If x and y are real numbers, what is the value of x + y? (1) x^2 + y^2 = 25 - 2xy (2) x^3 … — GMAT

If $x$ and $y$ are real numbers, what is the value of $x + y$ ?

(1) $x^2 + y^2 = 25 - 2xy$
(2) $x^3 + y^3 = 125 - 3xy(x+y)$

A
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.Cevap
C
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D
EACH statement ALONE is sufficient.
E
Statements (1) and (2) TOGETHER are NOT sufficient.

Statement (2) ALONE is sufficient to determine that

x + y = 5

, but Statement (1) ALONE is not sufficient because it allows

x + y = 5

x + y = -5

Statement (2) alone allows us to factor the expression into

(x+y)^3 = 125

. Because real numbers have a unique real cube root,

x+y

must equal

5

. Thus, Statement (2) alone provides a single, definitive answer to the question stem. Statement (1) yields

(x+y)^2 = 25

, which leads to

x+y = 5

x+y = -5

, making it insufficient on its own.

Rephrase the target question

The target is to find a unique numerical value for the expression

(x + y)

. We do not need individual values for

x

and

y

Data Sufficiency targets involving expressions often do not require solving for individual variables.

Evaluate Statement (1):

x^2 + y^2 = 25 - 2xy

Rearranging terms gives

x^2 + 2xy + y^2 = 25

, which factors as

(x + y)^2 = 25

. Taking the square root gives

x + y = 5

x + y = -5

Since there are two distinct real values for

x + y

, Statement (1) alone is insufficient.

Evaluate Statement (2):

x^3 + y^3 = 125 - 3xy(x+y)

Rearranging terms gives

x^3 + 3xy(x+y) + y^3 = 125

, which is the expanded form of

(x + y)^3 = 125

. Taking the cube root gives

x + y = 5

For real numbers, every real number has exactly one real cube root. Thus,

x + y = 5

uniquely. Statement (2) alone is sufficient.

Algebraic Rephrasing and Degree of Real Polynomial Identities

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