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

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In the xy-plane, if line k has negative slope and passes [#permalink]  02 Jul 2012, 01:45
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Difficulty:

85% (hard)

Question Stats:

49% (02:25) correct 51% (01:26) wrong based on 413 sessions
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

Diagnostic Test
Question: 39
Page: 26
Difficulty: 650
[Reveal] Spoiler: OA

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Re: In the xy-plane, if line k has negative slope and passes [#permalink]  02 Jul 2012, 01:47
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SOLUTION

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?

This question can be done with graphic approach (just by drawing the lines) or with algebraic approach.

Algebraic approach:

Equation of a line in point intercept form is $$y=mx+b$$, where: $$m$$ is the slope of the line, $$b$$ is the y-intercept of the line (the value of $$y$$ for $$x=0$$), and $$x$$ is the independent variable of the function $$y$$.

We are told that slope of line $$k$$ is negative ($$m<0$$) and it passes through the point (-5,r): $$y=mx+b$$ --> $$r=-5m+b$$.

Question: is x-intercept of line $$k$$ positive? x-intercep is the value of $$x$$ for $$y=0$$ --> $$0=mx+b$$ --> is $$x=-\frac{b}{m}>0$$? As we know that $$m<0$$, then the question basically becomes: is $$b>0$$?.

(1) The slope of line $$k$$ is -5 --> $$m=-5<0$$. We've already known that slope was negative and there is no info about $$b$$, hence this statement is insufficient.

(2) $$r>0$$ --> $$r=-5m+b>0$$ --> $$b>5m=some \ negative \ number$$, as $$m<0$$ we have that $$b$$ is more than some negative number ($$5m$$), hence insufficient, to say whether $$b>0$$.

(1)+(2) From (1) $$m=-5$$ and from (2) $$r=-5m+b>0$$ --> $$r=-5m+b=25+b>0$$ --> $$b>-25$$. Not sufficient to say whether $$b>0$$.

Graphic approach:

If the slope of a line is negative, the line WILL intersect quadrants II and IV. X and Y intersects of the line with negative slope have the same sign. Therefore if X and Y intersects are positive, the line intersects quadrant I; if negative, quadrant III.

When we take both statement together all we know is that slope is negative and that it crosses some point in II quadrant (-5, r>0) (this info is redundant as we know that if the slope of the line is negative, the line WILL intersect quadrants II). Basically we just know that the slope is negative - that's all. We can not say whether x-intercept is positive or negative from this info.

Below are two graphs with positive and negative x-intercepts. Statements that the slope=-5 and that the line crosses (-5, r>0) are satisfied.

$$y=-5x+5$$:

$$y=-5x-20$$:

For more on Coordinate Geometry check: math-coordinate-geometry-87652.html

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Re: In the xy-plane, if line k has negative slope and passes [#permalink]  02 Jul 2012, 02:53
Hi,

Difficulty level: 650

With reference to below diagram:
Attachment:

r.jpg [ 5.36 KiB | Viewed 7458 times ]

Even after using (1) & (2), we can't find the x-intercept.

Regards,
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Kudos [?]: 50033 [0], given: 7519

Re: In the xy-plane, if line k has negative slope and passes [#permalink]  06 Jul 2012, 02:30
Expert's post
SOLUTION

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?

This question can be done with graphic approach (just by drawing the lines) or with algebraic approach.

Algebraic approach:

Equation of a line in point intercept form is $$y=mx+b$$, where: $$m$$ is the slope of the line, $$b$$ is the y-intercept of the line (the value of $$y$$ for $$x=0$$), and $$x$$ is the independent variable of the function $$y$$.

We are told that slope of line $$k$$ is negative ($$m<0$$) and it passes through the point (-5,r): $$y=mx+b$$ --> $$r=-5m+b$$.

