In the xy-plane, does the line with equation y = 3x + 2 contains the

Thanks for your reply, I get it now. I missed that Bunuel actually wrote "OR both". I happened to think that he came to the conclusion that only either one of them could be 0 only from statement 1 alone.

key to cracking the question is in understanding the fact when an equation leads to zero both factors can be zero or each individually can be zero keeping that and moving forward
r and s can be considered as x and y individually
state 1 :
(3r+2-s)(4r+9-s)=0 => (3r+2-s)=0 or (4r+9-s)=0
Not sufficient

State 2 :
(4r-6-s)(3r+2-s)=0 => (3r+2-s)=0 or (4r-6-s)=0
Not suff
When combined 1&2
results in (3r+2-s)=0
Suff
Hence IMO C

If the point (r,s ) lies in the equation y= 3x+2 then after substituting r,s in the line equation, we get.
S= 3r + 2
Or 3r+ 2 – S =0

So our Question here is to find whether 3r+2-S = 0 or not? Yes or No
Statement1.
(3r+2−s)(4r+9−s)=0
From Statement 1, We can infer that either 3r+2-s =0 or 4r+9-s =0 or both could be zero. Hence, we cannot answer the Question with a definite Yes or No.
So Statement 1 is insufficient. Answer option A and D can be eliminated. Possible answer options are B, C and E.

Statement 2.
(4r−6−s)(3r+2−s)=0
From Statement 2, We can infer that either 4r−6−s =0 or 3r+2−s or both could be zero. Hence, we cannot answer the Question with a definite Yes or No
So, Statement 2 is also insufficient. Answer option B can be eliminated. Possible answer options are C and E.


Let’s Try combining both statements.
4r+9-s =0 => 4r-s =-9
4r−6−s =0 => 4r-s = 6
So, both 4r+9-s =0 and 4r−6−s =0 cannot be 0 at the same time. As 4r-s cannot be both -9 and 6.
Hence, we can conclude that in one of the statements, 3r+2−s will be definetly 0.
So, we can answer the question by combining the both statements.
Option C

Thanks,
Clifin J Francis,
GMAT SME

Hi Bunuel,

From 1 and 2, what if 4r−s was equal to one of them? (-9 or 6)

CrackverbalGMAT
Thank you for your helpful reply. I am hoping to confirm my understanding of factoring quadratics with you.

Bunuel says "either 3r+2−s=03r+2−s=0 OR 4r+9−s=04r+9−s=0 OR both"

So, when you factor out the quadratic, you can have just one of the factors be the answer or both of the factors be an answer?

Take another example if I have (x+2)(x-4) --> x=-2 and x=4, depending on the constraints of the original problem, just x=-2 could be the answer or just 4 could be the answer or -2 and 4 could both be the answer?

Is my understanding correct?

Hi IanStewart - i actually chose (E) on this one and not (C)

From (s1) :
s = 3r + 2
OR
s = 4r +9

From (s2) :
s = 3r + 2
OR
s = 4r -6

When you combine the statements, there are 3 options for "s"

(i) s = 3r + 2
or
(ii) s = 4r +9
or
(iii) s = 4r -6

I see : s = 3r + 2 is a common factor but why does that imply, we can just "drop" option (ii) and option (iii) as possible values of "s" ?

When we combine the statements, can we really say s DOES NOT EQUAL 4r +9 | S does not equal 4r -6 ?

I thought upon combining the statement, there are 3 options for the value of "S"

hence "E"

Everything is correct to this point. When we combine the statements, clearly one possibility is that s = 3r + 2. If s is not equal to 3r + 2, then from Statement 1, the only remaining possibility is that s = 4r + 9, and from Statement 2, the only remaining possibility is that s = 4r - 6. So if s is not equal to 3r + 2, s must be equal to both 4r + 9 and 4r - 6, but those are definitely two different numbers (4r + 9 is exactly 15 greater than 4r - 6), so s could never simultaneously be equal to both of them, which is why the only possibility is that s = 3r + 2, and the answer is C.

The answer could have been E here if there was a legitimate second possibility -- it's because 4r - 6 and 4r + 9 can never be equal that s = 3r + 2 needs to be true.
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(r,s) is a point on the XY plane e.g (1, 2) or (2, 8) or (3, 21) etc. There is a defined relation between r and s.
Say if (r, s) is (1, 2), we can say that r + s = 3, s - r = 1, 2r + s = 4 etc. Each of these expressions will take only one value because both r and s have defined values.
This is an important point to understand in this question.

We want to know whether (r, s) lies on y = 3x + 2 i.e. whether s = 3r + 2. So we want to know whether the value of 3r - s is -2.
Note that all points (r, s) that satisfy this condition (3r - s = -2) will lie on the line y = 3x + 2. So the line represents the locus of all such points. e.g. (1, 5), (2, 8), (3, 11) etc.

Coming back to the question, we want to figure out whether 3r - s is -2.

(1) \((3r+2-s)(4r+9-s)=0\)

This tells us that either 3r - s = -2 or 4r-s = -9. At least one of these relations certainly holds.
Not sufficient alone.

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

This tells us that either 4r - s = 6 or 3r - s = -2. At least one of these relations certainly holds.
Not sufficient alone.

Using both statements, now we know that one of 3r - s = -2 and 4r-s = -9 certainly holds and one of 4r - s = 6 and 3r - s = -2 certainly holds.

Now there are two possibilities.
Either 3r - s = -2 in which case, all works out.
Or 3r - s is not equal to -2 and both 4r-s = -9 and 4r - s = 6 hold. But here is the problem: both 4r-s = -9 and 4r - s = 6 cannot hold. As discussed above, 4r-s can take only one value. It cannot be both -9 and 6. Hence this case is not possible.

Then 3r - s must be equal to -2 and hence (r, s) must lie on the line y = 3x + 2.

Answer (C)
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Hi Karishma – not sure I agree with the red.

Per the red – you are implying – one of the two statements - S1 or S2 is lying

But we know the statement DON’T lie.

(S1) / (S2) will never give factually incorrect information.

(s1) or (S2) could give us -- INCOMPLETE or COMPLETE information, but never FACTUALLY INCORRECT information.

Note that S1 says that EITHER 3r - s = -2 OR 4r-s = -9 (at least one of them must be true)
If we decide that 4r-s cannot be -9, then we get that 3r - s must be -2. Hence S1 is satisfied. One of them is true.
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