If x^2+y^2=1, is x+y=1 ? (1) xy=0 (2) y=0

Question

If x^2+y^2=1, is x+y=1 ?

(1) xy=0
(2) y=0

Bunuel · Accepted Answer

I think answer B is not correct.

(1)  xy=0  --> either  x=0  or  y=0 :
if  x=0 , then  x^2+y^2=y^2=1  and  y=1  or  y=-1 , so  x+y=0+1=1  (answer YES) or  x+y=0-1=-1  (answer NO);
if  y=0 , then  x^2+y^2=x^2=1  and  x=1  or  x=-1 , so  x+y=1+0=1  (answer YES) or  x+y=-1+0=-1  (answer NO);
Two different answers. No sufficient.

(2)  y=0  -->  x^2+y^2=x^2=1  and  x=1  or  x=-1 , so  x+y=1+0=1  (answer YES) or  x+y=-1+0=-1  (answer NO). Two different answers. No sufficient.

(1)+(2)  xy=0  and  y=0  -->  y=0  and  x=1  or  x=-1 , so  x+y=1+0=1  (answer YES) or  x+y=-1+0=-1  (answer NO). Two different answers. No sufficient.

Answer: E.

saxenashobhit · Answer

D can be true only when problem says x and y are positive.

Bunuel · Answer

Both  x  and  y  cannot be positive as it would contradict the statements.

For D to be the answer the question should ask "is  x+y>1 ?" instead of "is  x+y=1 ?".

pratk · Answer

IMO Answer is D.. Here's why:

x^2+y^2= 1

X^2+y^2 can be written as (x+y)^2-2xy

Therefore (x+y)^2-2xy = 1

From statement 1, xy=0 we get (x+y)^2 = 1
From statement 2, y=0 we get (x+y)^2 = 1

Thus (x+y)=1-----> Square root of both sides

Both statements are sufficient.

Bunuel · Answer

The answer to this question is E, not D.

Consider two sets of numbers, which satisfy stem, as well as both statements and give different values of x+y:
If  y=0  and  x=1  then  x+y=1+0=1 ;
If  y=0  and  x=-1  then  x+y=-1+0=-1 .

Two different answers. No sufficient.

Answer: E.

Now, the problem in your solution (the red part) is that (x+y)^2=1 means that x+y=1  OR  x+y=-1 (you forgot to consider negative root). Basically the same way as x^2=4 means that x=2 or x=-2.

Hope it's clear.

pratk · Answer

Yes Bunuel, what you mention is correct and I also thought about it that way and this would hold true if ther question would have been phrased differently- perhaps something like : "What is the value of x?" However the question simply asks: is x+y=1? And based on my post above, the answer to that question is Yes using both statements independently.

Not sure if my thinking is correct, guess I have been doing alot of critical reasoning questions so my mind is working in a different way.

Any thoughts?

Not sure if my thinking is correct, guess I have been doing alot of critical reasoning questions so my mind is working in a different way.

Any thoughts?

Bunuel · Answer

No, your thinking is not correct. It's seems that you have some problem with this type of DS question. It's a YES/NO DS question.  In a Yes/No Data Sufficiency question, statement(s) is sufficient if the answer is “always yes” or “always no” while a statement(s) is insufficient if the answer is "sometimes yes" and "sometimes no" .

Now, we have that even when statements are taken together x+y can equal to 1 as well as -1. So, both statements are not sufficient to give definite YES or definite NO answer to the question whether x+y=1.

Hope it's clear.

pratk · Answer

Thanks for the explanation. I get it now.
Guess I need to practice more of these Always Yes/No type.

RohanKhera · Answer

Hi Bunuel,

In one of posts, I read that "square root function can not give negative result"

So in the solution above, is it ok to assume that Under root Y Square (or X Square) will have 2 values: one positive and one negative.

Regards

Rohan

Bunuel · Answer

I guess you are confused by the part where we have  x=1  or  x=-1  from  x^2=1 .

Square root function cannot give negative result  -->  \sqrt{some \ expression}\geq{0} , for example  \sqrt{x^2}\geq{0}  -->  \sqrt{4}=2  (not +2 and -2).

In contrast, the equation  x^2=25  has TWO solutions, +5 and -5,  because both 5^2 and (-5)^2 equal to 25.

Hope it's clear.

RohanKhera · Answer

So you mean that a square root operation results in 2 solution (positive and negative) only in case of an equation ? And otherwise (in case of non equation) there is only one solution i.e. positive ?

Regards,

Rohan

Regards,

Rohan

Bunuel · Answer

Not sure I understand what you mean. Anyway:

x^2=4  -->  x=2  or  x=-2 .

\sqrt{x}=4  -->  x=16 . Or  x=\sqrt{4}  -->  x=2 .

gaurav1418z · Answer

Bunuel

Similar question but couldnt find any thread

If y is not equal to 1, is x=1?

(1) x^2 + y^2 = 1
(2) y= 1-x

Statement 1 is clearly not sufficient, as y can be 1/2 or 0, so x can be +3/4 , -3/4 or +1/-1
Similar statement 2 alone is not sufficient

Even when you combine both

y = 1-x
x+y =1 
squaring both sides
(x+y)^2 = 1
x^2 +y^2 + 2xy = 1

from (1), 
1 + 2xy = 1, hence xy =0 
so x could be 1, 2, 3... and y could be 0, not sufficient.

But is OA is C. I am not sure how

Bunuel · Answer

From xy=0, x=0, y=0 or both. But if x=0, then from y=1-x, we get that y=1 but we are told that y≠1, thus x≠0. Hence y=0 and from y=1-x, we get that x=1. This question is discussed here: if-y-1-is-x-161421.html Hope it helps.

gaurav1418z · Answer

Thanks as always Bunuel, yes it helps. Cheers and have a good day

ParvathyRV · Answer

Please tell me if my assumption is wrong: 
Given: x^2+y^2=1 
Assume: x^2+y^2+2xy = 1+2xy
 (x+y)^2 = 1+2xy
 x+y = sqrt(1+2xy)

Stmt1: xy = 0; So, plugging this in: x+y = sqrt(1) = 1 --> hence sufficient?!

Bunuel · Answer

From  (x+y)^2 = 1+2xy  we'd have  x+y=-\sqrt{1+2xy}  or  x+y=\sqrt{1+2xy} .

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If x^2+y^2=1, is x+y=1 ? (1) xy=0 (2) y=0

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