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

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Kudos [?]: 19 [0], given: 65

In the xy-plane, if line k has negative slope and passes [#permalink]  29 Feb 2012, 12:18
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90% (01:40) correct 10% (00:00) wrong based on 22 sessions
Whenever I see a similar question, my mind freezes and I go into fetal position. I am unable to picture them and even if sketch them, I am still confused as to how approach them.

I know how to get a slope.
I know the basic equation of a line.
I know how a prep bisector works.
I know the distance between two points.
I know how to get the mid between two points.

I am just unable to lump all these together and solve these questions quick enough.

Here are some examples of these questions:

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

Official answer for this one is :
[Reveal] Spoiler:
E

In the rectangular coordinate system, are the points
(r,s) and (u,v ) equidistant from the origin?
(1) r + s = 1
(2) u = 1 – r and v = 1 – s

OA:
[Reveal] Spoiler:
C

If line k in the xy-plane has equation y = mx + b, where
m and b are constants, what is the slope of k ?
(1) k is parallel to the line with equation
y = (1 – m)x + b + 1.
(2) k intersects the line with equation y = 2x + 3 at
the point (2,7).

OA :
[Reveal] Spoiler:
A

In the XY plane, region R consists of all the points (x,y) such that 2x+3y<=6. Is the point (r,s) in region R?
1. 3r+2s=6
2. r<=3 & s<=2

OA :
[Reveal] Spoiler:
E

These questions seem to pop up on every GMAT prep I took.

Any helps or tricks are appreciated. If I can't thank you in a post, I will make sure I kudos you.

Last edited by Bunuel on 29 Feb 2012, 12:59, edited 1 time in total.
Topic is locked. The links to the open discussions of these questions are given in the posts below.
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Math Expert
Joined: 02 Sep 2009
Posts: 27526
Followers: 4327

Kudos [?]: 42559 [5] , given: 6038

Re: These questions really scare me - coordinate Geom. [#permalink]  29 Feb 2012, 12:34
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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?

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.

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

1.png [ 9.73 KiB | Viewed 6038 times ]

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

2.png [ 10.17 KiB | Viewed 6037 times ]

More on this please check Coordinate Geometry chapter of Math Book: math-coordinate-geometry-87652.html

In case of any question please post it here: in-the-xy-plane-if-line-k-has-negative-slope-and-passes-110044.html

Hope it helps.
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Math Expert
Joined: 02 Sep 2009
Posts: 27526
Followers: 4327

Kudos [?]: 42559 [2] , given: 6038

Re: These questions really scare me - coordinate Geom. [#permalink]  29 Feb 2012, 12:35
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2. In the rectangular coordinate system, are the points (r,s) and (u,v) equidistant from the origin?

(1) r + s = 1

(2) u = 1 - r and v = 1 - s

Distance between the point A (x,y) and the origin can be found by the formula: $$D=\sqrt{x^2+y^2}$$.

Basically the question asks is $$\sqrt{r^2+s^2}=\sqrt{u^2+v^2}$$ OR is $$r^2+s^2=u^2+v^2$$?

(1) $$r+s=1$$, no info about $$u$$ and $$v$$;

(2) $$u=1-r$$ and $$v=1-s$$ --> substitute $$u$$ and $$v$$ and express RHS using $$r$$ and $$s$$ to see what we get: $$RHS=u^2+v^2=(1-r)^2+(1-s)^2=2-2(r+s)+ r^2+s^2$$. So we have that $$RHS=u^2+v^2=2-2(r+s)+ r^2+s^2$$ and thus the question becomes: is $$r^2+s^2=2-2(r+s)+ r^2+s^2$$? --> is $$r+s=1$$? We don't know that, so this statement is not sufficient.

(1)+(2) From (2) question became: is $$r+s=1$$? And (1) says that this is true. Thus taken together statements are sufficient to answer the question.

In case of any question please post it here: in-the-rectangular-coordinate-system-are-the-points-r-s-92823.html
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Kudos [?]: 42559 [1] , given: 6038

Re: These questions really scare me - coordinate Geom. [#permalink]  29 Feb 2012, 12:37
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If Line k in the xy-plane has equation y = mx + b, where m and b are constants, what is the slope of k?

$$y=mx+b$$ is called point-intercept form of equation of a line. Where: $$m$$ is the slope of the line; $$b$$ is the y-intercept of the line; $$x$$ is the independent variable of the function $$y$$.

So we are asked to find the value of $$m$$.

(1) k is parallel to the line with equation y = (1-m)x + b +1 --> parallel lines have the same slope --> slope of this line is $$1-m$$, so $$1-m=m$$ --> $$m=\frac{1}{2}$$. Sufficient.

(2) k intersects the line with equation y = 2x + 3 at the point (2, 7) --> so line k contains the point (2,7) --> $$7=2m+b$$ --> can not solve for $$m$$. Not sufficient.

In case of any question please post it here: if-line-k-in-the-xy-plane-has-equation-y-mx-b-where-m-100295.html
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Posts: 27526
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Kudos [?]: 42559 [2] , given: 6038

Re: These questions really scare me - coordinate Geom. [#permalink]  29 Feb 2012, 12:41
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In the xy-plane, region R consists of all the points (x, y) such that $$2x + 3y =< 6$$ . Is the point (r,s) in region R?

I'd say the best way for this question would be to try boundary values.

Q: is $$2r+3s\leq{6}$$?

(1) $$3r + 2s = 6$$ --> very easy to see that this statement is not sufficient:
If $$r=2$$ and $$s=0$$ then $$2r+3s=4<{6}$$, so the answer is YES;
If $$r=0$$ and $$s=3$$ then $$2r+3s=9>6$$, so the answer is NO.
Not sufficient.

(2) $$r\leq{3}$$ and $$s\leq{2}$$ --> also very easy to see that this statement is not sufficient:
If $$r=0$$ and $$s=0$$ then $$2r+3s=0<{6}$$, so the answer is YES;
If $$r=3$$ and $$s=2$$ then $$2r+3s=12>6$$, so the answer is NO.
Not sufficient.

(1)+(2) We already have an example for YES answer in (1) which valid for combined statements:
If $$r=2<3$$ and $$s=0<2$$ then $$2r+3s=4<{6}$$, so the answer is YES;
To get NO answer try max possible value of $$s$$, which is $$s=2$$, then from (1) $$r=\frac{2}{3}<3$$ --> $$2r+3s=\frac{4}{3}+6>6$$, so the answer is NO.
Not sufficient.

Number picking strategy for this question is explained here: in-the-xy-plane-region-r-consists-of-all-the-points-x-y-102233.html#p795613

In case of any question pleas post it here: in-the-xy-plane-region-r-consists-of-all-the-points-x-y-102233.html
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Re: These questions really scare me - coordinate Geom.   [#permalink] 29 Feb 2012, 12:41
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