If line L has a positive slope, what is the x-intercept of L

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gmattesttaker2: Legendary Member; Posts: 641; Joined: Tue Feb 14, 2012 3:52 pm; Thanked: 11 times; Followed by:8 members

If line L has a positive slope, what is the x-intercept of L

by gmattesttaker2 » Mon Sep 02, 2013 5:29 pm

Hello,

Can you please assist with this question? This is from MGMAT:

OA: D

Thanks for your help.

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Source: Beat The GMAT — Data Sufficiency | Mon Sep 02, 2013 5:29 pm

GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pm; Location: New York, NY; Thanked: 13060 times; Followed by:1906 members; GMAT Score:790

by GMATGuruNY » Mon Sep 02, 2013 9:11 pm

If line L in the xy-coordinate plane has a positive slope, what is the x-intercept of L ?

(1) There are different points (a, b) and (c, d) on line L such that ad = bc.

(2) There are constants m and n such that the points (m, n) and (-m, -n) are both on line L

Many slope questions can be solved efficiently by DRAWING.

Statement 1: There are different points (a, b) and (c, d) on line L such that ad = bc.
Case 1: a=-1, d=4, b=-2, c=2
Here, line L contains (a,b) = (-1,-2) and (c,d) = (2,4):

Since line L passes through the origin, its x-intercept is 0.

Case 2: a=-4, d=2, b=-1, c=8
Here, line L contains (a,b) = (-4,-1) and (c,d) = (8,2):

Since line L passes through the origin, its x-intercept is 0.

The two random cases above illustrate that -- when ad=bc -- line L must pass through the origin and thus will have an x-intercept of 0.
SUFFICIENT.

Statement 2: There are constants m and n such that the points (m, n) and (-m, -n) are both on line L
Case 3: m=1 and n=1
Here, line L contains (m,n) = (1,1) and (-m,-n) = (-1,-1):

Since line L passes through the origin, its x-intercept is 0.

Case 4: m=-3, n=-3
Here, line L contains (m,n) = (-3,-3) and (-m,-n) = (3,3):

Since line L passes through the origin, its x-intercept is 0.

The two random cases above illustrate that -- since Line L contains both (m,n) and (-m,-n) -- it must pass through the origin and thus will have an x-intercept of 0.
SUFFICIENT.

The correct answer is D.

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kevincanspain: GMAT Instructor; Posts: 613; Joined: Thu Mar 22, 2007 6:17 am; Location: madrid; Thanked: 171 times; Followed by:64 members; GMAT Score:790

by kevincanspain » Tue Sep 03, 2013 3:51 am

Mitch's method is fantastic!

Also, for (1), you could reason as follows:

Any two different points on the line can be written as (a,b) and (c,d)=(a+k, b+mk), where m is the slope of the line, and k is a positive number.

That ad=bc tells us that ab +amk = ba + bk

Thus amk =bk , meaning that am=b.
So (a,b) = (a,am)
If (x,0) is also on this line with slope m, (am)/(a-x) = m ,and assuming that a is not equal to x, a= a - x, and therefore x=0. Thus the x-intercept has to be 0, which is to say that the line passes through the origin

If a=x, (a,b) and (x,0) are the same point, and the line passes through the origin.

In any event, the x-intercept is 0.

Perhaps there is a shorter way to arrive at this conclusion mathematically, but NO way will be more efficient that Mitch's! It's amazing how many questions are best done by means of examples

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GMAT Instructor
Gmatclasses
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kevincanspain: GMAT Instructor; Posts: 613; Joined: Thu Mar 22, 2007 6:17 am; Location: madrid; Thanked: 171 times; Followed by:64 members; GMAT Score:790

by kevincanspain » Tue Sep 03, 2013 3:51 am

Kevin Armstrong
GMAT Instructor
Gmatclasses
Madrid

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Mission2012: Master | Next Rank: 500 Posts; Posts: 184; Joined: Sun Aug 19, 2012 10:04 pm; Thanked: 10 times; Followed by:2 members

by Mission2012 » Tue Sep 03, 2013 4:24 am

Hi Mitch,

I had a little different approach for (1) -

as (a,b) and (c,d) are different points on the line and ad = bc or b/a = d/c

As for any point (m,n) the slope of line joining the point and origin will have slope = n/m
Hence slope of these two points with line joining them to origin b/a and d/c respectively.

