# DS Equations, Roots

A DS question on equations and roots.

### Question

Is y = 7?
1.    (x - 3) = 0
2.    (x - 3)(y - 7) = 0

Correct Answer : Choice E. Data is insufficient

The given question is an "Is" question. The data provided is sufficient to answer the question if we are able to conclusively answer with an Yes or a No to the question. Else the data is insufficient.

Question : Is y = 7?
If y = 7 and can be nothing else, then our answer will be a conclusive Yes.
Alternatively, if y is not equal to 7, then our answer will be a conclusive No.

However, if we answer saying "y could be 7" - but need not necessarily be 7, then we have not given a definitive answer.

Let us evaluate the statements given.

Statement 1:  (x - 3) = 0.
The statement does not provide any information about y.
Hence, INSUFFICIENT.

Statement 2: (x - 3)(y - 7) = 0.
The product of (x - 3) and (y - 7) is 0.
The product of 2 numbers is 0 when at least one of them is 0.

We have 3 possibilities.
a. (x - 3) = 0 and (y - 7) is not equal to 0. Then, we answer the question "Is y = 7?" with a No.
b. (x - 3) is not equal to 0 and (y - 7) = 0. Then, we answer the question "Is y = 7" with an Yes.
c. (x - 3) = 0 and (y - 7)= 0. Then, we answer the question "Is y = 7" with an Yes.

Using the information in statement 2, we answer No sometimes and Yes sometimes. We have not been able to find a definitive answer to the question.
Hence, INSUFFICIENT.

Now let us combine the information in the two statements.
We know, (x - 3) = 0 from statement 1. We know (x - 3)(y - 7) = 0. from statement 2.

We are still left with two of three possibilities that we evaluated for statement 2
a. (x - 3) = 0 and (y - 7) is not equal to 0. Then, we answer the question "Is y = 7?" with a No.

c. (x - 3) = 0 and (y - 7)= 0. Then, we answer the question "Is y = 7" with an Yes.

Because we have not been able to find a conclusive answer using the statements independently or together, the correct answer is Choice E.

Now here is an alternative version of the same question.

### Alternative Version

Is y = 7?
1.    (x - 4) = 0
2.    (x - 3)(y - 7) = 0