If x and y are non-zero real numbers, what is the value of \frac{x}{y}? (1) x^2 y - x y^2 … — GMAT

If $x$ and $y$ are non-zero real numbers, what is the value of $\frac{x}{y}$ ?

(1) $x^2 y - x y^2 = 2xy$
(2) $x^2 + y^2 = 5xy$

A
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
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.
Statements (1) and (2) TOGETHER are NOT sufficient.Cevap

Cevap

Statements (1) and (2) together are NOT sufficient.

The statement that both statements together are not sufficient is correct because combining Statement (1) (

x = y + 2

) and Statement (2) (

x^2 + y^2 = 5xy

) leads to a quadratic equation

3y^2 + 6y - 4 = 0

with two distinct real roots for

y

. Evaluating

\frac{x}{y} = 1 + \frac{2}{y}

for both roots yields two distinct possible values,

\frac{5 + \sqrt{21}}{2}

and

\frac{5 - \sqrt{21}}{2}

. Because a unique numerical value cannot be determined, the information provided is not sufficient.

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Evaluate Statement (1) independently.

Simplifies to

x - y = 2 \implies x = y + 2

Since

x, y \neq 0

xy \neq 0

. Dividing

x^2 y - x y^2 = 2xy

xy

yields

x - y = 2

. Thus

\frac{x}{y} = \frac{y+2}{y} = 1 + \frac{2}{y}

, which varies with

y

. Not sufficient.

Evaluate Statement (2) independently.

Yields

(\frac{x}{y})^2 - 5(\frac{x}{y}) + 1 = 0

Dividing

x^2 + y^2 = 5xy

y^2

gives a quadratic in

k = \frac{x}{y}

k^2 - 5k + 1 = 0

. Solving for

k

gives

k = \frac{5 \pm \sqrt{21}}{2}

. Because there are two distinct real solutions for the ratio, Statement (2) alone is not sufficient.

Evaluate Statements (1) and (2) combined.

Two distinct pairs of real numbers satisfy both conditions, giving two distinct values for

\frac{x}{y}

Substitute

x = y + 2

into

x^2 + y^2 = 5xy

(y+2)^2 + y^2 = 5y(y+2) \implies 3y^2 + 6y - 4 = 0

. This gives two real roots for

y

y = \frac{-3 \pm \sqrt{21}}{3}

. Substituting each

y

back into

\frac{x}{y} = 1 + \frac{2}{y}

yields

\frac{x}{y} = \frac{5 + \sqrt{21}}{2}

and

\frac{x}{y} = \frac{5 - \sqrt{21}}{2}

. Since two distinct ratios remain possible, combined statements are not sufficient.

Anahtar Kavram

Non-linear systems in Data Sufficiency often result in multiple valid solutions, requiring explicit verification of solution uniqueness rather than assuming two equations with two variables yield a single solution.

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