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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+y)2(x + y)^2?

(1) x2+y2=25x^2 + y^2 = 25
(2) x2−y2=7x^2 - y^2 = 7

  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 correct response states that Statements (1) and (2) together are not sufficient because solving the system yields x2=16x^2 = 16 and y2=9y^2 = 9. This gives x=±4x = \pm 4 and y=±3y = \pm 3, allowing xyxy to be either 1212 or −12-12. Consequently, (x+y)2=x2+y2+2xy(x + y)^2 = x^2 + y^2 + 2xy can equal either 4949 or 11, which prevents finding a single unique value.

Step-by-Step Solution

1
Rephrase the target expression using algebraic identities
(x+y)2=x2+y2+2xy(x + y)^2 = x^2 + y^2 + 2xy
Expanding the square shows that we need the sum of squares x2+y2x^2 + y^2 and the product term 2xy2xy (or the individual values of xx and yy) to determine a unique value.
2
Evaluate Statement (1) alone
x2+y2=25x^2 + y^2 = 25, but xyxy can take multiple values.
For example, if x=5x = 5 and y=0y = 0, then (x+y)2=25(x + y)^2 = 25. If x=4x = 4 and y=3y = 3, then (x+y)2=49(x + y)^2 = 49. Multiple values are possible, so Statement (1) alone is insufficient.
3
Evaluate Statement (2) alone
x2−y2=7x^2 - y^2 = 7, but (x+y)2(x + y)^2 is not uniquely determined.
If x=4x = 4 and y=3y = 3, then x2−y2=16−9=7x^2 - y^2 = 16 - 9 = 7 and (x+y)2=49(x + y)^2 = 49. If x=7x = \sqrt{7} and y=0y = 0, then x2−y2=7x^2 - y^2 = 7 and (x+y)2=7(x + y)^2 = 7. Multiple values are possible, so Statement (2) alone is insufficient.
4
Evaluate Statements (1) and (2) combined
x2=16x^2 = 16 and y2=9y^2 = 9, which leads to x=±4x = \pm 4 and y=±3y = \pm 3.
Adding the two equations gives 2x2=32  ⟹  x2=162x^2 = 32 \implies x^2 = 16. Subtracting gives 2y2=18  ⟹  y2=92y^2 = 18 \implies y^2 = 9. Thus x=4x = 4 or −4-4, and y=3y = 3 or −3-3.
5
Check for sign ambiguity in the cross-term xyxy
If x=4,y=3x = 4, y = 3, then (x+y)2=72=49(x + y)^2 = 7^2 = 49. If x=4,y=−3x = 4, y = -3, then (x+y)2=12=1(x + y)^2 = 1^2 = 1.
Because the system of non-linear equations determines x2x^2 and y2y^2 but not the relative signs of xx and yy, there are two distinct valid values for (x+y)2(x + y)^2. Therefore, Statements (1) and (2) together are NOT sufficient.

Key Concept

Non-linear systems of equations and degree ambiguity in Data Sufficiency
Estimated Time:2m 0s
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Topics

Algebraic Equations and Systems in Data Sufficiency

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