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Zorluk: OrtaAlgebraic 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.Cevap

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

Non-Linear Systems and Multiple Solution Ambiguity in Data Sufficiency
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Algebraic Equations and Systems in Data Sufficiency

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