Question: is x-intercept of line $$k$$ positive? x-intercep is the value of $$x$$ for $$y=0$$ --> $$0=mx+b$$ --> is $$x=-\frac{b}{m}>0$$? As we know that $$m<0$$, then the question basically becomes: is $$b>0$$?.

(1) The slope of line $$k$$ is -5 --> $$m=-5<0$$. We've already known that slope was negative and there is no info about $$b$$, hence this statement is insufficient.

(2) $$r>0$$ --> $$r=-5m+b>0$$ --> $$b>5m=some \ negative \ number$$, as $$m<0$$ we have that $$b$$ is more than some negative number ($$5m$$), hence insufficient, to say whether $$b>0$$.

(1)+(2) From (1) $$m=-5$$ and from (2) $$r=-5m+b>0$$ --> $$r=-5m+b=25+b>0$$ --> $$b>-25$$. Not sufficient to say whether $$b>0$$.

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Re: In the xy-plane, if line k has negative slope and passes [#permalink]  06 Jul 2012, 02:32
Expert's post
SOLUTION

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

Graphic approach:

If the slope of a line is negative, the line WILL intersect quadrants II and IV. X and Y intersects of the line with negative slope have the same sign. Therefore if X and Y intersects are positive, the line intersects quadrant I; if negative, quadrant III.

When we take both statement together all we know is that slope is negative and that it crosses some point in II quadrant (-5, r>0) (this info is redundant as we know that if the slope of the line is negative, the line WILL intersect quadrants II). Basically we just know that the slope is negative - that's all. We can not say whether x-intercept is positive or negative from this info.

Below are two graphs with positive and negative x-intercepts. Statements that the slope=-5 and that the line crosses (-5, r>0) are satisfied.

$$y=-5x+5$$:
Attachment:

1.png [ 9.73 KiB | Viewed 8186 times ]

$$y=-5x-20$$:
Attachment:

2.png [ 10.17 KiB | Viewed 8175 times ]

For more on Coordinate Geometry check: math-coordinate-geometry-87652.html

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Re: In the xy-plane, if line k has negative slope and passes [#permalink]  12 Jul 2012, 22:33
what addition information should we have to make it sufficient.. ???
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Re: In the xy-plane, if line k has negative slope and passes [#permalink]  14 Aug 2013, 08:18
sanjoo wrote:
what addition information should we have to make it sufficient.. ???

By regarding both statements:
For any r greater than 25 the x intercept of k is positive
For any r smaller than 25 it is negative.

This is the information which is missing for answer C to be correct.

Why 25?
--> Because for a slope of -5, you have to go down 25 steps if you want to go 5 steps to the right.
So if r was 25, the x intercept would be 0....
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Re: In the xy-plane, if line k has negative slope and passes [#permalink]  14 Aug 2013, 10:01
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i solved like this-

line passes thru (-5,r). Eqn r= -5m + c. since slope is -ve, if c is +ve , x intercept = +ve. otherwise -ve.

statement 1. m=-5. r= 25 +c . c= r-25. c could be +ve or -ve depending on r.

statement 2. r> 0. c = r+5m. c could be +ve or -ve depending on value of m.

together- c= r-25; r>0. c could still be +ve or - ve . Hence x intercept could be +ve or -ve.

Hence ans = E

Is it ok?
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Re: In the xy-plane, if line k has negative slope and passes [#permalink]  05 Sep 2014, 14:25
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Re: In the xy-plane, if line k has negative slope and passes [#permalink]  28 Apr 2015, 05:59
Line (d): y = kx + b
Passed (-5;r) <=> r = -5k + b
Question when y=0, x>0? <=> x= (r-b)/k >0?
(1) k= -5
x= (r-b)/-5: insufficiant.
(2) r>0: insufficient.
(1)+(2): r=25+b>0 <=>b>-25 <=>b= -25+|m|
y=0; x= (r+|m|-25)/-5?: insufficiant.
Re: In the xy-plane, if line k has negative slope and passes   [#permalink] 28 Apr 2015, 05:59
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