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# In the xy-plane, if line k has negative slope, is the

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Senior Manager
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In the xy-plane, if line k has negative slope, is the  [#permalink]

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14 Mar 2012, 01:04
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In the xy-plane, if line k has negative slope, is the y-intercept of line k positive?

(1) The x-intercept of line k is less than the y-intercept of line k.

(2) The slope of line k is less than -2.

It is a DS question, can you help and explain the answer?
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Re: In the xy-plane, if line k has negative slope, is the  [#permalink]

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15 Mar 2012, 09:15
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In the xy-plane, if line k has negative slope, is the y-intercept of line k positive?

Equation of a line in point intercept form is $$y=mx+b$$, where: $$m$$ is the slope of the line and $$b$$ is the y-intercept of the line (the value of $$y$$ for $$x=0$$). So, basically we are asked whether $$b>0$$.

(1) The x-intercept of line k is less than the y-intercept of line k --> x-intercept is value of $$x$$ for $$y=0$$, so it's $$-\frac{b}{m}$$. The statement says that: $$-\frac{b}{m}<b$$ --> multiply by negative $$m$$ and flip the sign of the inequality: $$-b>bm$$ --> $$b(m+1)<0$$. Now, in order $$b>0$$ to be true $$m+1$$ should be negative, so the question becomes: is $$m+1<0$$? --> is $$m<-1$$. We don't know that. Not sufficient.

(2) The slope of line k is less than -2. Insufficient on its own.

(1)+(2) From (1) the question became: "is $$m<-1$$?" and (2) says that $$m<-2$$. Sufficient.

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Re: DS co-ordinate geometry question  [#permalink]

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14 Mar 2012, 01:51
2
1
In the xy-plane, if line k has negative slope, is the y-intercept of line k positive?

(1) The x-intercept of line k is less than the y-intercept of line k.

(2) The slope of line k is less than -2.

It is a DS question, can you help and explain the answer?

let me try:
we have line k say: y=mx+c
we need to find if c>0

1) x-intercept, i.e. y=0
x=-c/m
y-intercept, i.e. x=0
y=c

hence -c/m<c
=> c*((1/m)+1)>0

i.e. for different value of "m", "c" can be both positive and negative

hence insufficient

2) cant infer anything about c
insufficient

1+2

if m<-2

c*(m+1)<0 (m<0 hence sign change)
as m<-2
hence m+1<-1
i.e. negative
i.e. c>0

Sufficient

hence C

hope it helps..!!!
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Re: DS co-ordinate geometry question  [#permalink]

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14 Mar 2012, 02:07
thanks , i also got the following explanation and dont understand the logic behind their deduction from 1 ,

Explanation

If a line has negative slope, the intercepts will have the same sign. So if we can find the sign of the x-intercept, we can answer the question.

Statement (1) is insufficient. It's possible that both intercepts are negative, for instance if the x-intercept is -4, the y-intercept could be -2. This is a relatively flat slope--as it turns out, it's true if the slope is greater than -1. It's also possible that both intercepts are positive. For instance, if the x-intercept is 3, the y-intercept could be 5. The negative slope here is steeper--in general, less than -1.

Statement (2) is also insufficient. Such a slope is relatively steep, but it could result in positive or negative intercepts--the slope of the line doesn't determine the location of the line.

Taken together, the statements are sufficient. In (1), we learned that if the slope is less than -1, both intercepts are positive. Since the slope is less than -2, both intercepts must be positive. Choice (C) is correct.

Can you help ?

thanks
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Re: DS co-ordinate geometry question  [#permalink]

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14 Mar 2012, 20:11
thanks , i also got the following explanation and dont understand the logic behind their deduction from 1 ,

Explanation

If a line has negative slope, the intercepts will have the same sign. So if we can find the sign of the x-intercept, we can answer the question.

