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Manager  Joined: 24 Nov 2010
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Lines m and n lie in the xy-plane and intersect at the point (-2; 4).  [#permalink]

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Lines m and n lie in the xy-plane and intersect at the point (-2; 4). Is the slope of line m less than the slope of line n?

(1) The x-intercept of line m is greater than the x-intercept of line n.
(2) The y-intercept of line n is greater than the y-intercept of line m.
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Re: Lines m and n lie in the xy-plane and intersect at the point (-2; 4).  [#permalink]

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karthiksms wrote:
could someone pl explain ? i'm unable to understand answer.

ALGEBRAIC WAY:
Lines m and n lie in the xy-plane and intersect at the point (-2; 4). Is the slope of line m less than the slope of line n?

Equation of a line in point intercept form is $$y=mx+b$$, where: $$m$$ is the slope of the line, $$b$$ is the y-intercept of the line (the value of $$y$$ for $$x=0$$), $$-\frac{b}{m}$$ is the x-intercept of the line (the value of $$x$$ for $$y=0$$). (Check Coordinate Geometry chapter of Math Book for more on this topic: math-coordinate-geometry-87652.html)

We are given two lines: $$y_m=mx+b$$ and $$y_n=nx+c$$. Now, as they intersect at the point (-2; 4) then: $$4=-2m+b$$ and $$4=-2n+c$$ (this point is common for both of the lines) --> $$b=4+2m$$ and $$c=4+2n$$.

Question: is $$m<n$$?

(1) The x-intercept of line m is greater than the x-intercept of line n --> $$-\frac{b}{m}>-\frac{c}{n}$$ --> $$-\frac{4+2m}{m}>-\frac{4+2n}{n}$$ --> $$\frac{1}{n}-\frac{1}{m}>0$$ --> insufficient to answer whether $$m<n$$: if $$n=2$$ and $$m=-4$$ then YES but if $$n=2$$ and $$m=4$$ then NO. Not sufficient.

(2) The y-intercept of line n is greater than the y-intercept of line m --> $$c>b$$ --> $$4+2n>4+2m$$ --> $$n>m$$. Sufficient.

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Re: Lines m and n lie in the xy-plane and intersect at the point (-2; 4).  [#permalink]

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I would go with D.
We are already given one point at the point of intersection.
Statement 1 gives us an extra point therefore we can determine which line is steeper. sufficient
Statement 2 also gives us the same information. sufficient
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Re: Lines m and n lie in the xy-plane and intersect at the point (-2; 4).  [#permalink]

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liftoff wrote:
I would go with D.
We are already given one point at the point of intersection.
Statement 1 gives us an extra point therefore we can determine which line is steeper. sufficient
Statement 2 also gives us the same information. sufficient

I was thinking if determining which one is steeper is enough or not.

Per (1) The x-intercept of line m is greater than the x-intercept of line n.

so line m can have a slope of lets say -2. If line n also has a -ve slope then it will need to be 'flatter' than line m for its x intercept to be greater than that of line m. So its slope will need to be > -2 (for -ve slopes flatter line are closer to 0 that slopes of steeper lines). so in this case slope of line n can be something like -1. So slope of n > slope of m (-1>-2).
however, if slope of m is lets say 2, slope of n can be -ve like -2 with a greater x intercept. This satisfies the condition 1, but slope of m> slope of n in this case (2>-2).
Hence, (1) is insufficient.

Similarly we can prove for the y intercept in case of 2nd statement.

The 2 statements taken together, they should still be insufficient.

OA is D. But i'm having trouble understanding it.

Can someone please explain if I'm wrong?
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Re: Lines m and n lie in the xy-plane and intersect at the point (-2; 4).  [#permalink]

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karthiksms wrote:
could someone pl explain ? i'm unable to understand answer.

GRAPHIC APPROACH:
Lines m and n lie in the xy-plane and intersect at the point (-2; 4). Is the slope of line m less than the slope of line n?

(1) The x-intercept of line m is greater than the x-intercept of line n. Draw lines:
Case A:
Attachment: graph.png [ 8.32 KiB | Viewed 9840 times ]
Red lines represent possible position of line m and blue line possible position of line n. (You can see that x-intercept of red lines>x-intercept of blue line, so the condition in the statement is satisfied).
In the first case, both slopes are positive and red line (m) is steeper than blue line (n) which means that slope of line m>slope of line n (a steeper incline indicates a higher slope absolute value).

