In the xy-plane, does the line...

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In the xy-plane, does the line...

by nchaswal » Tue May 31, 2016 9:49 am
I honestly could not understand the question at all.

What I am supposed to do?

Solve the equations given in statements and check if they fit into the line equation ? or take relationship of r & s from line equation and put them into statement to see if it fits?
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by GMATGuruNY » Tue May 31, 2016 10:30 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 and 3r - s = ANY VALUE.
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 and 3r - s = ANY VALUE.
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+2-s = 0, implying that 3r - s = -2.
SUFFICIENT.

The correct answer is C.
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by OptimusPrep » Mon Jun 06, 2016 8:17 pm
nchaswal wrote:
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

I honestly could not understand the question at all.

What I am supposed to do?

Solve the equations given in statements and check if they fit into the line equation ? or take relationship of r & s from line equation and put them into statement to see if it fits?
Required: Does the line with equation y=3*X +2 contain the point (r,s)
Or simply put, Is 3r - s +2 = 0?

Statement 1: (3r + 2 - s)(4r + 9 - s) = 0
This means either
(3r + 2 - s) = 0
Or (4r + 9 - s) = 0
INSUFFICIENT

Statement 2: (4r - 6 - s)(3r + 2 - s) = 0
This means either
(4r - 6 - s) = 0
Or (3r + 2 - s) = 0
INSUFFICIENT

Combining both statements:
We know that (3r + 2 - s) = 0
SUFFICIENT

Correct Option: C
For such questions, you should take relationship of r & s from line equation and put them into statement to see if it fits

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by Brent@GMATPrepNow » Tue Jun 07, 2016 8:42 am

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
One more solution to show that rephrasing the target question works in many (slightly) different forms:

Target question: Does the line with equation y = 3x + 2 contain the point (r,s)

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 REPHRASE the target question as "Does s = 3r + 2?"

Statement 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 REPHRASED target question is yes
If (4r+9-s) = 0 then s = 4r+9, in which case the answer to our REPHRASED target question is no
Since we cannot answer the REPHRASED target question with certainty, statement 1 is NOT SUFFICIENT

Statement 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 REPHRASED target question is no
If (3r+2-s) = 0 then s = 3r+2, in which case the answer to our REPHRASED target question is yes
Since we cannot answer the REPHRASED target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Since (3r+2-s) is the only expression common to BOTH equations, it MUST be true that 3r+2-s = 0, in which case s MUST equal 3r+2
Since we can answer the REPHRASED target question with certainty, the combined statements are SUFFICIENT

Answer = C


Cheers,
Brent

Aside: We have a free video with tips on rephrasing the target question: https://www.gmatprepnow.com/module/gmat- ... cy?id=1100
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by Matt@VeritasPrep » Tue Jun 07, 2016 11:17 pm
nchaswal wrote:I honestly could not understand the question at all.

What I am supposed to do?

Solve the equations given in statements and check if they fit into the line equation ? or take relationship of r & s from line equation and put them into statement to see if it fits?
Yeah, you have the right idea. If (r, s) is on the line, then its x-coordinate (which has value r) and its y-coordinate (which has value s) must satisfy the line equation. In other words, we'd need

s = 3r + 2

to be true.

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by renatofarias » Tue Apr 18, 2017 12:05 pm
Hi everyone. I did not understand why each statement is not true by itself.

I followed the instructions in the beggining reaching s = 3r + 2. So I went to substitute s in the Statements:

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

When I reached (3r + 2 - (3r + 2))(4r + 9 - (3r + 2)) = 0 I wento on to say that the equation was true, thus making (r, s) part of the line.

The same for statement (2)

The question is, why does (r, s) have to satisfy "both parenthesis" if satisfying one should be enough to make the equation true?

Thank you very much

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by DavidG@VeritasPrep » Tue Apr 18, 2017 12:49 pm
renatofarias wrote:Hi everyone. I did not understand why each statement is not true by itself.

I followed the instructions in the beggining reaching s = 3r + 2. So I went to substitute s in the Statements:

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

When I reached (3r + 2 - (3r + 2))(4r + 9 - (3r + 2)) = 0 I wento on to say that the equation was true, thus making (r, s) part of the line.

The same for statement (2)

The question is, why does (r, s) have to satisfy "both parenthesis" if satisfying one should be enough to make the equation true?

Thank you very much
Think of a simple example with the same logic. Imagine the following question "Is x = 2?"

If you had a statement that read (x - 2)(x + 4) = 0, you'd determine that x could be 2 or x could be -4, so because you could get a YES or a NO, the statement alone would not be sufficient to answer the question.

But if you plugged x = 2 into the equation, you'd get (2 - 2)(2 +4) = 0, which of course, is true. But it doesn't make the statement sufficient, right? (Remember x could be -4 too.) The issue is that we're not testing to see if the statement is true, given the initial question. We're testing to see if we can answer the original question, given a statement, which we have to take as true.
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by DavidG@VeritasPrep » Tue Apr 18, 2017 1:02 pm
(Similarly, if you're given (3r + 2 - s)(4r + 9 - s) = 0, it could be the case that 3r + 2 - s = 0, but it could also be true that 4r + 9 - s = 0. The concept is the same, it's just that the expressions are slightly more complicated in this example.)
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