If x and y are integers, does x = y?

[Seyitbek]

1. xy=y2
2. x2=y2

Guys, could you please suggest the thought process that goes into answering the question. I understand the reasons why they A and B are insufficient. I don't understand why the answer is C. Would really appreciate your answer.

Seyit

[avik.ch]

Lets start with the easy one, B for me.

1. X^2 = Y^2

X and Y can take both positive and nagative value. This means that |x| = |y|, the modulus value of both X and Y is same. Insufficient

2. XY = Y^2

Y^2 is positive, hence XY is positive.
Y can be both positive and nagative , and since XY is positive, either X and Y is both positive or both nagative.

an example : Y = -2, x=y=-2 -- IS X = Y ? no
Y = 2, X=Y=2 -- IS X=Y ? yes

Combining both :

X^2 = Y^2 and XY = Y^2

from XY=Y^2, X and Y should be of same sign

and X^2 = Y^2 : |x|=|Y|

we can reasonable infer that both x and Y are of same sign and x and y modulus values are same. Hence sufficient.

Hope this helps !

[GMATGuruNY]

If x and y are integers, does x=y?

1. xy=y2
2. x2=y2

Guys, could you please suggest the thought process that goes into answering the question. I understand the reasons why they A and B are insufficient. I don't understand why the answer is C. Would really appreciate your answer.

Seyit

Combining the two statements:
xy = y²
x² = y².

Since both xy and x² are equal to y²:
xy = x²
x² - xy = 0.
x(x-y) = 0.

Thus, either x=0 or x-y=0.
If x=0, then y=0 -- since x²=y² -- with the result that x=y.
If x-y=0, then x=y.
Since in each case x=y, SUFFICIENT.

The correct answer is C.

[[email protected]]

Gmatguru I did not understand your explanation... Why did u even go to the option C.
According to me the answer is A.

the first statement itself is giving me the answer...

y^2 = xy
y^2 - xy = 0
y(y-x) = 0
hence y = 0 or (y-x) = 0
If y = 0 then x also = 0
Hence y = x

why even look at the option C when option A is giving the answer???
please help me understand the question...

[GMATGuruNY]

Gmatguru I did not understand your explanation... Why did u even go to the option C.
According to me the answer is A.

the first statement itself is giving me the answer...

y^2 = xy
y^2 - xy = 0
y(y-x) = 0
hence y = 0 or (y-x) = 0
If y = 0 then x also = 0
Hence y = x

why even look at the option C when option A is giving the answer???
please help me understand the question...

Statement 1: xy = y².
y²- xy = 0
y(y-x) = 0.
Thus, either y=0 or x=y.
If y=0, then x can be ANY value.
It's possible that y=0 and x=0, in which case x=y.
It's possible that y=0 and x=1, in which case x≠y.
Hence, statement 1 on its own is INSUFFICIENT.

However, when the two statements are combined, if y=0, then x=0, since statement 2 requires that x²=y².
Thus, when we combine the two statements, x=y in every possible case, making the correct answer C.