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Manager  Joined: 25 Dec 2009
Posts: 75
In the xy-plane, if line k has negative slope and passes  [#permalink]

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Difficulty:   55% (hard)

Question Stats: 63% (02:00) correct 37% (02:03) wrong based on 449 sessions

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In the xy-plane, if line k has negative slope and passes through the point (-5,r) , is the x-intercept of line k positive?

(1) The slope of line k is -5.
(2) r > 0

OPEN DISCUSSION OF THIS QUESTION IS HERE: http://gmatclub.com/forum/in-the-xy-pla ... 35197.html
Math Expert V
Joined: 02 Sep 2009
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Re: In the xy-plane, if line k has negative slope and passes  [#permalink]

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9
2
In the xy-plane, if line k has negative slope and passes
through the point (-5,r) , is the x-intercept of line k
positive?

(1) The slope of line k is -5.
(2) r> 0

What is the answer and why ?

I am for B but it is not OG answer. Based on second condition r > 0, x-intercept for this line is negative, so we can determine definitively sign of x-intercept.

Let the $$x$$ intercept be the point $$(x,0)$$. Slope $$m$$ is rise over run and for two points $$(-5,r)$$ and $$(x,0)$$ would be $$m=\frac{r-0}{-5-x}=\frac{r}{-5-x}$$ --> $$x=\frac{-r-5m}{m}$$.

Question: is $$x>0$$? --> is $$x=\frac{-r-5m}{m}>0$$?

(1) $$m=-5$$ --> $$x=\frac{-r-5m}{m}=\frac{-r+25}{-5}>0$$? $$x=\frac{-r+25}{-5}=\frac{r}{5}-5>0$$? We can not determine whether $$\frac{r}{5}-5>0$$ or not. Not sufficient.

(2) $$r>0$$ and $$m<0$$ --> $$x=\frac{-r-5m}{m}=\frac{-r}{m}-5>0$$? $$\frac{-r}{m}$$ is some positive value (as $$m<0$$) but we don't know whether it's more than $$5$$ or not. Not sufficient.

(1)+(2) $$x=\frac{r}{5}-5$$ and $$r>0$$ --> $$r=5x+25>0$$ --> $$x>-5$$. $$x$$ can be positive as well as negative. Not sufficient.

This can be done by visualizing the question. Statement (2) tells us that the point $$(-5,r)$$, as $$r>0$$, is in the II quadrant. Line with negative slope through the point in the II quadrant can have $$x$$ intercept positive as well as negative.

Taken together: as we don't know the exact location of the point $$(-5,r)$$ in II quadrant we can not say even knowing the slope whether the $$x$$ intercept would be positive or negative.
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Re: In the xy-plane, if line k has negative slope and passes  [#permalink]

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18
In the xy-plane, if line k has negative slope and passes
through the point (-5,r) , is the x-intercept of line k
positive?

(1) The slope of line k is -5.
(2) r> 0

What is the answer and why ?

I am for B but it is not OG answer. Based on second condition r > 0, x-intercept for this line is negative, so we can determine definitively sign of x-intercept.

Best Approach to use a graph!
Attachments OG 12 DS Question.png [ 27.25 KiB | Viewed 18131 times ]

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General Discussion
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GMAT 1: 670 Q49 V31 GMAT 2: 710 Q50 V35 Re: In the xy-plane, if line k has negative slope and passes  [#permalink]

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Letme give you one alternate solution and correct me if m wrong.

Suppose line is y=mx+c

1. put value of y and x as -5,r

we get r =25+c

now put y=x=0 in y-mx-c which gives -c
This i hav done to check the side on which origin lies.
This is dependent on value of -c

2. r>0

For r >0, we cannot say anything about value of C even using r=25+c also.

thus we cannot c on which side origin lies.
For +ve intercept it should lie on left side of line and vice versa.

Thus ans E
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Manager  Joined: 25 Dec 2009
Posts: 75
Re: In the xy-plane, if line k has negative slope and passes  [#permalink]

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yes, good way too Gurpreet.
lets play with it further to strenghtne the concepts .

If I modify the second statement to

2. r < 0 ,

let's try to solve it in two ways i.e. First by visualising it and second, by using the basic equation y = mx +c

Using the second method,

The equation is r = 25+c and we know r < 0, so only possibility is c < -25 , so this can tell definitely the answer. Am I correct ?
But if we want to visualise it, is there any possibility it can pass through First quadrant ? Second quadrant i can imagine easy .

