GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 19 Mar 2019, 03:18 ### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here. ### Request Expert Reply # In the xy-plane, if line k has negative slope and passes post reply Question banks Downloads My Bookmarks Reviews Important topics
Author Message
Manager  Joined: 08 Jun 2011
Posts: 81
In the xy-plane, if line k has negative slope and passes  [#permalink]

### Show Tags

9 00:00

Difficulty:

(N/A)

Question Stats: 83% (01:04) correct 17% (01:03) wrong based on 76 sessions

### HideShow timer Statistics

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 :

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:

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 :

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 :

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.

--== Message from the GMAT Club Team ==--

THERE IS LIKELY A BETTER DISCUSSION OF THIS EXACT QUESTION.
This discussion does not meet community quality standards. It has been retired.

If you would like to discuss this question please re-post it in the respective forum. Thank you!

To review the GMAT Club's Forums Posting Guidelines, please follow these links: Quantitative | Verbal Please note - we may remove posts that do not follow our posting guidelines. Thank you.

Originally posted by Lstadt on 29 Feb 2012, 13:18.
Last edited by Bunuel on 29 Feb 2012, 13:59, edited 1 time in total.
Topic is locked. The links to the open discussions of these questions are given in the posts below.
Math Expert V
Joined: 02 Sep 2009
Posts: 53698
Re: These questions really scare me - coordinate Geom.  [#permalink]

### Show Tags

7
7
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 16224 times ]

$$y=-5x-20$$:
Attachment: 2.png [ 10.17 KiB | Viewed 16224 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.
_________________
Math Expert V
Joined: 02 Sep 2009
Posts: 53698
Re: These questions really scare me - coordinate Geom.  [#permalink]

### Show Tags

2
1
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
_________________
Math Expert V
Joined: 02 Sep 2009
Posts: 53698
Re: These questions really scare me - coordinate Geom.  [#permalink]

### Show Tags

2
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
_________________
Math Expert V
Joined: 02 Sep 2009
Posts: 53698
Re: These questions really scare me - coordinate Geom.  [#permalink]

### Show Tags

2
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
_________________
Non-Human User Joined: 09 Sep 2013
Posts: 10132
Re: In the xy-plane, if line k has negative slope and passes  [#permalink]

### Show Tags

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________ Re: In the xy-plane, if line k has negative slope and passes   [#permalink] 23 Dec 2018, 09:16
Display posts from previous: Sort by

# In the xy-plane, if line k has negative slope and passes post reply Question banks Downloads My Bookmarks Reviews Important topics Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.  