Is the solution for this GMAT data sufficiency question correct?

[Xori]

This is a problem from my GMAT practice exam that gives me the right answer but not an explanation as to why. The question is formatted as a data sufficiency question, which means that it asks you whether a pair of statements would be individually and / or together sufficient to answer the question. For the GMAT itself, it's not necessary to actually answer the question, only to know whether you have sufficient data.

In the xy-plane, does the line with the equation y = 3x + 2 contain the point (r,s)?

Statement 1: (3r + 2 - s)(4r + 9 - s) = 0
Statement 2: (4r - 6 - s)(r3 + 2 - s) = 0

a) Statement 1 alone is sufficient, but Statement 2 alone is not sufficient
b) Statement 2 alone is sufficient, but Statement 1 alone is not sufficient
c) Both statements together are sufficient, but neither statement alone is sufficient
d) Each statement alone is sufficient
e) Statements 1 & 2 together are not sufficient


So the way I approached this is to say that in order to answer the original question, one must know the values of both r & s.

Since it would appear that each statement allows me to solve for both r & s, both statements should be individually sufficient. To double check my work, I solved for both statements.



Solution of Statement 1:
(3r + 2 - s)(4r + 9 - s) = 0

I set each of the factors equal to 0
(3r + 2 - s) = 0
(4r + 9 - s) = 0

Then I solve for the variables:

Rewrite both equations:
r3 + 2 = s
4r + 9 = s

Substitute:
3r + 2 = 4r + 9
-r = 7
r = -7
Solve for s:
4 * (-7) + 9 = s
s = -19

Now, knowing that r = -7 and s = -19, I can deduce that (-7, -19) does lie in y = 3x +2. Thus, my conclusion would be that (Statement 1) is Sufficient to answer the question.

For Statement 2, the calculations look identical, just different values, so I won't bother writing it out. Likewise, Statement 2 is then sufficient to answer the original question.

I also see an obviously easier way to solve the problem, in the fact that (r3 + 2 - s) is a factor in both equations, which is the original y = 3x + 2 equation rewritten when we plug in r for x and s for y:
s = r3 +2, or 0 = r3 + 2 - s

To me, this is further evidence that both statements obviously are sufficient to answer the question individually.
So my choice was (D), that "Each Statement alone is sufficient to answer the question". However, I got it wrong; apparently, the correct answer is (C), that "Both statements together are sufficient, but neither statement alone is sufficient". Can someone please tell me what I'm doing wrong?

 

[LCKurtz]

Neither equation alone will suffice because the product of two factors can be zero if either factor is. So, for example if you know just the first one:

(3r + 2 - s)(4r + 9 - s) = 0

it might be true because r = 1, s = 13 makes the second factor zero, but doesn't make the first factor zero. Similarly for the second equation.

But if you know both are true you have:

(3r + 2 - s)(4r + 9 - s) = 0
(4r - 6 - s)(r3 + 2 - s) = 0

are both true. The only way this would not imply (r,s) is on your line is if neither of the (3r+2-s) factors is zero. That would mean the other factors would have to be zero:

4r + 9 - s = 0
4r - 6 - s = 0

But these obviously can't both be true. So the only way both equations work is if one of the 3r + 2 - s in the two equations is zero.

 

[Xori]

Ah, I get it, thanks!

Forgot that factoring only gives you one possible solution, not the only solution :P