Also, How a line can make more than 180° angle ? I am from good maths background but do not remember exacly how a line makes more than 180° . So little help will break the code.
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GMAT 1: 670 Q49 V31 GMAT 2: 710 Q50 V35 Re: In the xy-plane, if line k has negative slope and passes  [#permalink]

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yes, good way too Gurpreet.
lets play with it further to strenghtne the concepts .

If I modify the second statement to

2. r < 0 ,

let's try to solve it in two ways i.e. First by visualising it and second, by using the basic equation y = mx +c

Using the second method,

The equation is r = 25+c and we know r < 0, so only possibility is c < -25 , so this can tell definitely the answer. Am I correct ?
But if we want to visualise it, is there any possibility it can pass through First quadrant ? Second quadrant i can imagine easy .

Also, How a line can make more than 180° angle ? I am from good maths background but do not remember exacly how a line makes more than 180° . So little help will break the code.

r = 25+c and r>0 is given that means 25+c >0
=> 25> -c

=> -25 <c
thus we cannot determine whether c is -ve or +ve

r-25-c should have same or different sign with -C to know whether that number is on same or diff side of origin.
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Manager  Joined: 25 Dec 2009
Posts: 75
Re: In the xy-plane, if line k has negative slope and passes  [#permalink]

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Let's take it a step further.
Adding one more question that will take discussion further . (Source: GMATprep)

Lnes 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 the point (5,1)
2. The y intercept of line n is greater than the y intercept of line p.

1. Clearly insufficient : For equation n y=m1.x+c1 and for equation p y=m2.x+c2 =>
First equation of line :1 = 5.m1+c1
Second equation of line: 1= 5.m2+c2

Clearly, m1 and m2 Can take any vset of values that m1>m2 and m1<m2, both are possible

2. Insufficent again : y=m1.x+c1 and y=m2.x+c2 => given is c1 > c2 => m1 and m2 can take any set of values again

1 and 2 statement togather :

First line : 1 = 5.m1+c1
Second Line: 1= 5.m2+c2

From this we can conclude that 5.m1+c1 = 5.m2+c2 or 5 (m1 - m2) = c2 - c1
now we are given c1 > c2 , I beleive that Statement refers to absolute values so
c1 could be 4 and c2 as 3 => c2-c1 gives -1
or c1 could be -4 and c2 as 3 => c2-c1 gives 7
or c1 could be 4 and c2 as -3 => c2-c1 gives -7
or c1 could be -4 and c2 -3 => c2-c1 gives 1

that means m1 and m2 can be greater or lesser than each other.So both together not sufficinet.

If I ignore my assumption that the values are absolute :
In that case, c2-c1 has to be negative if c1 > c2 . so m1 - m2 has to be negative too .

m1 - m2 < 0 or m1 < m2 , Sufficient

Question 1 : Which of the case one should assume ? answer is C that both together are sufficient that means we can't assume absolute values.

Question 2 : Sloe -1 and slope -2 , generally speaking -1 is greater than -2 but slope if we look at -2 is greater than -1.

What is the rule one should follow ?
Intern  Joined: 15 Nov 2009
Posts: 3
Re: In the xy-plane, if line k has negative slope and passes  [#permalink]

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I still cant visualize.

In stmt 2: we know that like passes through second quadrant and it has negative slope. How can it intercept at positive x?

Negative slope always goes bottom to top from left to right.

Manager  Joined: 26 May 2005
Posts: 183
Re: In the xy-plane, if line k has negative slope and passes  [#permalink]

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jshah wrote:
I still cant visualize.

In stmt 2: we know that like passes through second quadrant and it has negative slope. How can it intercept at positive x?

Negative slope always goes bottom to top from left to right.

take a line that is passing through the origin and bisects the II quadrant and IV quadrant. Now draw parallel lines above and below that line .. one will have positive x intercept and other will have negative x intercept and the original line will have 0 x-intecept.
Manager  Joined: 25 Dec 2009
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Re: In the xy-plane, if line k has negative slope and passes  [#permalink]

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@testprep2010 : Your response is E. I do not understand what values such posts bring.

1. This is OG question so answer is known.
2. The post does not just ask to answer just an option
3. How do I know your answer is wrong or right ?
4. Even if everyone votes A for this, it will not change the answer.

