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

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Lines n and p lie in the xy plane. Is the slope of line n [#permalink] New post 10 Jul 2010, 00:05
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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 > 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
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Re: DS : Coordinate Geometery [#permalink] New post 10 Jul 2010, 05:27
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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: is m_1<m_2?

(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.

Answer: 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
1.PNG [ 14.29 KiB | Viewed 1990 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
2.PNG [ 13.66 KiB | Viewed 1990 times ]


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

Answer: C.

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

Hope it helps.
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Re: DS : Coordinate Geometery [#permalink] New post 10 Jul 2010, 00:46
1. Point of intersection is (5,1) This cannot help us determine the slopes of the lines as they could be pretty much any lines that pass through this point, all of which have different slopes.

INSUFFICIENT.

2. In the slope-intercept eqn of a line, we have that y = mx + b, where m is the slope and b is the y-intercept. As you can see the slope of a line is completely independent of its y-intercept.

INSUFFICIENT

Both 1 and 2.

Both lines n and p pass through (5,1)
n has y-intercept (0,f) and p has (0,g) where f > g

Slope can be found using the formula of two-points: (y2 - y1) / (x2 - x1)

=> Slope of n is (1 - f)/5 and slope of p is (1 -g)/5

Since f > g
=> -f < -g
=> 1 - f < 1 - g
=> (1 - f)/5 < (1 - g)/5
=> Slope of n < Slope of p

SUFFICIENT

Pick C
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Re: problem solving [#permalink] New post 06 Feb 2011, 10:09
y1= m1 * x1 = b1
y2= m2 * x2 = b2

1)
1 = m1 * 5 + b1
1 = m2 * 5 + b2
Not suff. as we do not know anything about b1 and b2
2) b1 > b2 Not suff. as we do not know anything about other values

Combining,
as b1 > b2, m1 has to be less than m2 to satifsy m1 * 5 + b1 = m2 * 5 + b2
So, C
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Re: DS : Coordinate Geometery [#permalink] New post 09 Feb 2011, 03:51
My take on how C is right :

Let the co-ord of n be (x1,y1) and the co-ord of p be (x1,y1) on y intercept

Slope of n = (5 - y1)/(1 - x1) = 5-y1 (as x1 is 0 on y-axis)




Slope of p = (5 - y2)/(1 - x2) = 5-y2 (as x2 is 0 on y-axis)



Now y1 > y2, slope of n < slope of p.

Regards,
Subhash
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Re: DS : Coordinate Geometery [#permalink] New post 19 May 2011, 13:08
If we try to draw lines, we can clearly see why c is true.
Re: DS : Coordinate Geometery   [#permalink] 19 May 2011, 13:08
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