Case B:
Attachment: graph 2.png [ 8.56 KiB | Viewed 9833 times ]
In this case, the slope of n is positive ans the slope of m is negative, hence slope of line n>slope of line m.

Two different answers. Not sufficient.

(2) The y-intercept of line n is greater than the y-intercept of line m. Draw lines:
Case A:
Attachment: graph 3.png [ 8.28 KiB | Viewed 9829 times ]
In the first case, both slopes are positive and blue line (n) is steeper than red line (m) which means that slope of line n>slope of line m (a steeper incline indicates a higher slope absolute value).Sufficient.

Case B:
Attachment: graph 4.png [ 7.92 KiB | Viewed 9818 times ]
In this case, both slopes are negative and red line (m) is steeper than blue line (m), so the slope of m is more negative than slope of n (|m|>|n| --> -m>-n --> n>m), which again means that slope of line n>slope of line m.Sufficient.

Similar question: lines-n-and-p-lie-in-the-xy-plane-is-the-slope-of-line-n-97007.html?hilit=more%20negative%20slope%20steeper#p747640

Hope it's clear.
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Lines m and n lie in the xy-plane and intersect at the point (-2; 4).  [#permalink]

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I have seen a lot of examples of this question, and every time it comes down to the Y intercept.
Can we use this as a shortcut? can we make a generalization here?
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Re: Lines m and n lie in the xy-plane and intersect at the point (-2; 4).  [#permalink]

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ronr34 wrote:
Bunuel wrote:
Bumping for review and further discussion.

I have seen a lot of examples of this question, and every time it comes down to the Y intercept.
Can we use this as a shortcut? can we make a generalization here?

Yes we can use something else other than the graph method, which I find a little bit difficult to follow on many cases.

Let's see we need to know if m>n or if m-n>0, 'm and 'n' being the slopes of the respective lines

We know that they intersect at point (-2,4)

Then we have Line m= y = mx + b ---> 4 = -2m +b
Line n = y = nx+c---> 4=-2n+c

Both equal, -2n+c= -2m+b
2m-2n = b-c
m-n = (b-c)/2

Now going back to the question.

Is b-c/2 > 0?

is b-c>0, is b-c?

Statement 1

We are given that -b/m>-c/n
-bn>-cm

We can't tell whether b>c
Insuff

Statement 2

This is exactly what we were looking for
b>c

Sufficient

B stands

Hope its clear
Cheers!
J YES YOU CAN

Originally posted by jlgdr on 28 Dec 2013, 16:14.
Last edited by jlgdr on 18 May 2014, 09:05, edited 1 time in total.
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Re: Lines m and n lie in the xy-plane and intersect at the point (-2; 4).  [#permalink]

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amaxing Jlgdr!! thank you buddy. it is so clear now
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Re: Lines m and n lie in the xy-plane and intersect at the point (-2; 4).  [#permalink]

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Bunuel wrote:

(1) The x-intercept of line m is greater than the x-intercept of line n --> $$-\frac{b}{m}>-\frac{c}{n}$$

hi Bunuel, I'm not sure how can we come to this equation mentioned above. Would you please explain a bit on this ?
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Re: Lines m and n lie in the xy-plane and intersect at the point  [#permalink]

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musunna wrote:
Bunuel wrote:

(1) The x-intercept of line m is greater than the x-intercept of line n --> $$-\frac{b}{m}>-\frac{c}{n}$$

hi Bunuel, I'm not sure how can we come to this equation mentioned above. Would you please explain a bit on this ?

I think this is explained here: slopes-of-m-and-n-124025.html#p1029568

Equation of a line in point intercept form is $$y=mx+b$$, where: $$m$$ is the slope of the line, $$b$$ is the y-intercept of the line (the value of $$y$$ for $$x=0$$),$$-\frac{b}{m}$$ is the x-intercept of the line (the value of $$x$$ for $$y=0$$). (Check Coordinate Geometry chapter of Math Book for more on this topic: math-coordinate-geometry-87652.html)
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Re: Lines m and n lie in the xy-plane and intersect at the point (-2; 4).  [#permalink]

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_________________ Re: Lines m and n lie in the xy-plane and intersect at the point (-2; 4).   [#permalink] 03 Oct 2019, 13:37
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