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Coordinate geometry, last question

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Coordinate geometry, last question

by sapuna » Sun Aug 10, 2014 1:07 am
In the xy plane , the line k passes through a point T with coordinates (a;b) where axb does not equal zero. Is b positive ?

1)The slope of line k is negative

2) a< b

I answered 5 , but the correct question is 3.

Here`s what I did:

From statement 1 we know that the slope is negative but thats not enough. Point T can obviousy be (1;1) but also ( -2;-4) so we don`t know from b

2) Statement 2 tells us that a < b. It`s not enough as b can equal 4 and a can equal 2 but point T can also be with coordinates ( -3;-2) and so again two different cases for b so we don`t know.

Combined statements 1) and 2) are given as enough even though the two possibilities I listed right above I still think are possible. What did I do wrong ?
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by GMATGuruNY » Sun Aug 10, 2014 4:42 am
The problem should read as follows:
In the xy-plane, the line k passes through THE ORIGIN and through the point (a,b), where ab does not equal 0. Is b positive?

(1) The slope of line k is negative

(2) a < b
Clearly neither statement is sufficient on its own.

Statements combined:
If b is negative, and a<b, then a and b are both negative.
Since line k passes not only through (a, b) but also through the origin, a figure like the following is implied:
Image
Not viable.
The line in the figure above has a positive slope, but statement 1 indicates that line k must have NEGATIVE slope.
Implication:
To satisfy the constraint that line k has a negative slope, b cannot be negative.
Thus, since b≠0, b must be POSITIVE.
SUFFICIENT.

The correct answer is C.
Last edited by GMATGuruNY on Sun Aug 10, 2014 8:43 am, edited 3 times in total.
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by sapuna » Sun Aug 10, 2014 5:49 am
Umm, I don`t want to embarass myself but I study in a prestigious European business university and the lecturers teach us that for a line to have a negative slope when the y decreases the x must increase. The lane that you drew, doesn`t it have a positive slope as both x and y increase ?
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by GMATGuruNY » Sun Aug 10, 2014 7:02 am
sapuna wrote:The lane that you drew, doesn`t it have a positive slope as both x and y increase ?
Precisely.
If b is negative, the result will be a line with a POSITIVE slope, as illustrated in my figure above.
But statement 1 indicates that line k must have a NEGATIVE slope.
It is not possible for line k to have a negative slope if b<0.
Thus, we know that b must be POSITIVE.
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by GMATGuruNY » Sun Aug 10, 2014 8:45 am
In the xy-plane, the line k passes through THE ORIGIN and through the point (a,b), where ab does not equal 0. Is b positive?

(1) The slope of line k is negative

(2) a < b
Here's how the statements can be combined algebraically:

The equation of a line is y = mx + b, where m is the slope and b is the y-intercept.
Since line k passes through (0,0), its y-intercept is 0.
Thus, the equation for line k is as follows:
y = mx.

Since (a, b) is on the line, we get:
b = ma
a = b/m.

Substituting a = b/m into a < b, we get:
b/m < b.
(b/m) - b < 0
(b)(1/m - 1) < 0.

Since the slope of line k is negative, m<0.
Implication:
1/m - 1 = negative - 1 = negative.

Since (b)(1/m - 1) < 0 and 1/m - 1 < 0, we get;
(b)(negative) < 0.
The resulting inequality holds true only if b>0.
SUFFICIENT.

The correct answer is C.
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by GMATGuruNY » Sun Aug 10, 2014 9:06 am
In the xy-plane, the line k passes through THE ORIGIN and through the point (a,b), where ab does not equal 0. Is b positive?

(1) The slope of line k is negative

(2) a < b
One more way to combine the two statements is to TEST CASES.
The slope of a line = (y₂ - y�)/(x₂ - x�).
Test whether it is possible that b<0.

Case 1: b=-1, a=-2
Since the line passes through (0, 0) and (-2, -1), we get:
Slope = (-1-0)/(-2-0) = 1/2.
Not possible, since statement 1 indicates that the slope must be negative.

Case 2: b=-3, a =-10
Since the line passes through (0, 0) and (-10, -3), we get:
Slope = (-3-0)/(-10-0) = 3/10.
Not possible, since statement 1 indicates that the slope must be negative.

The cases above illustrate that it is not possible that b<0.
Thus, b>0.
SUFFICIENT.

The correct answer is C.
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by sapuna » Tue Aug 12, 2014 2:27 am
I carefully reread what you wrote the first time and got it finally. Thank you !

I`d like to ask you something though. How do we know that the lane k passes through the origin (0;0) The mistake I made was that when you drew a lane with point T with a and b both negative , the lane you drew has a positive slope cuz it passed through the origin. However , if it didnt apss trhough the origin a and b could both be negative with b > a if it doesnt pass through the origin.
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by GMATGuruNY » Tue Aug 12, 2014 5:55 am
sapuna wrote:I carefully reread what you wrote the first time and got it finally. Thank you !

I`d like to ask you something though. How do we know that the lane k passes through the origin (0;0) The mistake I made was that when you drew a lane with point T with a and b both negative , the lane you drew has a positive slope cuz it passed through the origin. However , if it didnt apss trhough the origin a and b could both be negative with b > a if it doesnt pass through the origin.
The question stem that you posted is incorrectly worded.
Where did you find this version?
In my initial post, I corrected the question stem to read as follows:
In the xy-plane, the line k passes through THE ORIGIN and through the point (a,b), where ab does not equal 0. Is b positive?
In the correct version, we know that line k passes through both (0, 0) and (a, b).
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by sapuna » Tue Aug 12, 2014 6:15 am
GMATGuruNY wrote:
sapuna wrote:I carefully reread what you wrote the first time and got it finally. Thank you !

I`d like to ask you something though. How do we know that the lane k passes through the origin (0;0) The mistake I made was that when you drew a lane with point T with a and b both negative , the lane you drew has a positive slope cuz it passed through the origin. However , if it didnt apss trhough the origin a and b could both be negative with b > a if it doesnt pass through the origin.
The question stem that you posted is incorrectly worded.
Where did you find this version?
In my initial post, I corrected the question stem to read as follows:
In the xy-plane, the line k passes through THE ORIGIN and through the point (a,b), where ab does not equal 0. Is b positive?
In the correct version, we know that line k passes through both (0, 0) and (a, b).
I don`t actually remember. Either 800 score or the new gmat prep , avaiable for download from the official website.

To sum it all up , if the point passes through the origin , then answer C is correct. But if it doesn`t , then both statements together are not sufficient , correct ?
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by GMATGuruNY » Tue Aug 12, 2014 6:54 am
sapuna wrote:
To sum it all up , if the line passes through the origin , then answer C is correct. But if it doesn`t , then both statements together are not sufficient , correct ?
Correct!
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by sapuna » Tue Aug 12, 2014 7:36 am
GMATGuruNY wrote:
sapuna wrote:
To sum it all up , if the line passes through the origin , then answer C is correct. But if it doesn`t , then both statements together are not sufficient , correct ?
Correct!
I`m a machine as well now , a junior machine :D

Thumbs up for me tomrorow. Hope there will be no hard coordinate geometry questions lol
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