Solution



You can use this fact to examine the information given in the first two statements. Remember that you need to evaluate each statement by itself.

Choice A states that the x-intercept is twice the y-intercept, so you can conclude that both intercepts have the same sign, and thus the slope of line k is negative. So the information in Choice A is sufficient to determine that the slope of line k is negative.

Choice B states that the product of the x-intercept and the y-intercept is positive. You know that the product of two numbers is positive if both factors have the same sign. So this information is also sufficient to determine that the slope of line k is negative.

Choice C, it is helpful to recall the definition of the slope of a line passing through two given points. You may remember it as “rise over run.”
If the two points are (a, b) and (r, s), then the slope is \(\frac{b-s}{a-r}\)
Choice C states that the product of the quantities (a−r) and (b−s) is negative. Note that these are the denominator and the numerator, respectively, of \(\frac{b-s}{a-r}\), the slope of line k. So you can conclude that (a−r) and (b−s) have a−r opposite signs and the slope of line k is negative. The information in Choice C is sufficient to determine that the slope of line k is negative.
So each of the three statements individually provides sufficient information to determine whether the slope of line k is negative.

The correct answer are \(A, B, and C\).

"Choice A states that the x-intercept is twice the y-intercept, so you can conclude that both intercepts have the same sign, and thus the slope of line k is negative."
How can you assure that slope of line k is negative?? please can you explain it with let the x and y cor-ordiante?

This is actually a tricky one!

What you are thinking about is x intercept = -4 and y intercept = 2.

However \(-4 \neq 2*2\) hence even the first statement is sufficient for a -ve slope line K.
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Carcass I didnt understand choice A


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No matter value you consider for x then y is 1/2 of x

Of course, if x is -4 y must be -2 because Y cannot be for instance 2 because 2 is NOT 1/2 of -4

Hope now is more clear

For option A, if the X and Y intercepts are 0, then X intercept will indeed be twice that of Y intercept.
So couldn't a line have a positive slope like y = x and still satisfy the A condition?

Good idea, but the x- and y-intercepts cannot both be 0, since the question explicitly tells us that "line k does NOT pass through the origin"
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That's a very valid point, thank you. I guess this scenario is one of the main reasons to include the "not origin" clause in the question.

Bump

Got it wrong in the first attempt because of option C.

It's tricky because I'm used to the formula of slope of a line being written as y2-y1/x2-x1.
Anyway, since the questions says x1-x2/y1-y2<0 if we take the reciprocal of that and reverse the terms,

y2-y1/x2-x1<0 which basically means that the slope is negative.

Hence C is sufficient!

For A and B, the explanation given above by the moderators is spot-on!

for option b without having idea about x and y intercept how how we will know that product is positive...

Had a doubt if (a-r)(b-s)<0 then we can say that a<r and b<s correct so using the formula of line passing to 2 points it will s-b/r-a. Both of them are positive so how come C option is correct?

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In option B, both X and Y intercepts can be negative or positive. It does not mean the slope will be negative. I am a little bit confused.

"It's the hardest problem in the GRE official book. 90% of test takers missed it." -- per a Manhhtan GRE blog.
So congratulations to folks who got it right, and that too within a min or two.


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coming back to ques:

given: line doesn't pass through origin. i.e. m or c Not Equal to 0.
ques: is slope negative. i.e. is \(m < 0\) ?


A). The x-intercept of line k is twice the y-intercept of line k.
\(- c/m = 2c\)
\(c(2m+1)/m = 0\)
since, c & m are not zero.
thus, \(2m + 1 = 0\)
\(m = -1/2\)
\(m < 0\)


B). The product of the x-intercept and the y-intercept of line k is positive.
\((-c/m) * c > 0\)
\(c^2 / m < 0\)
since, \(c^2 >= 0\)
thus, \(m < 0\)


C). Line k passes through the points and where (a, b) (r, s), where (a − r)(b − s) < 0.
line eqn for 1st point --> \(b = ma + c\)
line eqn for 2nd point --> \(s = mr + c\)
since, both can be equated at C,
thus, \(b - ma = s -mr\)
\(m = (b-s) / (a-r)\)
now since, \((a − r)(b − s) < 0\).
thus, each term is opposite in sign to other.
\(m < 0\)