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Point on the line

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Source: — Data Sufficiency |
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by Matt@VeritasPrep » Thu Aug 08, 2013 4:04 pm
The point (r,s) has an x-coordinate of r and a y-coordinate of s. It will be on the line y = 3x + 2 if its coordinates satisfy that equation; in other words, if s = 3r + 2.

S1::

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

implies that either

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

If (3r + 2) - s = 0, then (3r + 2) = s ... on the line!
If (4r + 9) - s = 0, then (4r + 9) = s ... NOT (necessarily) on the line

We don't know which is true, so insufficient.

S2::

Same as S1 - two equations, one of which gives s = 3r + 2, the other of which gives s = 4r - 6. (r,s) could be on the line, or it could not be. Insufficient again.

S1 + S2::

Either (3r + 2 - s) = 0, or it does not, in which case BOTH (4r + 9 - s) and (4r - 6 - s) = 0.

But if (4r + 9 - s) = (4r - 6 - s) = 0, then 4r + 9 = s and 4r - 6 = s, meaning that 4r + 9 = 4r - 6. That equation is IMPOSSIBLE for any value of r. So (4r + 9 - s) and (4r - 6 - s) can't equal 0.

Thus (3r + 2 - s) must equal 0, and (r,s) is on the line!

Let me know if I can fill in any more details on this; the interpretation can be tricky.
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by GMATGuruNY » Thu Aug 08, 2013 5:08 pm

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

1) (3r+2-s)(4r+9-s) = 0
2) (4r-6-s)(3r+2-s) = 0
If (r,s) is a point on the line y = 3x + 2, then s = 3r + 2, implying that 3r - s = -2.

Question stem rephrased: Does 3r - s = -2?

Statement 1: (3r+2-s)(4r+9-s) = 0
Either 3r+2-s = 0 or 4r+9-s = 0.
If 3r+2-s = 0, then 3r - s = -2.
If 4r+9-s = 0, then 4r - s = -9, in which case it cannot be determined whether 3r - s = -2.
INSUFFICIENT.

Statement 2: (4r-6-s)(3r+2-s) = 0
Either 4r-6-s=0 or 3r+2-s = 0.
If 3r+2-s = 0, then 3r - s = -2.
If 4r-6-s = 0, then 4r - s = 6, in which case it cannot be determined whether 3r - s = -2.
INSUFFICIENT.

Statements 1 and 2 combined:
4r - s = -9 (from statement 1) and 4r - s = 6 (from statement 2) cannot both be true, since 4r - s cannot be equal to more than one value.
Thus, the only way the equations in the two statements can both be equal to 0 is if 3r - s = -2.
SUFFICIENT.

The correct answer is C.
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by Brent@GMATPrepNow » Thu Aug 08, 2013 9:35 pm
satish_iitg wrote:In the xy plane, does the line with equation y = 3x+2 contain the point (r,s)

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

1) (4r-6-s) (3r+2-s) = 0
If (r,s) is on the line defined by the equation y=3x+2, then (r,s) must satisfy the equation y=3x+2. In other words, it must be true that s=3r+2
For example: We know that the point (5, 17) is on the line y=3x+2, because when we plug x=5 and y=17 into the equation, we get 17 = 3(5)+2 and the equation holds true.

So, we can reword the target question to be "Does s = 3r + 2?"

1. (3r+2-s)(4r+9-s) = 0
From this, we know that either (3r+2-s) = 0 or (4r+9-s) = 0
If (3r+2-s) = 0 then s = 3r+2, in which case the answer to our new target question is yes
If (4r+9-s) = 0 then s = 4r+9, in which case the answer to our new target question is no
Since we get two different answers to the target question, statement 1 is NOT SUFFICIENT

2. (4r-6-s)(3r+2-s) = 0
From this, we know that either (4r-6-s) = 0 or (3r+2-s) = 0
If (4r-6-s)) = 0 then s = 4r-6, in which case the answer to our new target question is no
If (3r+2-s) = 0 then s = 3r+2, in which case the answer to our new target question is yes
Since we get two different answers to the target question, statement 2 is NOT SUFFICIENT

Statements 1&2 combined: Since (3r+2-s) is the only expression common to both statements, it must be true that 3r+2-s = 0, in which case s MUST equal 3r+2
As such the answer is C

Cheers,
Brent
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by satish_iitg » Thu Aug 08, 2013 11:37 pm
Thanks every one. this was second question on my Gmat Prep test.
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