In the xy-plane, does the line y = 2x - 1 contain the point (p,q)?

This question is a part of Are You Up For the Challenge: 700 Level Questions collection.

Hey! I also feel C is the answer. DO you know why E?

In the xy-plane, does the line y = 2x - 1 contain the point (p,q)?

The question basically asks whether q = 2p - 1. Or, which is the same, whether 2p - 1 - q = 0.

(1) (2p - 1 - q)(3p + 7 - q) = 0

2p - 1 - q = 0 or 3p + 7 - q = 0 (or both).

If the first equation is true, then we have an YES answer to the question.
If the second equation is true, then we can have an YES answer to the question (if p = -8, q = -17) as well as a NO answer to the question (for any other values of p and q, which satisfy 3p + 7 - q = 0).

Not sufficient.

(2) (5p + 2 - q)(1 - 2p + q) = 0.

1 - 2p + q= 0 or 5p + 2 - q = 0 (or both).

If the first equation is true, then we have an YES answer to the question.
If the second equation is true, then we can have an YES answer to the question (if p = -1, q = -3) as well as a NO answer to the question (for any other values of p and q, which satisfy 5p + 2 - q = 0).

Not sufficient.

(1)+(2) 2p - 1 - q = 0 is possible from each of the statements, so if it's true, then we have an YES answer to the question.

But what if 2p - 1 - q = 0 is NOT true? Then 3p + 7 - q = 0 (from 1) and 5p + 2 - q = 0 (from 2) must be simultaneously true (these two equations must have a solution). If you solve the system of equations 3p + 7 - q = 0 and 5p + 2 - q = 0 you get p = 5/2 and q = 29/2. So, we can have the case when 2p - 1 - q ≠ 0. This would happen when p = 5/2 and q = 29/2 (in this case both 3p + 7 - q = 0 (from 1) and 5p + 2 - q = 0 (from 2) will be true) and in this case we'd have a NO answer to the question.

Answer: E.

Good question. Fell for the C trap here.