2 + xy + y^2), substitute the given values: 26=(x−y)(13)  ⟹  x−y=226 = (x - y)(13) \implies x - y = 2. Next, expand (x−y)2=x2−2xy+y2=4(x - y)^2 = x^2 - 2xy + y^2 = 4. Subtracting this from x2+xy+y2=13x^2 + xy + y^2 = 13 gives 3xy=9  ⟹  xy=33xy = 9 \implies xy = 3. Then (x+y)2=(x−y)2+4xy=4+4(3)=16(x + y)^2 = (x - y)^2 + 4xy = 4 + 4(3) = 16, which implies x+y=4x + y = 4 or x+y=−4x + y = -4. Thus x2−y2=(4)(2)=8x^2 - y^2 = (4)(2) = 8 or (−4)(2)=−8(-4)(2) = -8.
Because Statement (1) produces two distinct possible values (88 and −8-8), Statement (1) ALONE is NOT sufficient.
3
Evaluate Statement (2) independently
x+y=4x + y = 4
Without any information about x−yx - y, x2−y2=4(x−y)x^2 - y^2 = 4(x - y) can take infinitely many values. Statement (2) ALONE is NOT sufficient.
4
Evaluate Statements (1) and (2) combined
From Statement (1), x−y=2x - y = 2. From Statement (2), x+y=4x + y = 4. Multiplying these equations yields x2−y2=(x+y)(x−y)=(4)(2)=8x^2 - y^2 = (x + y)(x - y) = (4)(2) = 8.
A single, unique numerical value is determined. Therefore, BOTH statements TOGETHER are sufficient.

Key Concept

Algebraic Factoring and Non-Linear System Ambiguity in Data Sufficiency
Estimated Time:2m 0s
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Topics

Algebraic Equations and Systems in Data Sufficiency

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