As b/a = d/c, it implies both the point are collinear with origin. Hence x intercept it 0

Is this approach right?

GMATGuruNY wrote:
If line L in the xy-coordinate plane has a positive slope, what is the x-intercept of L ?

(1) There are different points (a, b) and (c, d) on line L such that ad = bc.

(2) There are constants m and n such that the points (m, n) and (-m, -n) are both on line L
Many slope questions can be solved efficiently by DRAWING.

Statement 1: There are different points (a, b) and (c, d) on line L such that ad = bc.
Case 1: a=-1, d=4, b=-2, c=2
Here, line L contains (a,b) = (-1,-2) and (c,d) = (2,4):

Since line L passes through the origin, its x-intercept is 0.

Case 2: a=-4, d=2, b=-1, c=8
Here, line L contains (a,b) = (-4,-1) and (c,d) = (8,2):

Since line L passes through the origin, its x-intercept is 0.

The two random cases above illustrate that -- when ad=bc -- line L must pass through the origin and thus will have an x-intercept of 0.
SUFFICIENT.

Statement 2: There are constants m and n such that the points (m, n) and (-m, -n) are both on line L
Case 3: m=1 and n=1
Here, line L contains (m,n) = (1,1) and (-m,-n) = (-1,-1):

Since line L passes through the origin, its x-intercept is 0.

Case 4: m=-3, n=-3
Here, line L contains (m,n) = (-3,-3) and (-m,-n) = (3,3):

Since line L passes through the origin, its x-intercept is 0.

The two random cases above illustrate that -- since Line L contains both (m,n) and (-m,-n) -- it must pass through the origin and thus will have an x-intercept of 0.
SUFFICIENT.

The correct answer is D.

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faraz_jeddah: Master | Next Rank: 500 Posts; Posts: 358; Joined: Thu Apr 18, 2013 9:46 am; Location: Jeddah, Saudi Arabia; Thanked: 42 times; Followed by:7 members; GMAT Score:730

by faraz_jeddah » Tue Sep 03, 2013 9:53 pm

GMATGuruNY wrote:

Statement 1: There are different points (a, b) and (c, d) on line L such that ad = bc.
Case 1: a=-1, d=4, b=-2, c=2
Here, line L contains (a,b) = (-1,-2) and (c,d) = (2,4):

Since line L passes through the origin, its x-intercept is 0.

Case 2: a=-4, d=2, b=-1, c=8
Here, line L contains (a,b) = (-4,-1) and (c,d) = (8,2):

Since line L passes through the origin, its x-intercept is 0.

The two random cases above illustrate that -- when ad=bc -- line L must pass through the origin and thus will have an x-intercept of 0.
SUFFICIENT.

.
Mitch,

For evaluating Statement 1 - How would you know that the line passes through the origin? You would have to find that out using equation of a line which you havent mentioned in your solution.

A good question also deserves a Thanks.

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pm; Location: New York, NY; Thanked: 13060 times; Followed by:1906 members; GMAT Score:790

by GMATGuruNY » Tue Sep 03, 2013 10:07 pm

faraz_jeddah wrote:
For evaluating Statement 1 - How would you know that the line passes through the origin? You would have to find that out using equation of a line which you havent mentioned in your solution.

Connect (-1,-2) and (2.4), and you get a line that passes through the origin.
Connect (-4,-1) and (8,2), and you get a line that passes through the origin.
There is no need to determine the equation of either line.