Statement (1) is insufficient. It's possible that both intercepts are negative, for instance if the x-intercept is -4, the y-intercept could be -2. This is a relatively flat slope--as it turns out, it's true if the slope is greater than -1. It's also possible that both intercepts are positive. For instance, if the x-intercept is 3, the y-intercept could be 5. The negative slope here is steeper--in general, less than -1.

Statement (2) is also insufficient. Such a slope is relatively steep, but it could result in positive or negative intercepts--the slope of the line doesn't determine the location of the line.

Taken together, the statements are sufficient. In (1), we learned that if the slope is less than -1, both intercepts are positive. Since the slope is less than -2, both intercepts must be positive. Choice (C) is correct.

Can you help ?

thanks

proceed graphically and check the slope,
1) when the intercepts are in first quadrant, you will see the slope should be less than tan(135)
i.e. less than -1 to satisfy the condition y>x intercept. (at -1 you will see x=y intercept)
similarly, when in third quadrant slope should be greater than tan (135) i.e. -1

insufficient

2) insufficient

both 1 and 2 slope less than -2 i.e. less than -1 hence both intercept are positive.

hope this clarifies
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In the xy-plane, if line k has negative slope, is the  [#permalink]

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11 Aug 2014, 05:13
Bunuel,

I dont understand

-b/m<b --> multiply by negative m and flip the sign of the inequality: -b>bm --> b(m+1)<0... can you explain?

IF -b/m<b, then -b<bm....b(m+1)>0...Can you explain how b(1+m) < 0?
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Re: In the xy-plane, if line k has negative slope, is the  [#permalink]

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12 Aug 2014, 08:30
saikrishna123 wrote:
Bunuel,

I dont understand

-b/m<b --> multiply by negative m and flip the sign of the inequality: -b>bm --> b(m+1)<0... can you explain?

IF -b/m<b, then -b<bm....b(m+1)>0...Can you explain how b(1+m) < 0?

When you multiply by a negative value you must flip the sign of the inequality.

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Re: In the xy-plane, if line k has negative slope, is the  [#permalink]

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13 Aug 2014, 07:49
-b/m<b --> If we multiply by negative m, we have to flip the sign as well as multiply by -m on both sides. Isnt this correct?

If I multiply (left side equation) -b/m by -m => -b/m*-m => b
If I multiply (right side equation) b by -m => -b*m
If I flip the sign,

(Left side) b > -b*m (right side) => b(1+m) > 0....Where did I go wrong? Please clarify my concept.
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Re: In the xy-plane, if line k has negative slope, is the  [#permalink]

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01 Nov 2015, 03:02
hi bunuel plz explain, why the sing of m is not considered here( at x intersept) y = mx +b , x = -b/m, why its not x = b/m(taking -m, negative slope)
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Re: In the xy-plane, if line k has negative slope, is the  [#permalink]

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01 Nov 2015, 09:07
vipulgoel wrote:
hi bunuel plz explain, why the sing of m is not considered here( at x intersept) y = mx +b , x = -b/m, why its not x = b/m(taking -m, negative slope)

You do not substitute a variable, say x, by -x if you know that x is negative. This does not make sense.
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Re: In the xy-plane, if line k has negative slope, is the  [#permalink]

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04 Nov 2015, 08:19
Hi Bunuel,

I have the same doubt as vipulgoel, however i couldn't understand your follow-up explanation. Since we know m is negative, shouldnt we take the sign into consideration ? Could you please explain what do you mean by "you do not substitute a variable say, x by -x" ?

Thanks.
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Re: In the xy-plane, if line k has negative slope, is the  [#permalink]

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04 Nov 2015, 09:50
1
Swaroopdev wrote:
Hi Bunuel,

I have the same doubt as vipulgoel, however i couldn't understand your follow-up explanation. Since we know m is negative, shouldnt we take the sign into consideration ? Could you please explain what do you mean by "you do not substitute a variable say, x by -x" ?

Thanks.

