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# Lines n and p lie in the xy plane. Is the slope of the line

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Manager
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Lines n and p lie in the xy plane. Is the slope of the line [#permalink]

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15 Jun 2006, 07:05
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Lines n and p lie in the xy plane. Is the slope of the line n less than the slope of line p?

(1) Lines n and p intersect at the point (5,1)
(2) The y-intercept of line n is greater than the y-intercept of line p

OPEN DISCUSSION OF THIS QUESTION IS HERE: lines-n-and-p-lie-in-the-xy-plane-is-the-slope-of-line-n-127999.html
[Reveal] Spoiler: OA

Last edited by Bunuel on 28 Aug 2013, 10:31, edited 4 times in total.
Edited the question and added the OA
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15 Jun 2006, 07:32
Line n and p lie in the xy-plane.Is the slope of the line n less than the slope of line p?

I'm getting E

If itâ€™s correct, will explain later, if not, then someone plz explain to me
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15 Jun 2006, 13:03
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The point of intersection is known. Draw a rough sketch and the answer is before you

<phew> time for a beer...
Attachments

This is the only configuration when you take both conditions into acct. If only one condition is taken, then you can have multiple configurations.

untitled.GIF [ 3.8 KiB | Viewed 24202 times ]

Last edited by paddyboy on 16 Jun 2006, 08:05, edited 2 times in total.
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15 Jun 2006, 14:08
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I am getting C and here's why.
1) Lines n and p intersect at the point (5,1)
2) The y-intercept of line n is greater than the y-intercept of line p

Stmt1: Intersection point doesn't tell us anything. Not sufficient.

Stmt2: Yn > Yp .. not sufficient either.

Together:

Let intersection point (x2, y2) = (5, 1)
Let the other intersection points be on the y-axis so that x = 0. Given Yn > Yp

Sn = Slope of line n
Sp = Slope of line p

Sn = (y2-y1)/(x2-x1) = (1-Yn)/(5-0) = (1-Yn)/5..(a)
Sp = (y2-y1)/(x2-x1) = (1-Yp)/(5-0) = (1-Yp)/5..(b)

From (a) and (b)
Yn = 1-5Sn
Yp = 1-5Sp

Given Yn > Yp,
1-5Sn > 1-5Sp => Sn > Sp

Therefore together sufficient.
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15 Jun 2006, 15:17
(C) for me.

I suggest to draw quickly an XY plan to figure the problem.

for n: Y = a(n)*X + b(n)
for p: Y = a(p)*X + b(p)

1) intersecting on 1 points <=> a(n) <> a(p)
INSUFF

2) b(n) > b(p)
INSUFF

(1) and (2), it exists 3 cases:

case 1 : b(n) > 1 and b(p) >= 1
To pass by the point (5,1):
> The line n has a negative slop, a(n) < 0
> The line p has a negative slop, a(p) =< 0
as b(n) > b(p), a(p) > a(n)

case 2 : b(n) >= 1 and b(p) < 1
To pass by the point (5,1):
> The line n has a negative slop, a(n) =< 0
> The line p has a positive slop, a(p) > 0
thus a(p) > a(n)

case 3 : b(n) < 1 and b(p) < 1
To pass by the point (5,1):
> The line n has a positive slop, a(n) > 0
> The line p has a positive slop, a(p) > 0
as b(n) > b(p), a(p) > a(n)
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16 Jun 2006, 06:39
[quote="haas_mba07"]I am getting C and here's why.
1) Lines n and p intersect at the point (5,1)
2) The y-intercept of line n is greater than the y-intercept of line p

Stmt1: Intersection point doesn't tell us anything. Not sufficient.

Stmt2: Yn > Yp .. not sufficient either.

Together:

Let intersection point (x2, y2) = (5, 1)
Let the other intersection points be on the y-axis so that x = 0. Given Yn > Yp

Sn = Slope of line n
Sp = Slope of line p

Sn = (y2-y1)/(x2-x1) = (1-Yn)/(5-0) = (1-Yn)/5..(a)
Sp = (y2-y1)/(x2-x1) = (1-Yp)/(5-0) = (1-Yp)/5..(b)

From (a) and (b)
Yn = 1-5Sn
Yp = 1-5Sp

Given Yn > Yp,
1-5Sn > 1-5Sp => Sn > Sp

Therefore together sufficient.[/quote]

Can you please explain usng the equation y=mx+c, where m=slope and c=intercept.
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08 Jan 2007, 12:35
its actually quite easy.
y=mx+c
Put (5,1) in both eqns. substract eqn of n from eqn of p.
then see urself..
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27 May 2007, 15:32
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Line n and p lie in the xy-plane.Is the slope of the line less than the slope of line p?
1) Lines n and p intersect at the point (5,1)
2) The y-intercept of line n is greater than the y-intercept of line p

For line n: y = m1x + b1
For line p: y = m2x + b2

at (5,1), for line n: 5 = m1+b1
at (5,1), for line p: 5 = m2+b2

Conclusion from statement 1: m1+b1 = m2+b2
When using statement 2 in addition to statement 1,
b1 > b2 ---> m1 < m2
[ where m1 is the slope of line n and m2 is the slope of line p ]

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27 May 2007, 16:02
Its C. Heres my two cents.

