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# Lines r and s lie in the xy-plane. Is the y-intercept of lin

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Intern
Joined: 19 Apr 2013
Posts: 3
Lines r and s lie in the xy-plane. Is the y-intercept of lin  [#permalink]

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Updated on: 08 Dec 2015, 02:38
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65% (hard)

Question Stats:

57% (01:57) correct 43% (01:45) wrong based on 269 sessions

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Lines r and s lie in the xy-plane. Is the y-intercept of line r less than the y-intercept of line s ?

(1) At the intersection point of r and s, the x-coordinate and y-coordinate are both negative.
(2) The slope of line r is greater than the slope of line s.

Source Jeff Sackmann

Please explain. Also, please explain how this would work if the were question were asking about the x-intercepts of lines 'r' and 's'

Originally posted by kzaveri17 on 07 Jun 2013, 02:35.
Last edited by chetan2u on 08 Dec 2015, 02:38, edited 2 times in total.
Edited the question.
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Joined: 11 Jun 2010
Posts: 74
Re: Lines r and s lie in the xy-plane. Is the y-intercept of lin  [#permalink]

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07 Jun 2013, 14:12
1
See attached images.

Ans C
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St1 ST2 Together.JPG [ 30.08 KiB | Viewed 4105 times ]

Manager
Joined: 11 Jun 2010
Posts: 74
Re: Lines r and s lie in the xy-plane. Is the y-intercept of lin  [#permalink]

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07 Jun 2013, 14:32
If we were to consider the same question for x-coordinates, I think the answer would be E.
See image attached.
Attachments

SameQ for Xco-ord.JPG [ 73.16 KiB | Viewed 4101 times ]

Intern
Joined: 19 Apr 2013
Posts: 3
Re: Lines r and s lie in the xy-plane. Is the y-intercept of lin  [#permalink]

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08 Jun 2013, 00:04
Thank you so much. Figuring that out for Y-axis was not much of a problem, however, not so for the x-axis. Thank for for the second opinion.

Warm regards
Math Expert
Joined: 02 Sep 2009
Posts: 53063
Re: Lines r and s lie in the xy-plane. Is the y-intercept of lin  [#permalink]

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08 Jun 2013, 02:28
kzaveri17 wrote:
Lines r and s lie in the xy-plane. Is the y-intercept of line r less than the y-intercept of line s ?

(1) At the intersection point of r and s, the x-coordinate and y-coordinate are both negative.
(2) The slope of line r is greater than the slope of line s.

Source Jeff Sackmann

Please explain. Also, please explain how this would work if the were question were asking about the x-intercepts of lines 'r' and 's'

Similar questions to practice:
slopes-of-m-and-n-124025.html
point-r-s-lies-on-the-line-l-is-the-line-s-intercept-with-94505.html
in-the-xy-plane-both-line-k-and-l-intersect-with-axis-y-is-94508.html
line-m-and-n-pass-through-point-1-2-is-the-slope-of-m-90629.html
if-lines-k-and-j-are-in-the-xy-coordinate-plane-is-the-84532.html
in-the-xy-plane-is-the-slope-of-line-l-greater-than-the-126941.html
lines-n-and-p-lie-in-the-xy-plane-is-the-slope-of-the-line-30553.html
line-l1-has-a-slope-a-and-line-l2-has-the-slope-b-is-a-b-140953.html
if-the-slopes-of-the-line-l1-and-l2-are-of-the-same-sign-is-126759.html
lines-n-and-p-lie-in-the-xy-plane-is-the-slope-of-line-n-127999.html

Hope it helps.
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Joined: 06 Sep 2013
Posts: 1694
Concentration: Finance
Re: Lines r and s lie in the xy-plane. Is the y-intercept of lin  [#permalink]

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15 Feb 2014, 04:30
Bunuel wrote:
kzaveri17 wrote:
Lines r and s lie in the xy-plane. Is the y-intercept of line r less than the y-intercept of line s ?

(1) At the intersection point of r and s, the x-coordinate and y-coordinate are both negative.
(2) The slope of line r is greater than the slope of line s.

Source Jeff Sackmann

Please explain. Also, please explain how this would work if the were question were asking about the x-intercepts of lines 'r' and 's'

Similar questions to practice:
slopes-of-m-and-n-124025.html
point-r-s-lies-on-the-line-l-is-the-line-s-intercept-with-94505.html
in-the-xy-plane-both-line-k-and-l-intersect-with-axis-y-is-94508.html
line-m-and-n-pass-through-point-1-2-is-the-slope-of-m-90629.html
if-lines-k-and-j-are-in-the-xy-coordinate-plane-is-the-84532.html
in-the-xy-plane-is-the-slope-of-line-l-greater-than-the-126941.html
lines-n-and-p-lie-in-the-xy-plane-is-the-slope-of-the-line-30553.html
line-l1-has-a-slope-a-and-line-l2-has-the-slope-b-is-a-b-140953.html
if-the-slopes-of-the-line-l1-and-l2-are-of-the-same-sign-is-126759.html
lines-n-and-p-lie-in-the-xy-plane-is-the-slope-of-line-n-127999.html

Hope it helps.