Say it's given that x=a, and you know that x is negative do you substitute x by -x in this case? No.
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Re: In the xy-plane, if line k has negative slope, is the  [#permalink]

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06 Nov 2015, 22:08
hi, Let me try , y = mx+ c is a general form, irrespective of slope, first just write x intercept (without considering - ve slope), now as Bunuel did multiply with -m(negative slope on both sides, that's how -ve slope comes in picture)
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Re: In the xy-plane, if line k has negative slope, is the  [#permalink]

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06 Sep 2017, 20:47
Bluelagoon wrote:
In the xy-plane, if line k has negative slope, is the y-intercept of line k positive?

(1) The x-intercept of line k is less than the y-intercept of line k.

(2) The slope of line k is less than -2.

It is a DS question, can you help and explain the answer?

Before we start let's revise a rule, which says that if a line has -ve slope, then either both the intercept will be -ve or both will be +ve. They cannot have different sign.

or

the line can pass from (0,0) i.e origin we are not considering this case.

Let's say X intercept is A and Y intercept is B

1)

says A < B ...(I)
now as the slope is -ve. A and B both can be -ve or +ve.
Insufficient.

2)

slope is -2.
Formula of slope is \frac{Y-intercept}{X-Intercept}
so, \frac{B}{A} = -2 ...(II)
again, A and B both can be -ve or +ve.
Insufficient

Together

from II, \frac{B}{-2} = A
substitute the above value in I,
\frac{B}{-2} < B
Multiply the above fraction by -2,
B > -2B
3B > 0, hence B is greater than 0.
sufficient
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Re: In the xy-plane, if line k has negative slope, is the  [#permalink]

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06 Sep 2017, 20:48
Bluelagoon wrote:
In the xy-plane, if line k has negative slope, is the y-intercept of line k positive?

(1) The x-intercept of line k is less than the y-intercept of line k.

(2) The slope of line k is less than -2.

It is a DS question, can you help and explain the answer?

Before we start let's revise a rule, which says that if a line has -ve slope, then either both the intercept will be -ve or both will be +ve. They cannot have different sign.

or

the line can pass from (0,0) i.e origin we are not considering this case.

Let's say X intercept is A and Y intercept is B

1)

says A < B ...(I)
now as the slope is -ve. A and B both can be -ve or +ve.
Insufficient.

2)

slope is -2.
Formula of slope is \frac{Y-intercept}{X-Intercept}
so, \frac{B}{A} = -2 ...(II)
again, A and B both can be -ve or +ve.
Insufficient

Together

from II, \frac{B}{-2} = A
substitute the above value in I,
\frac{B}{-2} < B
Multiply the above fraction by -2,
B > -2B
3B > 0, hence B is greater than 0.
sufficient
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In the xy-plane, if line k has negative slope, is the  [#permalink]

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05 Nov 2017, 11:29
Hi Bunuel

Please refer to the attached graph.

Line 1:
has slope < -2 and x intercept = -1.5 which is less than y-intercept = -1 => y-intercept is -ve

Line 2:
has slope < -2 and x intercept = 1 which is less than y-intercept = 1.5 => y-intercept is +ve

So the answer must E ..right?

Or am i missing anything?
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Re: In the xy-plane, if line k has negative slope, is the  [#permalink]

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05 Nov 2017, 11:39
hellosanthosh2k2 wrote:
Hi Bunuel

Please refer to the attached graph.

Line 1:
has slope < -2 and x intercept = -1.5 which is less than y-intercept = -1 => y-intercept is -ve

Line 2:
has slope < -2 and x intercept = 1 which is less than y-intercept = 1.5 => y-intercept is +ve

So the answer must E ..right?

Or am i missing anything?

The correct answer is C, as explained here: https://gmatclub.com/forum/in-the-xy-pl ... l#p1058889

The slope of a line passing through (1, 0) and (0, 1.5) is -1.5, which is not less than -2, as per (2).
The slope of a line passing through (-1.5, 0) and (0, -1) is -0.67, which is not less than -2, as per (2).
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In the xy-plane, if line k has negative slope, is the  [#permalink]

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05 Nov 2017, 12:02
Bunuel wrote:
hellosanthosh2k2 wrote:
Hi Bunuel

Please refer to the attached graph.

