If x and y are real numbers, what is the value of x - y? (1) x^2 - y^2 = 12(x + y) and x +… — GMAT

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

(1) $x^2 - y^2 = 12(x + y)$ and $x + y \neq 0$
(2) $x^2 + y^2 = 2xy + 144$

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.Answer
B
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
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 (1) ALONE is sufficient, but statement (2) alone is not sufficient.

The option stating that Statement (1) ALONE is sufficient while Statement (2) ALONE is not sufficient is correct. Factoring Statement (1) gives

(x - y)(x + y) = 12(x + y)

. Because

x + y \neq 0

, dividing both sides by

(x + y)

establishes

x - y = 12

uniquely. In contrast, Statement (2) simplifies to

(x - y)^2 = 144

, yielding two possible values (

12

and

-12

), which makes Statement (2) insufficient on its own.

Evaluate Statement (1) independently

The statement gives

x^2 - y^2 = 12(x + y)

. Factoring the difference of squares gives

(x - y)(x + y) = 12(x + y)

. Since

x + y \neq 0

, divide both sides by

(x + y)

to get

x - y = 12

Because a unique numerical value for

x - y

is obtained, Statement (1) ALONE is sufficient.

Evaluate Statement (2) independently

Rearranging

x^2 + y^2 = 2xy + 144

yields

x^2 - 2xy + y^2 = 144

, which contracts to

(x - y)^2 = 144

. Taking the square root gives

x - y = 12

x - y = -12

Because there are two distinct potential values for

x - y

, Statement (2) ALONE is not sufficient.

Combine the evaluations

Statement (1) alone is sufficient, whereas Statement (2) alone is not sufficient.

No combination of statements is required since Statement (1) alone uniquely determines the target value.

Algebraic Stem Simplification and Degree Ambiguity in Data Sufficiency

Estimated Time:2m 0s

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