Here is something a bit non-conformant to this blog.

The other day I came up with the following GMAT-type data sufficiency question:

What is the value of xy ?

(1) x^2 + y^2 = 0

(2) x + y = 5

[You need to determine whether (1) alone is sufficient to find out the value of xy or (2) alone is sufficient, both (1) and (2) are needed, both (1) and (2) are independently sufficient or neither is sufficient.]

Now this is a tricky question.

The 'obvious' approach is to square (2) and substitute (1) to get:

x^2 + y^2 + 2 x y = 25

=> 2 x y = 25

=> xy = 12.5

So you would say that both (1) and (2) are necessary to determine the value of xy. After all, there are two variables and don't you need two equations?

Not in this particular case!

Look closely at (1): x^2 + y^2 = 0. The sum of two squares equal zero. When can this happen? Remember that squares are non-negative. So the only way to add two squares and get zero is if both of them are individually zero, which means both x=0 and y=0! So obviously, xy = 0!

So the correct answer is only (1) alone is sufficient!

So what went wrong in our earlier calculation? Here we assumed that both (1) and (2) are correct. In fact, because of the above reasoning, (1) and (2) together constitute an inconsistent system of equations. (Can you prove it?)

[P.S. Thus, at the expense of increasing the degree and the number of potential roots, any set of simultaneous equations (for real variables) can be converted into a single equation by squaring them and equating the sum to zero.

E.g. if p(x, y, ...) = 0, q(x, y,...)=0, r(x, y,...)=0 are the equations, the single equation would be

p(x,y,...)^2 + q(x, y,...)^2 + r(x, y,...)^2 = 0]