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Lines and Graphs

by kartikshah » Thu Jul 26, 2012 9:31 am
In the xy plane, region R consists of all points (x,y) that satisfy the equation 2x+3y≤6. Is point (r,s) in the region?
Statement 01: 3r +2s = 6
Statement 02: r≤3, s≤2

Is there a way to solve this non-graphically?
OA is E
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Source: — Data Sufficiency |
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by eagleeye » Thu Jul 26, 2012 10:38 am
kartikshah wrote:In the xy plane, region R consists of all points (x,y) that satisfy the equation 2x+3y≤6. Is point (r,s) in the region?
Statement 01: 3r +2s = 6
Statement 02: r≤3, s≤2

Is there a way to solve this non-graphically?
OA is E
There is a way using algebra.

Statement 1:
3r +2s = 6
s = (6-3r)/2
substitute "s" in the inequality:
2r+3s <=6
=> 2r + 3(6-3r)/2 <=6
=> 4r+18-9r <=12
=> 5r >=6
=> r>=6/5

Now, we don't know whether r >=6/5. Statement 1 is insufficient.

At this point, we can see that, since statement 2 says r<=3, even if we were to combine the two statements, one says r<=3, and we need r>=6/5. Since we know that,
its true for r between 6/5 and 3, but false for r <6/5, the two together are insufficient.

Hence E.

Let me know if this helps :)
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by kartikshah » Fri Jul 27, 2012 5:20 am
Hi eagleeye,

I didn't completely understand.

1. We found that r≥(6/5) what does that mean? How can we conclude from this value of r whether or not it belongs to region R? What value would help us determine r's location?
2. How is it false for r<(6/5)? But doesn't combining the two equations tell us that R is between 6/5 and 3 both values included?

The problem with this question is not in solving it or doing the algebra. I just don't get the logic. And I don't think it's possible to draw graphs on the real test to solve! Surprisingly the official GMAT solution is with graphs. It's clear but not an efficient approach to the problem.
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by eagleeye » Fri Jul 27, 2012 5:52 pm
kartikshah wrote:Hi eagleeye,

I didn't completely understand.

1. We found that r≥(6/5) what does that mean? How can we conclude from this value of r whether or not it belongs to region R? What value would help us determine r's location?
r is an unknown variable (its the same thing as x here). Since we plug in the value of the equation in that of the region R, we can conclude from the answer that r is in the region if r>=6/5. I will write another DS eqn for you where it is always in the region. The algebraic method is correct here. r can be anything from -infinity to +infinity, hence r may or may not be in the region.
kartikshah wrote: 2. How is it false for r<(6/5)? But doesn't combining the two equations tell us that R is between 6/5 and 3 both values included?
No. The region R stays unchanged. This is what the two equations tell us. One tells us that r MUST be >=6/5 to be in the region. The other tells us that r IS <=3. If the second equation said r > 6/5, C would be the answer.
kartikshah wrote: The problem with this question is not in solving it or doing the algebra. I just don't get the logic. And I don't think it's possible to draw graphs on the real test to solve! Surprisingly the official GMAT solution is with graphs. It's clear but not an efficient approach to the problem.
Graphical solution works as well. I don't know what exactly you are missing. But the algebraic method above works!

Here's the re-worked question. OA is D.

In the xy plane, region R consists of all points (x,y) that satisfy the equation 2x+3y≤6. Is point (r,s) in the region?
Statement 01: 2r +3s = 7
Statement 02: r>3, s>0.

Try it out both algebraically and graphically.
Cheers!
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by GMATGuruNY » Fri Jul 27, 2012 9:19 pm
kartikshah wrote:In the xy plane, region R consists of all points (x,y) that satisfy the equation 2x+3y≤6. Is point (r,s) in the region?
Statement 01: 3r +2s = 6
Statement 02: r≤3, s≤2

Is there a way to solve this non-graphically?
OA is E
Question rephrased: Is 2r+3s≤6?

Try to plug in values that satisfy both statements.
Maximize r in one case and s in the other.

r maximized:
r=3 and s=0 satisfy both statements.
Is 2(3) + 3(0) ≤ 6?
YES.

s maximized:
r=2/3 and s=2 satisfy both statements.
Is 2(2/3) + 3(2) ≤ 6?
NO.

Since in the first case the answer is YES and in the second case the answer is NO, the two statements combined are INSUFFICIENT.

The correct answer is E.
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