Stmnt 1: Obviously insufficient. Line N and P could virtually go anyway you want them to so it could be yes and no = insuf.

Stmnt 2: Insuffc. Just make some points where N has a great y intercept (where x is 0) n: has point (0,2) p: (0,1) If u give the points (1,2) also for N. and (1,1) for p. then u can figure out the slopes. (the lines will be parallel) ul get a slope 0/1 for both. I dunno if u can have a slope of 0/1, but thats what it came out to. This means that p=n. You could also just pick some other points or just see that the lines could be made almost anyway. N could have a positive or negative slope and same with P. = insuf.

Stmnts Together.

Here N is forced to have a negative slope, while P is forced to have a positive slope. This is because the lines cant be perpendicular b/c they both have to have a y-intercept. They can't be parallel or be the same line since they have to have different y-intercepts. N has a greater y-intercept than P, which forces N to go from top left to bottom right since it has to meet at (5,1). Likewise it forces P to go from bottom left to top right.

Thus P's slope>N's slope.

Hope this wasn't too confusing. I wrote it like this to so I could better understand it myself.
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27 May 2007, 16:11
Hmmm lookin at other people's stmnts looks like the slopes of both lines can be negative or positive, etc... as stated by Fig.

Duh, shouldve seen that.

Makes sense, but that just again says that the answer should be C.
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29 May 2007, 21:29

The point of intersection is known. Draw a rough sketch and the answer is before you

<phew> time for a beer...

i agree that the answer is C... however i disagree with you that you have the only config. possible. i sketched two general situations to get my answer based one with negative slopes the other positve. I guess there should also be a third graph where slope of n=0 and slope of p is positive.

why is that the only configuration possible? you do not take
negative slopes into consideration, no? also, if the y intercept of n>y intercept of p don't you have your labels wrong (i think you labelebd your lines based on x intercepts)
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30 Jan 2010, 10:34
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Line N

y= m1x + b1

Line P

y=m2x + b2

Statement 1---Insufficient

1 = 5m1 + b1
m1 = (1-b1)/5---------------------------------------(1)

1=5m2 + b2
m2= (1-b2)/5---------------------------------------(2)

Since we dont know b1 or b2 ( y intercepts)-----INSUFFICIENT

Statement 2------INSUFFICIENT

b1>b2. No relationship established

Combining both,

From (1) and (2),

m1 = (1-b1)/5 and m2 = (1-b2)/5

Since, b1>b2

m1<m2

Hence Sufficient

Hope u guys find this method easier
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30 Jan 2011, 16:50
Has to be E. It is being seen that in the explanation provided by paddyboy that slope(n) > slope(p),
Fig proves that slope(p) > slope(n) and mishari again proves slope(p) > slope(n) so why is it C?
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30 Jan 2011, 17:47
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puneetj wrote:
Has to be E. It is being seen that in the explanation provided by paddyboy that slope(n) > slope(p),
Fig proves that slope(p) > slope(n) and mishari again proves slope(p) > slope(n) so why is it C?

Graphic approach:

Lines n and p lie in the xy plane. Is the slope of line n less than the slope of line p?

(1) Lines n and p intersect at (5,1)
(2) The y-intercept of line n is greater than y-intercept of line p

The two statements individually are not sufficient.

(1)+(2) Note that a higher absolute value of a slope indicates a steeper incline.

Now, if both lines have positive slopes then as the y-intercept of line n (blue) is greater than y-intercept of line p (red) then the line p is steeper hence its slope is greater than the slope of the line n:
Attachment:

1.PNG [ 14.29 KiB | Viewed 22472 times ]

If both lines have negative slopes then again as the y-intercept of line n (blue) is greater than y-intercept of line p (red) then the line n is steeper hence the absolute value of its slope is greater than the absolute value of the slope of the line p, so the slope of n is more negative than the slope of p, which means that the slope of p is greater than the slope of n:
Attachment:

2.PNG [ 13.66 KiB | Viewed 22474 times ]

So in both cases the slope of p is greater than the slope of n. Sufficient.

Algebraic approach:

Lines n and p lie in the xy-plane. Is the slope of line n less than the slope of line p?

We have two lines: $$y_n=m_1x+b_1$$ and $$y_p=m_2x+b_2$$. Q: $$m_1<m_2$$ true?

(1) Lines n and p intersect at the point (5,1) --> $$1=5m_1+b_1=5m_2+b_2$$ --> $$5(m_1-m_2)=b_2-b_1$$. Not sufficient.
(2) The y-intercept of line $$n$$ is greater than the y-intercept of line $$p$$ --> y-intercept is value of $$y$$ for $$x=0$$, so it's the value of $$b$$ --> $$b_1>b_2$$ or $$b_2-b_1<0$$. Not sufficient.