Hi Bunuel, is it worth giving algebraic approach a try here? Or how does one deal with the statement mentioning that they meet on the third quadrant?

Thanks!
Cheers
J
Intern
Joined: 16 Jun 2010
Posts: 18
Re: Lines r and s lie in the xy-plane. Is the y-intercept of lin  [#permalink]

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05 Jun 2014, 23:49
kzaveri17 wrote:
Lines r and s lie in the xy-plane. Is the y-intercept of line r less than the y-intercept of line s ?

(1) At the intersection point of r and s, the x-coordinate and y-coordinate are both negative.
(2) The slope of line r is greater than the slope of line s.

Source Jeff Sackmann

Please explain. Also, please explain how this would work if the were question were asking about the x-intercepts of lines 'r' and 's'

The line s: y = ax + b, b is the y-intercept of s
The line r: y = cx + d, d is the y-intercept of r

The question "Is the y-intercept of line r less than the y-intercept of line s?" can interpret as: Is d < b?

1. The intersection point of r and s will have the same x-coordinate and y-coordinate: ax + b = cx + d => $$x=$$ $$\frac{(d-b)}{(a-c)}$$Since this intersection point has negative x-intercept: $$\frac{(d-b)}{(a-c)}$$ $$<0$$
There are 2 scenario:
(a) d-b<0 and a-c>0 => d<b
or
(b) d-b>0 and a-c<0 => d>b
=> (1) is not sufficient
2. c>a or a - c <0: not sufficient

(1) + (2):
a - c < 0 and [d-b][/a-c]<0
=> d - b > 0 => d > b: Sufficient. The answer is C
VP
Joined: 05 Mar 2015
Posts: 1001
Re: Lines r and s lie in the xy-plane. Is the y-intercept of lin  [#permalink]

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11 Oct 2016, 07:31
kzaveri17 wrote:
Lines r and s lie in the xy-plane. Is the y-intercept of line r less than the y-intercept of line s ?

(1) At the intersection point of r and s, the x-coordinate and y-coordinate are both negative.
(2) The slope of line r is greater than the slope of line s.

Source Jeff Sackmann

Please explain. Also, please explain how this would work if the were question were asking about the x-intercepts of lines 'r' and 's'

Bunuel /experts

is my algebraic approach correct??
Combining both statements
let intersection coordinates be(-x,-y)
then equation for
line r----> -y=m1(-x)+b
line s------> -y=m2(-x)+c

slope r------>m1=b/x
slope s------->m2=c/x

as per option (2)---------> b/x>c/x------> b>c.........sufficient

thanks
Magoosh GMAT Instructor
Joined: 28 Dec 2011
Posts: 4486
Re: Lines r and s lie in the xy-plane. Is the y-intercept of lin  [#permalink]

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11 Oct 2016, 10:22
abhimahna wrote:
Hey Mike,

I am a Magoosh student. I need your help on the below question. Could you please help me get into the final solution. I marked C as the answer but the different responces of the people have confused me.

Lines r and s lie in the xy-plane. Is the y-intercept of line r less than the y-intercept of line s ?

(1) At the intersection point of r and s, the x-coordinate and y-coordinate are both negative.
(2) The slope of line r is greater than the slope of line s

Dear abhimahna,

I'm happy to respond. The answer is (C).

Your question is extremely unclear. Are you confused about what this OA is? Or would you like further explanation about some of the reasoning that other people have used? If the latter is the case, please let me know exactly what points confuse you. I recommend this blog:

Does all this make sense?
Mike
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Mike McGarry
Magoosh Test Prep

Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)

VP
Joined: 05 Mar 2015
Posts: 1001
Re: Lines r and s lie in the xy-plane. Is the y-intercept of lin  [#permalink]

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11 Oct 2016, 10:30
mikemcgarry wrote:
abhimahna wrote:
Hey Mike,

I am a Magoosh student. I need your help on the below question. Could you please help me get into the final solution. I marked C as the answer but the different responces of the people have confused me.

Lines r and s lie in the xy-plane. Is the y-intercept of line r less than the y-intercept of line s ?

