If √(xy) = xy, what is the value of x + y?

Responding to a pm:

"If \(\sqrt{XY} = XY\) what is the value of x + y?

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


What i did was,

(XY)^1/2 = XY
XY =(XY)^2

so, I cancelled out XY and finally I got the below rephrased equation

XY=1.

BUT in the MGMAT explanation, I found that they are not cancelling out XY.

Below is their rephrased equation.

XY = (XY)^2

XY-(XY)^2=0

XY [1-(XY)] = 0

so, XY = 0 or XY = 1.

My question is why we are not cancelling out and when we should use cancelling technique."

For the time being, forget this question. Look at another one.

Which values of x satisfy this equation: \(x^2 = x\)
Let's say we cancel out x from both sides. What do we get? x = 1.
So we get that x can take the value 1.

But is your answer complete? I look at the equation and I say, 'x can also take the value 0.' Am I wrong? No.
x = 0 also satisfies your equation. So why didn't you get it using algebra? It is because you canceled x.
Let me treat this equation differently now.

\(x^2 = x\)
\(x^2 - x = 0\)
\(x * (x - 1) = 0\)
x = 0 OR (x - 1) = 0 i.e. x = 1

Now I get both the possible values that x can take. I do not lose a solution.

When you cancel off a variable from both sides of the equation, you lose a solution so you should not do that. You can cancel off constants of course.
Rule of thumb: Do not cancel off variables. Take them common. In some cases, it may not matter even if you do cancel off but either ways, your answer will not be incorrect of you don't cancel. On the other hand, sometimes, your solution could be incomplete if you do cancel and that's a problem.

Show Spoiler

Answer: C

Statement 1: \(X=-1/2\).... substitute in main equation

Scenario A: \(Y = -2/1\) is a reciprocal
\(\sqrt{xy}=xy\)
LHS =\(\sqrt{-1/2*-2/1}= \sqrt{1}= 1\)
RHS =\(-1/2*-2/1= -1*-1 = 1\)
Therefore, LHS = RHS

Scenario B: \(Y = 0\)
LHS =\(\sqrt{-1/2*-0}= \sqrt{0}= 0\)
RHS =\(-1/2*0= 0\)
Therefore again, LHS = RHS

Hence, we have two possible solutions, therefore Statement 1 is insufficient

Statement 2 : \(Y\) is not equal to zero
But, X could be equal to zero or be the reciprocal of Y, therefore, statement 2 would fall under the same scenarios as statement 1

Hence, we have two possible solutions, therefore Statement 2 is insufficient

Both Statement 1 & 2 together confirms that \(X & Y\) both are not equal to zero, therefore, they have to be reciprocals and since statement gives us the value of \(X=-1/2\), we can also the value of \(Y=-2/1\) (Reciprocal of X) and thereafter we can find the value of \(X + Y = -1/2 + -2/1 = -5/2\)

Hence, both together are sufficient, therefore C is the correct Answer!

I did something similar to DMMK, except for one thing:
I know \(\sqrt{n}\) = \(n\) only when \(n\) = 0 or \(n\) = 1. For all other values of \(n\), the two would not be equal. Knowing this made it much simpler to plug in the statements. I just had to determine if from the statement(s) provided, can I rule out xy = 0 or xy=1.

Stament 1:
Knowing that x = -1/2, y could equal 0 or -2, and still make the premise true. Either value of y would make x + y a different value. Therefore, it is insufficient.

Statement 2:
Knowing that y is not equal to 0, x could equal zero or the inverse of y, and still make the premise true. Either value of x would make x + y a different value. Therefore, it is insufficient.

Now to evaluate both statements together.

The reason we rejected statement 1 by itself was because y could equal one of two possible values. Statement 2 eliminates one of those options. Therefore y must equal -2. And therefore both statements together are sufficient to answer "what is the value of x+y."

let xy=k, thus
k^(1/2) =k;
squaring both sides we have
k=k^2
k(k-1)=0
k=0 or 1
i.e. xy=0 or xy=1

st.1

for x=-1/2
both y=0 and y=-2 satisfy the possible value of xy i.e. 0 or 1
hence not sufficient

st.2

y is not equal to zero. clearly not sufficient.
as nothing is said about x, therefore x can take any value it can be zero, fraction, integer etc.


combining st.1 and st.2

we know that y cannot be equal to zero. thus y=-2 and x=-1/2
and x+y= -2-(1/2)=-5/2

hence sufficient.

Why can't x and y both be -1? Thanks

From the stem x = y = -1 is possible, in this case xy = 1 (check my solution above). But then the first statement says that x = -1/2, thus these values are no longer possible.

Hope it's clear.

Given: √(xy) = xy
(xy)^2 - xy = 0
xy = 0 or 1 - (i)

Required: x + y = ?

Statement 1: x = -1/2
From the given equation (i), we can have two different values of y.
Hence two different values of x + y
INSUFFICIENT

Statement 2: y is not equal to 0
No information about x
INSUFFICIENT

Combining Statement 1 and Statement 2: x = -1/2 and y is not equal to 0
From (i), xy cannot be 0 since both x and y are not = 0
Hence xy = 1
y = -2
x + y = \(-\frac{5}{2}\)
SUFFICIENT

Option C

Video solution from Quant Reasoning:
Subscribe for more: https://www.youtube.com/QuantReasoning? ... irmation=1

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.