(1)+(2) $$5(m_1-m_2)=b_2-b_1$$, as from (2) $$b_2-b_1<0$$ (RHS), then LHS (left hand side) also is less than zero $$5(m_1-m_2)<0$$ --> $$m_1-m_2<0$$ --> $$m_1<m_2$$. Sufficient.

For more on this topic check Coordinate Geometry Chapter of Math Book: math-coordinate-geometry-87652.html

Hope it helps.
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30 Jan 2011, 20:11
Thanks Bunuel. The absolute value reasoning made it clear. + 1
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30 Jan 2011, 22:51
Q:
If $$m_n$$ and $$m_m$$ are slopes for n and m respectively,

Is $$m_n<m_m$$

Generic formula for a line passing through a single point $$(x_1,y_1)$$is:
$$y-y_1=m(x-x_1)$$
where m is the slope the line

1. (x,y)=(5,1)
Point (5,1) statisfies equations for both lines. Plugging in the values, we get the equation for both the lines as:

Equation for the line n:
$$y-1=m_n(x-5)$$
$$y=m_n*x-m_n*5+1$$
$$y=m_n*x+(1-m_n*5)$$

Likewise for the line m:
$$y-1=m_m(x-5)$$
$$y=m_m*x-m_m*5+1$$
$$y=m_m*x+(1-m_m*5)$$

$$(1-m_n*5)$$ and $$(1-m_m*5)$$ are Y-intercepts for n and m respectively

But we don't know any comparable information about the slopes or intercepts to deduce anything concrete; NOT SUFFICIENT.

2. Y-intercept for n is more than Y-intercept for m.

If equation for line n is
$$y=m_n*x+C_n$$

If equation for line m is
$$y=m_m*x+C_m$$

where $$C_n$$ and $$C_m$$ are Y-intercepts for n and m respectively
We just know that $$C_n>C_m$$. We know nothing about the slopes. NOT SUFFICIENT.

However, using both statements;

Y Intercept for n $$C_n$$ is $$(1-m_n*5)$$
Y Intercept for m $$C_m$$ is $$(1-m_m*5)$$

And

$$C_n > C_m$$

$$(1-m_n*5) > (1-m_m*5)$$
$$-m_n*5 > -m_m*5$$ ###subtracting 1 from both sides###
$$m_n*5 < m_m*5$$ ###multiplying both sides by -ve sign###
$$m_n < m_m$$ ###Dividing both sides by 5###

SUFFICIENT.

Ans: C
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Re: Line n and p lie in the xy-plane.Is the slope of the line [#permalink]

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24 Apr 2012, 05:20
what condition in the question leads to conclude that either both slopes will be positive or both negative.
i think because difference of y intercept of n & y intercept of p is positive,so slopes of n & p will either be both positive or both negative.
Algebraically its fine

But how to satisfy this condition in graphs.
Can't it be one positive and one negative slope.

Thanks
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Re: Line n and p lie in the xy-plane.Is the slope of the line [#permalink]

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24 Apr 2012, 12:39
GMATD11 wrote:
what condition in the question leads to conclude that either both slopes will be positive or both negative.
i think because difference of y intercept of n & y intercept of p is positive,so slopes of n & p will either be both positive or both negative.
Algebraically its fine

But how to satisfy this condition in graphs.
Can't it be one positive and one negative slope.

Thanks

It's not so that the slopes of the lines are either both positive to both negative. For example it's possible line n (blue) to have negative slope and line p (red) to have positive slope (naturally in this case slope of n will be less than the slope of p). We are just considering the cases which are not that obvious to see that in ALL cases slope of n is less than the slope of p.

Hope it's clear.
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Re: Lines n and p lie in the xy plane. Is the slope of the line [#permalink]

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19 Jun 2012, 21:04
Hi Bunuel,
Request you to explain the following part. I'm unable to comprehend it.. though I am able to solve the questions by assuming the coordinates.
Thanks
H
Bunuel wrote:

If both lines have negative slopes then again as the y-intercept of line n (blue) is greater than y-intercept of line p (red) then the line n is steeper hence the absolute value of its slope is greater than the absolute value of the slope of the line p, so the slope of n is more negative than the slope of p, which means that the slope of p is greater than the slope of n:

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Re: Lines n and p lie in the xy plane. Is the slope of the line [#permalink]

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19 Jun 2012, 23:12
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C.

1. Says they intersect. Nothing about the lines.
2. Y intersect of n > p [This can narrow down, but there is a chance that lines could be parallel]

Both=> intersecting(not parallel) hence enough.
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Re: Lines n and p lie in the xy plane. Is the slope of the line   [#permalink] 19 Jun 2012, 23:12

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