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Difficulty: HardAlgebraic Equations and Systems in Data Sufficiency

If xx and yy are real numbers, what is the value of x3−y3x^3 - y^3?

(1) x−y=2x - y = 2
(2) x2+y2=10x^2 + y^2 = 10

  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.
Combining Statement (1) and Statement (2) allows us to determine the exact value of xy=3xy = 3. Substituting x−y=2x - y = 2, x2+y2=10x^2 + y^2 = 10, and xy=3xy = 3 into the factored form x3−y3=(x−y)(x2+xy+y2)x^3 - y^3 = (x - y)(x^2 + xy + y^2) yields 2×(10+3)=262 \times (10 + 3) = 26. Because this value is uniquely determined, both statements together are sufficient while neither statement alone is sufficient.

Step-by-Step Solution

1
Rephrase the target expression using algebraic identities.
The target expression x3−y3x^3 - y^3 factors into (x−y)(x2+xy+y2)(x - y)(x^2 + xy + y^2).
To find x3−y3x^3 - y^3, we need the values of (x−y)(x - y) and (x2+xy+y2)(x^2 + xy + y^2), or equivalently, (x−y)(x - y), (x2+y2)(x^2 + y^2), and xyxy.
2
Evaluate Statement (1) alone.
Statement (1) gives x−y=2x - y = 2.
Without knowing x2+y2x^2 + y^2 or xyxy, x3−y3x^3 - y^3 can take infinitely many values (for example, if x=2,y=0x=2, y=0, x3−y3=8x^3-y^3=8; if x=3,y=1x=3, y=1, x3−y3=26x^3-y^3=26). Thus, Statement (1) alone is NOT sufficient.
3
Evaluate Statement (2) alone.
Statement (2) gives x2+y2=10x^2 + y^2 = 10.
Without knowing x−yx - y or xyxy, x3−y3x^3 - y^3 can take multiple values (for example, if x=10,y=0x=\sqrt{10}, y=0, x3−y3=1010x^3-y^3=10\sqrt{10}; if x=3,y=1x=3, y=1, x3−y3=26x^3-y^3=26). Thus, Statement (2) alone is NOT sufficient.
4
Combine Statement (1) and Statement (2).
Square Statement (1): (x−y)2=22  ⟹  x2−2xy+y2=4(x - y)^2 = 2^2 \implies x^2 - 2xy + y^2 = 4. Substitute x2+y2=10x^2 + y^2 = 10: 10−2xy=4  ⟹  2xy=6  ⟹  xy=310 - 2xy = 4 \implies 2xy = 6 \implies xy = 3.
By squaring x−yx - y and using x2+y2x^2 + y^2, we isolate a unique value for the product term xyxy.
5
Substitute known values into the target expression.
x3−y3=(x−y)(x2+y2+xy)=2×(10+3)=26x^3 - y^3 = (x - y)(x^2 + y^2 + xy) = 2 \times (10 + 3) = 26.
Since the expression evaluates to a single, unique numerical value (2626), the two statements together are sufficient.

Key Concept

Algebraic Expression Rephrasing and Solution Set Invariance in Data Sufficiency
Estimated Time:2m 0s
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Topics

Algebraic Equations and Systems in Data Sufficiency

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