I feel sad why people put thier time answering like this on these forums.
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Re: In the xy-plane, if line k has negative slope and passes  [#permalink]

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Thanks Bunuel.. for the Kudos! A kudos from u.. means a lot! _________________
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Re: In the xy-plane, if line k has negative slope and passes  [#permalink]

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Bunuel, how do you know from statement 2 that m<0? Statement 2 only says that r>0. Could you please explain it to me? Thank you!
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Re: In the xy-plane, if line k has negative slope and passes  [#permalink]

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MariaBez wrote:
Bunuel, how do you know from statement 2 that m<0? Statement 2 only says that r>0. Could you please explain it to me? Thank you!

The question stem tells you that line k has negative slope so m < 0.

Also, you may want to look at the diagram made by jeeteshsingh in the first reply to this question. It uses both the statements together and shows you two possible cases (through two different dotted lines). In one case, line k has positive x intercept; in the second case it has negative x intercept. Hence both statements together are not sufficient.
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Re: In the xy-plane, if line k has negative slope and passes  [#permalink]

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VeritasPrepKarishma, thank you for your reply! I got it. Yeah, m<0 is given to us.
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Re: In the xy-plane, if line k has negative slope and passes  [#permalink]

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MariaBez wrote:
Bunuel, how do you know from statement 2 that m<0? Statement 2 only says that r>0. Could you please explain it to me? Thank you!

'm' is the slope of the line in the equation y = m*x + c
The question stem already states that the line has negative slope. Hence, m < 0.
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Re: In the xy-plane, if line k has negative slope and passes  [#permalink]

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1
In the xy-plane, if line k has negative slope and passes
through the point (-5,r) , is the x-intercept of line k
positive?

(1) The slope of line k is -5.
(2) r> 0

What is the answer and why ?

I am for B but it is not OG answer. Based on second condition r > 0, x-intercept for this line is negative, so we can determine definitively sign of x-intercept.

y=mx+c
substitute (-5,r)

r=-5m+c

C= r+5m

1) m=-5

y=mx+c

put y=0 to find x intercept

and m=-5

0= -5x+r+5(-5)
5x=r-25

Dont know r so insufficient

2) r> 0, certainly not sufficient
. .
. 0= mx+r+5m , we do not know m and r

combine both 1 and 2
5X=r-25 and r>0

r can be any thing so cannot determine X

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Posts: 1
Re: In the xy-plane, if line k has negative slope and passes  [#permalink]

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In the xy-plane, if line k has negative slope and passes through the point (-5,r) , is the x-intercept of line k positive?

(1) The slope of line k is -5.
(2) r > 0

Hi,
Can you please tell where I am wrong.
I assumed eq of line as y=mx+c.
Let x intercept be A(x,0)
At point A,
0=mx+c
=>x=-c/m
Hence, sign of x will depend on sign of c/m
Since m is negative,
i. if c<0, -c/m <0 =>x<0
ii. if c>0, -c/m >0 =>x>0

Statement 1 is not sufficient.
As per statement 2, r>0
=> -5m+c>0
=>-5m>-c
=>5m<c
=>5<c/m or c/m >5
=> both c and m have same sign i.e. m,c<0
=> x<0
=> Statement 2 is sufficient
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Re: In the xy-plane, if line k has negative slope and passes  [#permalink]

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PROMPT ANALYSIS
The equation of the will be y = r + m(x+5) where m is the slope. X intercept is -r/m -5

Superset
The answer will be either yes or no.

Translation
In order to find the answer we need:
1# Exact value of m and r
2# Equations to solve m and r

STATEMENT ANALYSIS

St 1: m = -5. For this x intercept is r/5 -5. Cannot say about the sign of the expression because r can be any real number.INSUFFICIENT. Hence option a and d eliminated.

St 2: r>0. The value of -r/m -5 will depend on the value of m, which can be any real number. INSUFFICIENT. Hence option b eliminated.

St 1&st 2: r/5 -5 will be positive for natural number r>25, these kind of constraints are not given. SO INSUFFICIENT. HEnce option c eliminated.
Option E.
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Re: In the xy-plane, if line k has negative slope and passes  [#permalink]

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In the xy-plane, if line k has negative slope and passes through the point (-5,r) , is the x-intercept of line k positive?

(1) The slope of line k is -5.
(2) r > 0

OPEN DISCUSSION OF THIS QUESTION IS HERE: http://gmatclub.com/forum/in-the-xy-pla ... 35197.html
_________________ Re: In the xy-plane, if line k has negative slope and passes   [#permalink] 19 Jan 2018, 10:39
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