If X*Y ≠ 0 and X^3*Y = X*Y^3, then which of the following must be true

Question

If  XY ≠ 0  and  X^3Y = XY^3 , then which of the following must be true?

I)  |X| = -|Y| 
II)  |X| = |Y| 
III)  X = Y = 1

A) None
B) II only
C) III only
D) I and II only
E) I, II, and III

KanishkM · Accepted Answer

X^3Y = XY^3 
 X^3Y - XY^3 = 0 
 XY (X^2 - Y^2) = 0 
 XY (X - Y) (X+Y) = 0

Now XY != 0(given) so X^2 = Y^2

I)  |X| = -|Y| 
Now this inequality can show 4 cases, We can't say anything about XY(This can be +ive or -ive), i wont go with this. Not sufficient
X= - Y, X= -(-Y), -X = -Y and -X = - (-Y)

II)  |X| = |Y| 
This is always true, square root of X^2 = Y^2 => |X| = |Y|

III)  X = Y = 1 
Not necessary, what if the value is different

Answer B

rajatvermaenator · Answer

If XY≠0XY≠0 and X3Y=XY3X3Y=XY3, then which of the following must be true?

I) |X|=−|Y||X|=−|Y|
II) |X|=|Y||X|=|Y|
III) X=Y=1X=Y=1

A) None
B) II only
C) III only
D) I and II only
E) I, II, and III

IMO B
1- |X| = -|Y|
Not sufficient coz modulus never gives a negative value ,

2- |X| = |Y|
4 cases but it will always end up in 2 saying x=Y or x= -Y
 Upon simplifying the original equation we will get
Xy= 0 - not possible
X= Y or x= -y
Sufficient

3- x=Y=1 
Not sufficient coz x and u could 2 and 2 or 3 and 3 or 2 and -2
Not sufficient

So answer is B

Posted from my mobile device

Diwakar003 · Answer

X^3Y = XY^3

X^3Y - XY^3  = 0

XY( X^2 -Y^2 ) = 0

Given  XY ≠ 0

So  X^2 -Y^2  = 0

X^2 = Y^2

Options II and III satisfy the above condition.

Also in I)  |X| = -|Y| ,

|X| is always positive, how can it be equal to a negative value? The case is only possible when X=Y=0 but questions says  XY ≠ 0 .

Are the given options correct or am I missing something?

Cheers!

sahibr · Answer

KanishkM

If C must be true ie X=Y=1 that means X & Y cant hold any other value, other than 1.

However, X=Y=2 or X=Y=3 also achieves the condition mentioned in the question.

MathRevolution · Answer

XY≠0 => Neither X nor Y is zero.

X^{3}Y = XY^{3}  => When X = Y = the same value ( It can be any value) -  So, Option III not necessarily must be true.

The absolute value of X cannot give negative value  Hence Option I is not true.

X^{3}Y = XY^{3}  => Dividing both the sides first by X and then by Y gives us:

=>  X^{2}= Y^{2}

=> This means |X|= |Y|  Hence option II must be true.

Answer B

aviddd · Answer

Bunuel , I have a question.

In the original equation, can't I simplify it by canceling X and Y with each other and end up with equation X^2 = y ^2? All the answers I have seen is a bit more complicated than this.

Bunuel · Answer

Yes, since xy ≠ 0, we can safely reduce by xy and get x^2 = y^2, which is the same as |x| = |y|.

Abhishek009 · Answer

X^3Y = XY^3

Or,  X^2 = Y^2

Or,  +  X  =  +  Y  , Among the given options only (II) fits in perfectly....

PalakDiwanji · Answer

Can anyone please explain why |X| = - |Y| is not possible?

Fdambro294 · Answer

Algebra Manipulation and the Zero Product Rule will get you to the Answer quickly:

(X)^3 * Y = X * (Y)^3

(X)^3 * Y - X * (Y)^3 = 0

----take X*Y as Common Factor----

(X * Y) *   = 0

(XY) * (X + Y) * (X - Y) = 0

Given that the Product of these 3 Factors gives a Result = 0, at least 1 of these factors MUST equal = 0

Given that X*Y does NOT equal = 0

so EITHER:

X + Y = 0 ------ X = - (Y)

OR

X - Y = 0 ------ X = Y

I.   = -

can never be true. the Output from the Absolute Value Modulus will ALWAYS be (+) or 0. the Absolute Value of X can NEVER equal a (-)Negative Value, which is what MINUS the Absolute Value of Y will give UNLESS they both = 0, which they can not

II. whether it is true that:

X = Y ---- or ------- X = - (Y)

the absolute Value of either of these will give the same value. II MUST be True

III. X = Y = 1

X and Y do NOT have to both equal 1

simply plugging in X = Y = - (1) will show you that this is not true

-B- II Only

Basshead · Answer

X^3Y = XY^3 
 X^3Y - XY^3 = 0 
 XY (X^2 - Y^2) = 0

We're told XY is not equal to 0. Therefore,  X^2 = Y^2 .

Lets take a look at the options:

I)  |X|=−|Y|

An absolute value will not give us a negative value. This is not true.

II)  |X|=|Y|

Since we know  X^2 = Y^2 , we can say this statement must be true.

III)  X=Y=1

Not true -- x = y = 2 is possible.

Answer is B.

luisdicampo · Answer

Deconstructing the Question

We are given

XY 
e 0

and

X^3Y = XY^3

We need to determine which statement must be true.

Since  XY 
e 0 , neither  X  nor  Y  is zero, so we can divide both sides by  XY .

Step-by-step

Start with

X^3Y = XY^3

Divide both sides by  XY

\frac{X^3Y}{XY} = \frac{XY^3}{XY}

X^2 = Y^2

If two squares are equal, then the numbers have equal absolute value.

So

|X| = |Y|

This proves statement II.

Now check statement I

|X| = -|Y|

Absolute values are always nonnegative, so this would force both sides to equal  0 .

But  XY 
e 0 , so  X  and  Y  cannot be zero.

So statement I cannot be true.

Now check statement III

X=Y=1

This is not required.

For example,  X=2  and  Y=2  also satisfy

X^3Y = XY^3

So statement III does not must be true.

Therefore only statement II must be true.

Answer B: II only

If XY ≠ 0 and X^3Y = X*Y^3, then which of the following must be true

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

If X*Y ≠ 0 and X^3*Y = X*Y^3, then which of the following must be true

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

If XY ≠ 0 and X^3Y = X*Y^3, then which of the following must be true