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

\(X^3Y = XY^3\)

\(X^3Y - XY^3\) = 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!

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.

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

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.

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

\(X^3Y = XY^3\)

Or, \(X^2 = Y^2\)

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

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

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) * [ (X)^2 - (Y)^2 ] = 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. [X] = - [Y]

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

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

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.

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