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equation of a line qs

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by DevB » Tue Jul 22, 2014 4:59 am
This question is from GMAT Prep so anyone planning to give the practice exam may avoid to see it here.

For others, Please help me in answering the below question:

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

Thanks in advance!
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Source: — Data Sufficiency |
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by GMATinsight » Tue Jul 22, 2014 5:51 am
DevB wrote:This question is from GMAT Prep so anyone planning to give the practice exam may avoid to see it here.

For others, Please help me in answering the below question:

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

Thanks in advance!
Question : Does the line with equation y = 3x + 2 contain the point (r,s) ?

if (r,s) are contained by line y = 3x + 2, then they must satisfy the equation
i.e. 's' should be equal to '3r+2'
therefore,
Question Rephrased : Is s = 3r+2

Statement 1) (3r + 2 - s) (4r + 9 - s) = 0
i.e. either (3r + 2 - s) = 0 or (4r + 9 - s)=0
i.e. either (3r + 2 = s) or (4r + 9 = s)
INSUFFICIENT

Statement 2) (4r - 6 - s) (3r + 2 - s) = 0
i.e. either (3r + 2 - s) = 0 or (4r - 6 - s)=0
i.e. either (3r + 2 = s) or (4r - 6 = s)
INSUFFICIENT

Combining the two statements

We get (3r + 2 = s)
i.e. Sufficient

Answer: Option C
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by Brent@GMATPrepNow » Tue Jul 22, 2014 5:54 am
DevB 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
2. (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 GMATGuruNY » Tue Jul 22, 2014 6:07 am
In the xy-plane, does the line with equation y=3*X +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, and 3r - s = -2. Thus, the question can be 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.
INSUFFICIENT.

Statement 2: (4r-6-s)(3r+2-s) = 0
Either 4r-6-s=0 or 3r+2-s = 0.
If 4r-6-s = 0, then 4r - s = 6.
If 3r+2-s = 0, then 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 that 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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