(1) At the intersection point of r and s, the x-coordinate and y-coordinate are both negative.
(2) The slope of line r is greater than the slope of line s

Dear abhimahna,

I'm happy to respond. The answer is (C).

Your question is extremely unclear. Are you confused about what this OA is? Or would you like further explanation about some of the reasoning that other people have used? If the latter is the case, please let me know exactly what points confuse you. I recommend this blog:

Does all this make sense?
Mike

Hi mike

I too have a confusion regarding the same..

both attached fig have slope R> slope S
but intercept of r > s in first but intercept of R , s in 2nd one

Thanks
Attachments

tutor.png [ 16.03 KiB | Viewed 1792 times ]

Magoosh GMAT Instructor
Joined: 28 Dec 2011
Posts: 4486
Re: Lines r and s lie in the xy-plane. Is the y-intercept of lin  [#permalink]

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11 Oct 2016, 10:36
rohit8865 wrote:
kzaveri17 wrote:
Lines r and s lie in the xy-plane. Is the y-intercept of line r less than the y-intercept of line s ?

(1) At the intersection point of r and s, the x-coordinate and y-coordinate are both negative.
(2) The slope of line r is greater than the slope of line s.

Source Jeff Sackmann

Please explain. Also, please explain how this would work if the were question were asking about the x-intercepts of lines 'r' and 's'

Bunuel /experts

is my algebraic approach correct??
Combining both statements
let intersection coordinates be(-x,-y)
then equation for
line r----> -y=m1(-x)+b
line s------> -y=m2(-x)+c

slope r------>m1=b/x
slope s------->m2=c/x

as per option (2)---------> b/x>c/x------> b>c.........sufficient

thanks

Dear rohit8865,
My friend, I would say that yours is an exceptionally elegant solution. Good work!
Mike
_________________

Mike McGarry
Magoosh Test Prep

Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)

Magoosh GMAT Instructor
Joined: 28 Dec 2011
Posts: 4486
Re: Lines r and s lie in the xy-plane. Is the y-intercept of lin  [#permalink]

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11 Oct 2016, 10:51
1
rohit8865 wrote:
Hi mike

I too have a confusion regarding the same..

both attached fig have slope R> slope S
but intercept of r > s in first but intercept of R , s in 2nd one

Thanks

Dear rohit8865,

I'm happy to respond.

My friend, in math, we have to treat even the words with mathematical precision. Consider statement #2:
The slope of line r is greater than the slope of line s.
Exactly what does this say? Slope is a number, a number on the number line.

Suppose line r has a slope of -0.1, and line s had a slope of -50. If looked at a picture, line r would look almost horizontal, with a very slight downward grade, but line s would have a massively steep slope down to the the right. Which has a bigger slope?

Well, (-50) < (-0.1). The question is asking for the literal order of the two numbers on the number line, and -0.1 is to the right of -50 on the number line. Therefore, m = -0.1 is a bigger slope than m = -50, even though the latter is steeper. Steeper in a negative direction is a lower slope, exactly as bigger in the negative direction is a smaller number.

Does all this make sense?
Mike
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Mike McGarry
Magoosh Test Prep

Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)

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Posts: 1001
Re: Lines r and s lie in the xy-plane. Is the y-intercept of lin  [#permalink]

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11 Oct 2016, 11:15
mikemcgarry wrote:
rohit8865 wrote:
Hi mike

I too have a confusion regarding the same..

both attached fig have slope R> slope S
but intercept of r > s in first but intercept of R , s in 2nd one

Thanks

Dear rohit8865,

I'm happy to respond.

My friend, in math, we have to treat even the words with mathematical precision. Consider statement #2:
The slope of line r is greater than the slope of line s.
Exactly what does this say? Slope is a number, a number on the number line.

Suppose line r has a slope of -0.1, and line s had a slope of -50. If looked at a picture, line r would look almost horizontal, with a very slight downward grade, but line s would have a massively steep slope down to the the right. Which has a bigger slope?

Well, (-50) < (-0.1). The question is asking for the literal order of the two numbers on the number line, and -0.1 is to the right of -50 on the number line. Therefore, m = -0.1 is a bigger slope than m = -50, even though the latter is steeper. Steeper in a negative direction is a lower slope, exactly as bigger in the negative direction is a smaller number.

Does all this make sense?
Mike

thanks mike
this is exactly what i also thought after solving algebraically as in above post
Just wanna confirm from experts...

Thanks
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Joined: 16 Oct 2010
Posts: 8891
Location: Pune, India
Re: Lines r and s lie in the xy-plane. Is the y-intercept of lin  [#permalink]

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12 Oct 2016, 03:55
2
rohit8865 wrote:
kzaveri17 wrote:
Lines r and s lie in the xy-plane. Is the y-intercept of line r less than the y-intercept of line s ?

