Graphical Solution

Expert replies

knight247: Legendary Member; Posts: 504; Joined: Tue Apr 19, 2011 1:40 pmFollowed by: 11 members; Thanked: 114 times

by knight247 » Tue Jan 20, 2015 5:31 am

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 and s=< 2

I was able to solve this simply by trial and error. The answer is E

However, when I solve it graphically, I encounter a problem.

I'm able to prove statement (1) is insufficient because the graphs of 3r + 2s = 6 and 2x + 3y =< 6 don't overlap. (SEE ATTACHED GRAPH)

However, in statement (2) r=< 3 and s=< 2, don't the values of r and s coincide EXACTLY with the graph of 2x + 3y =< 6? I mean, if r=3 and s=2, don't r and s fall exactly ON the graph of 2x + 3y =< 6? Also if we select values of r and s that are less than 3 and 2 respectively, both r and s will lie on the LEFT of the graph of 2x + 3y =< 6.

So how is statement (2) insufficient?

Detailed explanations would be appreciated. Thanks in advance.

Last edited by knight247 on Tue Jan 20, 2015 6:22 am, edited 1 time in total.

Like this post

Source: Beat The GMAT — Problem Solving | Tue Jan 20, 2015 5:31 am

MartyMurray: Legendary Member; Posts: 2135; Joined: Mon Feb 03, 2014 9:26 amFollowed by: 140 members; Location: https://martymurraycoaching.com/; Thanked: 955 times; GMAT Score:800

by MartyMurray » Tue Jan 20, 2015 6:16 am

knight247 wrote: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 and s=< 2

I was able to solve this simply by trial and error. The answer is E

However, when I solve it graphically, I encounter a problem.

I'm able to prove statement (1) is insufficient because the graphs of 3r + 2s = 6 and 2x + 3y =< 6 don't overlap. (SEE ATTACHED GRAPH)

However, in statement (2) r=< 3 and s=< 2, don't the values of r and s coincide EXACTLY with the graph of 2x + 3y =< 6. I mean, if r=3 and s=2, don't r and s fall exactly ON the graph of 2x + 3y =< 6? Also if we select values of r and s that are less than 3 and 2 respectively, both r and s will lie on the LEFT of the graph of 2x + 3y =< 6.

So how is statement (2) insufficient?

Detailed explanations would be appreciated. Thanks in advance.

Looks as if you are doing it more like 2x <= 6 and 3y <= 6.

When you combine them, for the point (r,s), if x = 3 and y = 2, 2x + 3y = 12.That does not match at all.

Also, (3,2) does not lie within the region defined by 2x + 3y <= 6. The region is to the left of the line, and (3,2) is to the the right of the line. You can confirm this on your graph.

So with r <= 3 and s <= 2 you can generate points both to the left and to the right of the line.

Thus Statement 2 is insufficient.

Marty Murray
Perfect Scoring Tutor With Over a Decade of Experience
MartyMurrayCoaching.com
Contact me at [email protected] for a free consultation.

Like this post

GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Tue Jan 20, 2015 7:37 am

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. râ‰¤3 & sâ‰¤2

Region R is composed of all the points on or below y=(-2/3)x + 2.

Statement 1: s = (-3/2)r + 3.

The figure above shows that some points on s=(-3/2)r + 3 lie BELOW y=(-2/3)x + 2, while others lie ABOVE y=(-2/3)x + 2.
INSUFFICIENT.

Statement 2: râ‰¤3 and sâ‰¤2.

Inside the green box are points such that râ‰¤3 and sâ‰¤2.
Some of the points inside the green box lie BELOW y=(-2/3)x + 2, while others lie ABOVE y=(-2/3)x + 2.
INSUFFICIENT.

Statements 1 and 2 combined:

Inside the green box are points on s=(-3/2)r + 3 such that râ‰¤3 and sâ‰¤2.
Some of these points lie BELOW y=(-2/3)x + 2, while others lie ABOVE y=(-2/3)x + 2.
INSUFFICIENT.

The correct answer is E.

Private tutor exclusively for the GMAT and GRE, with over 20 years of experience.
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.

As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.

For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3

Like this post

GMAT/MBA Expert

Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Tue Jan 20, 2015 8:05 am

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 and s < 2

Target question: Is the point (r, s) in region R?

Given: Region R consists of all the points (x,y) such that 2x + 3y <6
So, what does Region R look like?
To find out, let's first graph the EQUATION, 2x + 3y = 6

Since Region R is described as an INEQUALITY, we can choose any point on the coordinate plane to test whether or not it is in Region R. An easy point to test is (0,0).
So, does x=0 and y=0 satisfy the inequality 2x + 3y <6? YES
2(0) + (3)(0) is less than or equal to 6.
So, the point (0,0) is in Region R. More importantly, EVERY POINT on the same side of the line will also be in Region R.

Statement 1: 3r + 2s = 6
The target question refers to the point (r, s)
In other words, the x-coordinate is r and the y-coordinate is s.
So, all of the points (r, s) that satisfy the above equation can be found on the line 3x + 2y = 6
In other words, statement 1 tells us that the point (r,s) lies somewhere on the red line below.

As you can see, some points are in Region R, and some points are not in Region R
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: r < 3 and s < 2
There are many points that satisfy this condition.
In fact, the point (r,s) can be ANYWHERE inside the red box shown below.

As you can see, some points are in Region R, and some points are not in Region R
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
When we combine the statements, we are saying that the point (r,s) is on the red line (2x + 3y = 6) AND inside the red box.

As you can see by the two blue points below, it's possible to have a point in Region R, and it's possible to have a point not in Region R

Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Answer = E

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

Like this post

Post new topic Post Reply

• Page 1 of 1

GMAT Course Reviews

560 reviews

1168 reviews

431 reviews

359 reviews

147 reviews

5/5

5 Star (560 Reviews)

"Target Test Prep is the closest to the official version of the GMAT exam, about 99% accuracy in terms of the quality and quantity of information. The course has excellently created singular sets of focused lessons and tests for every possible topic that one could come across in the official GMAT exam."

"The TTP course maximizes the efficiency of the time you spend studying. It will take time and effort but I could almost guarantee that if you complete the course exactly as it is laid out you will get an amazing score. They also have a very responsive team willing to help with any questions you might have."

"TTP has two things that I think no other test prep company offers: A teaching approach that reinforces understanding and an attitude that will give you the mental preparedness needed to succeed on the test. TTP gives you a deep understanding of the concept you need to know while teaching you how to think."

Write a Review