What is the value of x? (1) X^2 x^2 = 0 (2) x/y + y/x = 0

Question

What is the value of x?
 
(1) X^2 – x^2 = 0
(2) x/y + y/x = 0

BrentGMATPrepNow · Answer

Awesome question! 
  Target question :    What is the value of x?

Statement 1 : x² – x² = 0   
This means that x² = x²
As you might imagine, there are several solutions to this equation (in fact, the equation holds true for ANY value of x. 
So,  x = 0 ,  x = 2 ,  x = -3  (etc) are all solutions to the equation x² - x² = 0
Since we cannot answer the  target question  with certainty, statement 1 is NOT SUFFICIENT

Statement 2 : x/y + y/x = 0  
Subtract y/x from both sides to get: x/y = -y/x
Cross multiply to get: x² = -(y²)
This equation should be somewhat perplexing. 
We know that y² must be greater than or equal to zero. 
So, this means that   -(y²) must be  less than or equal to zero .   
Likewise,   x² must be  greater than or equal to zero   .

So, if x² = -(y²), then we are saying:   A number that's  greater than or equal to zero    =   -(y²) A number that's  less than or equal to zero .   
This equation holds true ONLY IF x = 0 and y = 0 
So, we can conclude that  x = 0 
EDIT......
 STOP!! 
I have since been reminded that x = 0 and y = 0 does not satisfy the statement 2 equation x/y + y/x = 0, since 0/0 is undefined.

So, it seems, we have a faulty question.

Cheers,
Brent

Jayeshvekariya · Answer

Option a. Insufficient. X can be any number
Option b.
X/y + y/x =0
(x^2+y^2)/xy=0
X^2+y^2=0
Since
X^2 and y^2 can't be negative. Both must be zero to fulfill the condition.
So x=0

Ans. Option B

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Alexey1989x · Answer

(1) Insufficient, as long as subject condition is applicable for all values of x
(2) x/y+y/x=0
 x/y=-y/x
squaring
 x^2/y^2=y^2/x^2
 x^4=y^4
 x^4-y^4=0
 (x^2-y^2)(x^2+y^2)=0
 So, both these conditions can exist only if x=y=0 Sufficient.
 GMATPrepNow 
Would you please advise if my reasoning is correct

Alexey1989x · Answer

(1) Insufficient, as long as subject condition is applicable for all values of x
(2) x/y+y/x=0
 x/y=-y/x
squaring
 x^2/y^2=y^2/x^2
 x^4=y^4
 x^4-y^4=0
 (x^2-y^2)(x^2+y^2)=0
 So, both these conditions can exist only if x=y=0 Sufficient.
 GMATPrepNow 
Would you please advise if my reasoning is correct

BrentGMATPrepNow · Answer

Squaring both sides of an equation (as you did at the beginning of your solution) can yield unintended consequences. 
For example, consider the equation x = -x
This equation has  only one  solution: x = 0

However, if we take the equation x = -x and square both sides, we get: x² = (-x)²
This simplifies to be x² = x² 
This new equation has  infinitely many  solutions. In fact, the new equation (x² = x²) holds true for ANY value of x.

The unintended consequences in your solution becomes apparent when you get to x^4=y^4 in your solution. 
Notice that, with this new equation (x^4=y^4), we also have infinitely many solutions, including:
x = 1 and y = 1 
x = 2 and y = 2 
x = 2 and y = -2 
x = -5 and y = 5 
etc.

However, if you plug any of these solutions back into the original equation (x/y + y/x = 0), you'll find that they do not satisfy that equation.

Cheers,
Brent

Mike2805 · Answer

#GMATPrepNow

I'm missing something. Since in the GMAT, all results of an equation must be  for defined values  of the variable, how can x = 0 in statement 2 and provide a unique solution? In the statement, if x=0, then y/x is undefined (division by 0).
Please advise.
Thanks a lot.

dabhishek87 · Answer

I agree with  Mike2805  if x = 0 then y/x is undefined in B so this will not hold.
Answer should be E
Hi  Bunuel  ,  VeritasPrepKarishma  , Can you please help in understanding the OA?

arvind910619 · Answer

Imo B 
From statement 1 we have x^2 =x^2 so it can take any value. 
From statement 2
x^2=-y^2
There is only one value that satisfy this equation that is 0

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ankursethi297 · Answer

Also, nowhere it is written that x is a real number.

incase x=1i
and y=1
first equation stands

BrentGMATPrepNow · Answer

Arghh! It was such a nice question too. You're right, I neglect to remember that statement 2 says that x/y + y/x = 0 If x = 0 and y = 0, then we cannot say that x/y + y/x = 0, since 0/0 is undefined. Unfortunately, the question is faulty

I have edited my response accordingly. Cheers, Brent

BrentGMATPrepNow · Answer

All numbers that appear on the GMAT are real numbers.

ydmuley · Answer

What is the value of x? (1) x^2 – x^2 = 0 x in this case can take any value, x can be = 1, or 2 or 3 or 4 or any other number Hence, (1) ===== is NOT SUFFICIENT (2) x/y + y/x = 0 x^2 + y^2 = 0 This is only possible when x and y are both ZERO. So, x = 0 Hence, (2) ===== is SUFFICIENT Hence, Answer is B Did you like the answer? 1 Kudos Please :good

RK007 · Answer

So is it E or not? Can x be 0 or not?
Somebody explain it already please...

amanvermagmat · Answer

Hello

As it has already been discussed in this thread before, this is a faulty question. Thats because for second statement to be satisfied, the only solution is both x=y =0. And we cannot do 0/0.

So I suggest you ignore this question in particular, there are thousands of other good and brilliant questions to analyse. But this question also gives us a learning: division by 0 is not permitted.

What is the value of x? (1) X^2 x^2 = 0 (2) x/y + y/x = 0

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