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Difficulty: MediumAlgebraic Equations and Systems in Data Sufficiency

If mm and nn are real numbers, what is the value of m+nm + n?

(1) m2−n2=0m^2 - n^2 = 0
(2) m2+n2=50m^2 + n^2 = 50

  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 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.
Evaluating both statements together yields four possible coordinate pairs: (5,5)(5, 5), (−5,−5)(-5, -5), (5,−5)(5, -5), and (−5,5)(-5, 5). The sum m+nm + n can equal 1010, −10-10, or 00. Because a Data Sufficiency question requires a single unique value to be deemed sufficient, having three distinct possible values means the statements together are insufficient.

Step-by-Step Solution

1
Evaluate Statement (1) independently.
m2−n2=0  ⟹  (m−n)(m+n)=0  ⟹  m=nm^2 - n^2 = 0 \implies (m - n)(m + n) = 0 \implies m = n or m=−nm = -n.
If m=−nm = -n, then m+n=0m + n = 0. However, if m=nm = n, then m+n=2mm + n = 2m, which varies. Since we cannot determine a single numerical value for m+nm + n, Statement (1) alone is NOT sufficient.
2
Evaluate Statement (2) independently.
m2+n2=50m^2 + n^2 = 50.
Different pairs (m,n)(m, n) satisfy m2+n2=50m^2 + n^2 = 50. For example, if (m,n)=(5,5)(m, n) = (5, 5), then m+n=10m + n = 10. If (m,n)=(7,1)(m, n) = (7, 1), then m+n=8m + n = 8. Since m+nm + n is not uniquely determined, Statement (2) alone is NOT sufficient.
3
Evaluate Statements (1) and (2) combined.
From (1), m2=n2m^2 = n^2. Substitute into (2): n2+n2=50  ⟹  2n2=50  ⟹  n2=25  ⟹  n=5n^2 + n^2 = 50 \implies 2n^2 = 50 \implies n^2 = 25 \implies n = 5 or n=−5n = -5.
Since m2=25m^2 = 25, mm can also be 55 or −5-5. The possible ordered pairs (m,n)(m, n) are (5,5)(5, 5), (−5,−5)(-5, -5), (5,−5)(5, -5), and (−5,5)(-5, 5). Calculating m+nm + n for these pairs gives 1010, −10-10, and 00. Because multiple outcomes are possible, the combined statements are NOT sufficient.

Key Concept

Non-Linear Systems and Multiple Solution Ambiguity in Data Sufficiency
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Topics

Algebraic Equations and Systems in Data Sufficiency

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