If x and y are different integers and x^2 = xy, which of the

Question

If x and y are different integers and x^2 = xy, which of the following must be true?

I. x = 0
II. y = 0
III. x = -y

(A) l only
(B) II only
(C) III only
(D) I and III only
(E) I, II, and III

Bunuel · Accepted Answer

SOLUTION

If x and y are different integers and x^2 = xy, which of the following must be true?

I. x = 0
II. y = 0
III. x = -y

(A) l only
(B) II only
(C) III only
(D) I and III only
(E) I, II, and III

x^2=xy  -->  x(x-y)=0  --> either  x=0  or  x=y  but as given that  x  and  y  are different numbers than the second option is out and we have:  x=0 . So only I is always true (in fact because of the same reason that  x  and  y  are different numbers II and III are never true).

Answer: A.

Manofsteel · Answer

Given Facts -
1. x and y are different integers
2. x^2 = xy

I - x=0
then x^2 = 0
and xy = 0 , for any value of y
 thus x^2 = xy - True for any value of y

II - y = 0
if x=1 then x^2 = 1
and xy = 0 
 Thus x^2 not equal to xy and thus not true for any value of x

III - x = -y
x^2 = (-y)^2 = y^2
xy = (-y)*(y)=-y^2
 Thus x^2 not equal to xy and thus not true for any value of x,y

Hence since I only is true for all values Option (A) is the correct answer.

reetskaur · Answer

If x and y are different integers and x^2 = xy, which of the following must be true?

I. x = 0
II. y = 0
III. x = -y

Sol.: From the question stem, 
x^2 – xy = 0
x ( x – y) = 0
⇨ Either x = 0 or x = y
Since x and y are two different integers, => x= 0.
Hence (A)

ScottTargetTestPrep · Answer

Let’s simplify the given equation:
 
x^2 = xy
 
x^2 - xy = 0
 
x(x - y) = 0
 
x = 0 or x - y = 0
 
x = 0 or x = y
 
Notice that it is given in the question that x and y are different integers; therefore x = y is not possible. Thus, it must be true that x = 0. None of the other Roman numerals need to be true, as we can see by letting x = 0 and y = 1. 
 
Answer: A

dave13 · Answer

Bunuel why this option is not valid III. x = -y if y = -2 and x= y ---> x = -2 then -2^2 = (-2*)(-2) --> 4 =4 can you explain ?

pleaese

Bunuel · Answer

This does not make sense.

If y = -2 and x = -y, then x = -(-2) = 2. What x= y has to do with this? x = y is not possible at all because the stem says that x and y are different integers...

Also, the question asks which of the following MUST be true not COULD be true. Even if there would exist x and y, for which x = -y, would satisfy the stem, still this would not be sufficient to say that III MUST be true. MUST be true means true for ALL possible values, not for some specific values only.

Finally, you should be careful with brackets. It's math, they DO matter. -2^2 means -(2^2) = -4, while (-2)^2 = 4.

Kinshook · Answer

x^2-xy=0
x(x-y)=0
Since x &y are different 
x=0

IMO A

Posted from my mobile device

CrackverbalGMAT · Answer

Synthesising the equation given in the question stem,  x^2  = xy, we have,

x^2  – xy = 0

x(x-y) = 0.

This means x = 0 or x=y. But, question clearly says that x and y are different integers. Therefore, x=y is not true.

Therefore, x=0 is always true. 
Statement I is always true. We can eliminate answer options B and C.

Since x≠y, y can never be 0. Therefore, statement II is always false. Answer option E can be eliminated.

If x = -y, x+y = 0. But, from the question data, we know that x – y = 0. Therefore, statement III is always false. Answer option D can be eliminated.

The correct answer option is A.

Hope that helps!
Aravind B T

rushimehta · Answer

Bunuel   chetan2u

Why can't we cancel the x in "x^2 = xy" and get x = y ? (Since we don't have an inequality, and also we know that the sign of x will be the same)

Is it because we don't know that x is not equal to 0 here ?

Please can you explain this query to me?

Thank you in advance!

Bunuel · Answer

When dividing an equation by an unknown, we do not care about the sign; we care whether this unknown can be 0. Recall that we cannot divide by 0. So, we cannot divide x^2 = xy by x to get x = y, because x could be 0. By doing so, we would lose a potential valid case, namely x = 0. You should solve it the way shown in the solution above:

x^2=xy

x(x-y)=0

Either  x=0  or  x=y

Never reduce equation by a variable (or expression with a variable), if you are not certain that the variable (or expression with the variable) doesn't equal to zero. We cannot divide by zero.

Hope it helps.

totaltestprepNick · Answer

A https://www.youtube.com/watch?v=E_dvdwSRshQ Nick Slavkovich, GMAT/GRE tutor with 20+ years of experience tutoring@totaltestprep.net

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