What is the value of x?
(1) y² = x
(2) (x + 4)/4 = y + 1

A. Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by itself is not.
B. Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by itself is not.
C. Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, even though NEITHER statement BY ITSELF is sufficient.
D. Either statement BY ITSELF is sufficient to answer the question.
E. Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question, meaning that further information would be needed to answer the question.

(E) Statement (1) is insufficient: we cannot determine the value of x without knowing the value of y.

If we multiply both sides of the equation in Statement (2) by 4, we get:
x + 4 = 4(y + 1), or
x + 4 = 4y + 4.
Subtracting 4 from both sides gives us:
x = 4y.
This statement is also insufficient.

Combined, both statements are still insufficient. When we substitute the second equation into the first, we get:
y² = 4y.

We cannot simply divide each side of the equation by y because this is a quadratic equation, and we do not know that y is not equal to 0.

We must set the equation equal to zero and factor it:
y² – 4y = 0
y(y – 4) = 0
y = 0 or 4.

Possible solutions for (x,y) are (0,0) and (16,4). But, we aren't looking for "possible" solutions, we are interested in THE solution. Since we do not have sufficient information to determine the value of x, even after combining the statements, the correct answer is choice (E).
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The question doesn't ask for a real value of x, or a value not in terms of other variables. So x very literally could be y², or x could be y – 3. Just because the 2 are not reconcilable with the given information, does not mean that either is not an acceptable answer.

When in GMAT question it is asked to find a value there is no ambiguity in it. You should find a specific, definite and unique VALUE.

If a question accepts a formula or a set as an answer, it implies so additionally: in words or in answer choices.

"y = 4" does NOT follow from y² = 4y.
You must never divide by a variable, unless you've proven that it can not be 0. Or you must consider the case of zero value as well.

The proper solution for y² = 4y is:
y² – 4y = 0
y(y – 4) = 0
So y = 0 or y = 4.
Therefore x = 0 or x = 16.

The both statements combined are not sufficient, since x can have two distinct values. The correct answer is E.

"16 = x" does NOT follow from 16x = x² .
You must never divide by a variable, unless you've proven that it can not be 0. Or you must consider the case of zero value as well.

The proper solution for 16x = x² is:
x² – 16x = 0
x(x – 16) = 0
So x = 0 or x = 16.
Thus the two situations are possible: x = 0 and y = 0 , x = 16 and y = 4 (Plug them in the original statements to see that they both fit).

The both statements combined are not sufficient, since there are two possible different values of x. The correct answer is E.