In the xy-plane line k passes through the origin. Is the slope of line

Official Solution:


In the \(xy\)-plane line \(k\) passes through the origin. Is the slope of line \(k\) is greater than 1 ?

A line passing through the origin (through the point (0, 0) has the equation \(y=mx\).

A line passing through the origin AND with the slope greater than 1 must pass through any point from the green region shown below:



(1) Line \(k\) does not pass through any point \((a, \ b)\) where \(a\) and \(b\) are positive and \(a > b\).

This statement says that \(k\) does NOT pass through any point from the red region shown below:



Two cases are possible:

(i) Line \(k\) has a slope of 1, so the equation of line \(k\) is \(y=x\) (blue line). In this case the line won't be in the green region and we'd have a NO answer to the question.

(ii) Line \(k\) has a slope grater than 1, for example \(y=2x\), \(y=10x\), ... In this case the line will be in the green region and we'd have an YES answer to the question.

Not sufficient.

(2) Line \(k\) passes through the point \((c, \ d +1)\) where \(c\) and \(d\) are consecutive integers and \(c > d\).

\(c\) and \(d\) are consecutive integers and \(c > d\) mean that \(c=d+1\). So, this statement says that \(k\) passes through the point \((c, \ c)\)

If we knew that \(c \neq 0\), then we'd have that \(k\) passes through \((0, \ 0)\) and some point \((c, \ c)\) and this would mean that the equation of line \(k\) is \(y=x\). Which would give a NO answer to the question.

But if \(c = 0\), then we'd have that the line passes through the origin, which we already knew from the stem and in this case we could have an YES as well as a NO answers to the questions.

Not sufficient.

(1)+(2) The second statement is basically useless if \(c = 0\) so it adds no additional info to (1), which means that even taken together the statements are not sufficient.


Answer: E

Asked: In the xy-plane line k passes through the origin. Is the slope of line k is greater than 1 ?
Let the equation of the line k be : y = mx where m is the slope of the line.

(1) Line k does not pass through any point (a, b) where a and b are positive and a > b.
\(b \neq ma\)
\(m \neq b/a <1\)
But there is no mention about negative a & b
NOT SUFFICIENT

(2) Line k passes through the point (c, d +1) where c and d are consecutive integers and c > d.
d+1 = mc
m = (d+1)/c
c = d+1; since c and d are consecutive integers and c > d.
m = 1
But if c = 0; then the information does not provide any information since it is already given that (0,0) is on the line
NOT SUFFICIENT

IMO E

Statement 1 says more than that -- I've highlighted the additional information in red. Line k can pass through quadrant 1 here; it just can't pass through any point (a, b) where a > b > 0, which (since the line passes through the origin) means it can't have a slope between 0 and 1.

You're right that if Statement 1 did tell you the line contained no points in the first quadrant, then we could be sure its slope was negative (assuming the line is not vertical or horizontal), and then Statement 1 would be sufficient.

For a line to pass from (c,c) it's equation should be y=x and clearly it has slope =1. So this should be sufficient to tell that slope is not greater than 1. Hence, B is sufficient. Did I miss something?

From the stem we know that the line passes through (0, 0) and the question asks whether the slope of the line is greater than 1.

Now, from (2) IF c = 0, then this would mean that (2) says that the line passes through (0, 0). So, in case of c = 0, this statement would not add any new info to what we already knew from the stem. A line passing through (0, 0) can have any slope: 1 (y = x), less than 1 (y = -x), more than 1(y = 2x). So, we can have both an YES and a NO answers to the question.

If we knew that c ≠ 0, then this statement would be sufficient because a line passing through (0, 0) and (c, c), where c ≠ 0 would have the slope of 1.

Hope it's clear.

Given: The line passes through origin
Asked: Is slope >1

Case 1: Line does not pass through any point (a,b) = a>b
Meaning the line will pass through points (a,a) or (a,b) where a<b --> this means the slope is >=1
Thus case 1 = insufficient

Case 2: Line passes through (c,d+1) where c and d are consecutive integers c>d
Note that it is mentioned the line passes through such a point --> line can pass through (0,0) --> and still be y=x or y=-x --> thus slope can be 1 or -1 that is <1
Thus case 2 = insufficient

Case 1+2: The solution of case 1 is still valid without any change --> Thus 1+2 = insufficient. Thus answer: E

I solved it using the below

Question Stem: y = mx + 0 is m > 1? or y > x or y/x > 1 ? or y-x >0 ?

St1 : plugging x and y for a and b we get y >= x. True if y > x. False if y = x Hence, not sufficient
St2 : plugging x and y for c and d+1 we get x > y-1 or y-x < 1. True if 0<y-x< 1 False if y-x < 0. Hence, not sufficient

Combining St1 & St2 we get 0<= y-x < 1. Hence, not sufficient

Hey Bunuel, for statement B, we know line passes through (0,0) and (c,c), slope of a line is y2-y1/x2-x1 so in this case it can never be greater than 1.Hence B should be sufficient. Can you give an example where slope is greater than 1?

The answer to your question is in the solution. Please check the highlighted part.

Hey Bunuel, it may be silly of me for asking, but I didn't understand the solution. Why does it matter if for one case (0,0) statement repeats what we already know ? It doesn't prove that slope can be > 1 . What am I missing here ?