Data Sufficiency

Hi Folks. I came across a question through another GMAT website and have found a question that has a solution, but the solution provided by the author doesn't make any sense at all. We ultimately came to the right solution, but I don't think this person's logic made any sense. I want to ask you folks in the forum what your opinion is.

Here is the question

if Sqrt(xy) = xy, what is the value of x + y?

(1) x = -1/2
(2) y is not equal to 0

So I solved the question using the following methodology

1. Squared both sides of the question being asked to get xy = (xy)^2
2. Looked at statement one and plugged in x = -1/2 --- I got y = - 2 and can be nothing but -2 in order for this equation to work given the constraints that the -1/2 applies on the original question

So now I have my y and my x, now adding those two together gave me -5/2, which was a part of the author's solution. In my opinion, I think I have found the solution and the subsequent answer, which would be A.

The solution provided by the author is as follows.

Let's start by rephrasing the question. If we square both sides of the equation we get:


Now subtract xy from both sides and factor:
(xy)2 - xy = 0
xy(xy - 1) = 0
So xy = 0 or 1

To find the value of x + y here, we need to solve for both x and y.
If xy = 0, either x or y (or both) must be zero. - I don't agree with this, because I solved for my y by plugging in the x, which is -1/2

If xy = 1, x and y are reciprocals of one another.
While we can't come up with a precise rephrasing here, the algebra we have done will help us see the usefulness of the statements.

(1) INSUFFICIENT: Knowing that x = -1/2 does not tell us if y is 0 (i.e. xy = 0) or if y is -2 (i.e. xy = 1) ---- I agree that it doesn't tell you that y is 0, because it tells you that y is -2 definitively

(2) INSUFFICIENT: Knowing that y is not equal to zero does not tell us anything about the value of x; x could be zero (to make xy = 0) or any other value (to make xy = 1). ---- Given that I plugged in x, I have found the solution and no longer need to do any work, but he/she still takes this into consideration[/b]

(1) AND (2) SUFFICIENT: If we know that y is not zero and we have a nonzero value for x, neither x nor y is zero; xy therefore must equal 1. If xy = 1, since x = -1/2, y must equal -2. We can use this information to find x + y, -1/2 + (-2) = -5/2.

The correct answer is C.

What are your thoughts.

Thanks for the response, I truly appreciate it, but I still don't agree.

If the final solution for both solutions is -5/2, that takes into account the -1/2.

Statement 2 is superfluous. The solution is -5/2, which is derived from statement one in two steps. From all the text books I've written on this stuff is if you find a solution with the first statement, that is your answer and there is no need to go on to statement 2. I don't need to know that y cannot equal to 0, because I have already proven to you that y IS -2; and therefore x = -1/2

Text books I've read. Apologies for the typo.

Another thing to consider

Show me a solution that gives me a y = 0 if x = -1/2, because that is ultimately what this is asking us. If I am given an x (-1/2), and I need to solve for y, I should be able to isolate y and get 0, if this solution is correct, right?

If you try that exercise, you will not get y = 0.

Hi hsingh,

As Rich has already mentioned the problem in your steps towards solution, I will share with you my steps,which are "nearly" same as yours, to the solution:

Sqrt(xy) = xy
Square both sides
xy = (xy)^2
=> (xy)^2 - xy = 0
=> xy(xy - 1) = 0
=> (xy - 0)(xy - 1) = 0
So, either xy = 0 or xy = 1

TO find: x + y

Statement 1:
x = -1/2
if xy = 0
y = 0
x + y = -1/2

if xy = 1
y = -2
x + y = -1/2 + -2 == -2.5
INSUFFICIENT


Statement 2:
y != 0
but that doesn't tell us whether x is "0" or not
INSUFFICIENT

COmbining...
x = NOT 0
Y = NOT 0
that means possible solution is xy=1
So, y = -2
x + y = -2.5
SUFFICIENT
Answer {C}

IMP: Do not strike-off the same variable(s) from both the sides(LHS & RHS) when considering the various possible values.

Both of these solutions have cleared up my question. Thanks for pointing out that I did not complete the algebra by considering that sqrt(xy^2) has two solutions.