Line 1:
has slope < -2 and x intercept = -1.5 which is less than y-intercept = -1 => y-intercept is -ve

Line 2:
has slope < -2 and x intercept = 1 which is less than y-intercept = 1.5 => y-intercept is +ve

So the answer must E ..right?

Or am i missing anything?

The correct answer is C, as explained here: https://gmatclub.com/forum/in-the-xy-pl ... l#p1058889

The slope of a line passing through (1, 0) and (0, 1.5) is -1.5, which is not less than -2, as per (2).
The slope of a line passing through (-1.5, 0) and (0, -1) is -0.67, which is not less than -2, as per (2).

Thanks Bunuel , i realized my mistake slope of Line 1 (-1/1.5) is not less than (-2), Line 1 is not possible.

I must have been half asleep while solving this problem
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Re: In the xy-plane, if line k has negative slope, is the  [#permalink]

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06 Nov 2017, 02:25
Bunuel wrote:
In the xy-plane, if line k has negative slope, is the y-intercept of line k positive?

Equation of a line in point intercept form is $$y=mx+b$$, where: $$m$$ is the slope of the line and $$b$$ is the y-intercept of the line (the value of $$y$$ for $$x=0$$). So, basically we are asked whether $$b>0$$.

(1) The x-intercept of line k is less than the y-intercept of line k --> x-intercept is value of $$x$$ for $$y=0$$, so it's $$-\frac{b}{m}$$. The statement says that: $$-\frac{b}{m}<b$$ --> multiply by negative $$m$$ and flip the sign of the inequality: $$-b>bm$$ --> $$b(m+1)<0$$. Now, in order $$b>0$$ to be true $$m+1$$ should be negative, so the question becomes: is $$m+1<0$$? --> is $$m<-1$$. We don't know that. Not sufficient.

(2) The slope of line k is less than -2. Insufficient on its own.

(1)+(2) From (1) the question became: "is $$m<-1$$?" and (2) says that $$m<-2$$. Sufficient.

Hi Bunuel,

I have doubt here.. can you please help me to understand it
if -b/m<b, then can we write m>-1?? by cancelling b on both sides.
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Re: In the xy-plane, if line k has negative slope, is the  [#permalink]

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06 Nov 2017, 02:27
rahul16singh28 wrote:
Bunuel wrote:
In the xy-plane, if line k has negative slope, is the y-intercept of line k positive?

Equation of a line in point intercept form is $$y=mx+b$$, where: $$m$$ is the slope of the line and $$b$$ is the y-intercept of the line (the value of $$y$$ for $$x=0$$). So, basically we are asked whether $$b>0$$.

(1) The x-intercept of line k is less than the y-intercept of line k --> x-intercept is value of $$x$$ for $$y=0$$, so it's $$-\frac{b}{m}$$. The statement says that: $$-\frac{b}{m}<b$$ --> multiply by negative $$m$$ and flip the sign of the inequality: $$-b>bm$$ --> $$b(m+1)<0$$. Now, in order $$b>0$$ to be true $$m+1$$ should be negative, so the question becomes: is $$m+1<0$$? --> is $$m<-1$$. We don't know that. Not sufficient.

(2) The slope of line k is less than -2. Insufficient on its own.

(1)+(2) From (1) the question became: "is $$m<-1$$?" and (2) says that $$m<-2$$. Sufficient.

Hi Bunuel,

I have doubt here.. can you please help me to understand it
if -b/m<b, then can we write m>-1?? by cancelling b on both sides.

No. You cannot reduce an inequality by a variable unless you know its sign. If the variable is positive you should keep the sign but if the variable is negative you should flip the sign of the inequality.
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# In the xy-plane, if line k has negative slope, is the

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