(1) At the intersection point of r and s, the x-coordinate and y-coordinate are both negative.
(2) The slope of line r is greater than the slope of line s.

Source Jeff Sackmann

Please explain. Also, please explain how this would work if the were question were asking about the x-intercepts of lines 'r' and 's'

Bunuel /experts

is my algebraic approach correct??
Combining both statements
let intersection coordinates be(-x,-y)
then equation for
line r----> -y=m1(-x)+b
line s------> -y=m2(-x)+c

slope r------>m1=b/x
slope s------->m2=c/x

as per option (2)---------> b/x>c/x------> b>c.........sufficient

thanks

Responding to a pm:

I don't get how you calculated slopes as b/x and c/x.

If I have not misunderstood your variables, it should be:

r passes through (-x, -y) and (0, b) (since y intercept is b)
s passes through (-x, -y) and (0, c)

Slope = (y2 - y1)/(x2-x1)

Slope for r = (b + y)/(0 + x) = (b+y)/x
Slope for s = (c + y)/(0 + x) = (c+y)/x

Though in the case too,
(b+y)/x > (c+y)/x

Multiply both sides by x which is positive.

b + y > c + y

b > c
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Re: Lines r and s lie in the xy-plane. Is the y-intercept of lin  [#permalink]

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13 Oct 2016, 08:38
Dear Mike/ experts,

Considering stm 1 and 2 together: what if R is parallel to Y axis, hence its slope is infinity and it has no Y intercept. S can be any line with positive Y intercept. We can have a scenario where both these lines intersect such that both its coordinates are negative (stm 1) and considering that infinity is more than the slope of line S (say having positive slope of 50) , it also adheres to stm 2. But here intercept of R (nil) is not more than intercept of S (some positive value).
Magoosh GMAT Instructor
Joined: 28 Dec 2011
Posts: 4486
Re: Lines r and s lie in the xy-plane. Is the y-intercept of lin  [#permalink]

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13 Oct 2016, 09:57
Alchemist87 wrote:
Dear Mike/ experts,

Considering stm 1 and 2 together: what if R is parallel to Y axis, hence its slope is infinity and it has no Y intercept. S can be any line with positive Y intercept. We can have a scenario where both these lines intersect such that both its coordinates are negative (stm 1) and considering that infinity is more than the slope of line S (say having positive slope of 50) , it also adheres to stm 2. But here intercept of R (nil) is not more than intercept of S (some positive value).

Dear Alchemist87,

I'm happy to respond.

First, I will say that this is a very creative and out-of-the-box observation. This creativity will serve you well in your career.

In this particular problem, though, I think that case is eliminated by the question. Here's the text of the question again:
Lines r and s lie in the xy-plane. Is the y-intercept of line r less than the y-intercept of line s ?

(1) At the intersection point of r and s, the x-coordinate and y-coordinate are both negative.
(2) The slope of line r is greater than the slope of line s.

If the question is asking whether one quantity is less than another quantity. It would be devious and tricky in a way that the GMAT simply does not do if one of the quantities about which it was asking in that context simply did not exist. The fact that the GMAT asks about the two y-intercepts really implies, among other things, that they both exist.

Now, a vertical line R might be consistent with Statement #1 by itself, by this statement by itself is not sufficient anyway.

Once we get to statement #2, we get another prohibition: although stated verbally, the statement (slope R) < (slope S) is a well-defined mathematical statement. We can only make well-defined mathematical statements with numbers that exist on the number line. Metaphorically, we can say that infinity is "bigger" than any given number, but we can't actually state that fact as a well-defined mathematical statement. (There are many different kinds of infinity, but that's mathematics that is well beyond the GMAT!) In practice, any value that becomes infinite, such as 1/0, is called "undefined," precisely because it departs from the region for which we can make well-defined statements.

Therefore, both the prompt and statement #2 make a vertical line impossible in this problem.

I want to emphasize, though, what you asked was truly a brilliant question, my friend. Let me know if you have any more questions.

Mike
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Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)

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Re: Lines r and s lie in the xy-plane. Is the y-intercept of lin  [#permalink]

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13 Oct 2016, 10:39
Thxs Mike.. for taking time to revert to my query & for those kind words- just made my day!!
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Re: Lines r and s lie in the xy-plane. Is the y-intercept of lin  [#permalink]

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24 Dec 2017, 07:11
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Re: Lines r and s lie in the xy-plane. Is the y-intercept of lin   [#permalink] 24 Dec 2017, 07:11
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