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faraz_jeddah: Master | Next Rank: 500 Posts; Posts: 358; Joined: Thu Apr 18, 2013 9:46 am; Location: Jeddah, Saudi Arabia; Thanked: 42 times; Followed by:7 members; GMAT Score:730

by faraz_jeddah » Tue Sep 03, 2013 10:16 pm

GMATGuruNY wrote:
faraz_jeddah wrote:
For evaluating Statement 1 - How would you know that the line passes through the origin? You would have to find that out using equation of a line which you havent mentioned in your solution.
Connect (-1,-2) and (2.4), and you get a line that passes through the origin.
Connect (-4,-1) and (8,2), and you get a line that passes through the origin.
There is no need to determine the equation of either line.

Dont you think that would leave chance for an error? I mean I just drew a rough graph and I cannot be 100% sure that the x int would be 0. It appears as 0.5 as my grids are not accurate.

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pm; Location: New York, NY; Thanked: 13060 times; Followed by:1906 members; GMAT Score:790

by GMATGuruNY » Tue Sep 03, 2013 11:41 pm

faraz_jeddah wrote:
GMATGuruNY wrote:
faraz_jeddah wrote:
For evaluating Statement 1 - How would you know that the line passes through the origin? You would have to find that out using equation of a line which you havent mentioned in your solution.
Connect (-1,-2) and (2.4), and you get a line that passes through the origin.
Connect (-4,-1) and (8,2), and you get a line that passes through the origin.
There is no need to determine the equation of either line.
Dont you think that would leave chance for an error? I mean I just drew a rough graph and I cannot be 100% sure that the x int would be 0. It appears as 0.5 as my grids are not accurate.

Even a rudimentary drawing should be sufficient to see that (a.b) and (c,d) exhibit SYMMETRY about the origin.

Case 1:
(-1,-2) --> left 1, down 2.
(2, 4) --> right 2, up 4.
For each point, the horizontal distance from the origin is 1/2 the vertical distance from the origin.

Case 2:
(-4, -1) --> left 4, down 1.
(8,2) --> right 8, up 2.
For each point, the horizontal distance from the origin is 4 times the vertical distance from the origin.

Since in each case (a,b) and (c,d) exhibit symmetry about the origin, the line that connects them must pass THROUGH the origin.

Quote

gmattesttaker2: Legendary Member; Posts: 641; Joined: Tue Feb 14, 2012 3:52 pm; Thanked: 11 times; Followed by:8 members

by gmattesttaker2 » Thu Sep 05, 2013 9:03 pm

GMATGuruNY wrote:
If line L in the xy-coordinate plane has a positive slope, what is the x-intercept of L ?

(1) There are different points (a, b) and (c, d) on line L such that ad = bc.

(2) There are constants m and n such that the points (m, n) and (-m, -n) are both on line L
Many slope questions can be solved efficiently by DRAWING.

Statement 1: There are different points (a, b) and (c, d) on line L such that ad = bc.
Case 1: a=-1, d=4, b=-2, c=2
Here, line L contains (a,b) = (-1,-2) and (c,d) = (2,4):

Since line L passes through the origin, its x-intercept is 0.

Case 2: a=-4, d=2, b=-1, c=8
Here, line L contains (a,b) = (-4,-1) and (c,d) = (8,2):

Since line L passes through the origin, its x-intercept is 0.

The two random cases above illustrate that -- when ad=bc -- line L must pass through the origin and thus will have an x-intercept of 0.
SUFFICIENT.

Statement 2: There are constants m and n such that the points (m, n) and (-m, -n) are both on line L
Case 3: m=1 and n=1
Here, line L contains (m,n) = (1,1) and (-m,-n) = (-1,-1):

Since line L passes through the origin, its x-intercept is 0.

Case 4: m=-3, n=-3
Here, line L contains (m,n) = (-3,-3) and (-m,-n) = (3,3):

Since line L passes through the origin, its x-intercept is 0.

The two random cases above illustrate that -- since Line L contains both (m,n) and (-m,-n) -- it must pass through the origin and thus will have an x-intercept of 0.
SUFFICIENT.

The correct answer is D.