If x and y are real numbers such that x \neq 3y, what is the value of x + 3y? (1) x^2 + 6x… — GMAT

If $x$ and $y$ are real numbers such that $x \neq 3y$ , what is the value of $x + 3y$ ?

(1) $x^2 + 6xy + 9y^2 = 49$
(2) $x^2 - 9y^2 = 12(x - 3y)$

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, but statement (1) alone is not sufficient.

Statement (1) simplifies to

(x + 3y)^2 = 49

, which means

x + 3y

could be

7

-7

, so it is not sufficient. Statement (2) factors as

(x - 3y)(x + 3y) = 12(x - 3y)

. Given

x \neq 3y

, dividing both sides by

(x - 3y)

uniquely determines

x + 3y = 12

. Thus, Statement (2) alone is sufficient while Statement (1) alone is not.

Analyze Statement (1) algebraically

The expression

x^2 + 6xy + 9y^2

is a perfect square trinomial equal to

(x + 3y)^2

. Therefore,

(x + 3y)^2 = 49

, which gives

x + 3y = 7

x + 3y = -7

Because there are two distinct valid values for

x + 3y

, Statement (1) alone does not provide a unique value.

Analyze Statement (2) algebraically

The left side

x^2 - 9y^2

factors as

(x - 3y)(x + 3y)

. The equation becomes

(x - 3y)(x + 3y) = 12(x - 3y)

Since the stem specifies

x \neq 3y

, we know

x - 3y \neq 0

, allowing us to divide both sides by

(x - 3y)

without dividing by zero.

Solve the simplified equation from Statement (2)

Dividing gives

x + 3y = 12

This yields a single, unique value for the target expression

x + 3y

, making Statement (2) alone sufficient.

Algebraic Stem Rephrasing and Quadratic Degree Evaluation in Data Sufficiency

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