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# In the xy-plane, region R consists of all the points (x, y)

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04 Aug 2012, 05:06
Bunuel wrote:
metallicafan 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 $$s=< 2$$

Though the solution provided by shrouded1 above is perfectly OK, it's doubtful that can be done in 2-3 minutes.

So 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.

Hope it's clear.

Hi Bunuel,

I did not understand the the way u have proved taking points 1 and 2 in this problem....can u plz explain.... what i have always done is take the conditions which is satisfying both point 1 and 2 and then try to get a PASS and a fail in it to prove that the answer is OPTION is C or E....
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Re: In the xy-plane, region R consists of all the points (x, y) [#permalink]

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05 Aug 2012, 07:26
shirisha091 wrote:
How did you get 2r + 3s ≤ 6 in the beginning of this problem?

Since we are told that region R consists of ALL the points (x, y) such that $$2x+3y\leq{6}$$, then in order point (r,s) to be in region R the same must hold true for that point too, hence it must be true that $$2r+3s\leq{6}$$.

P.S. Edited the diagram in shrouded1's post: in-the-xy-plane-region-r-consists-of-all-the-points-x-y-102233.html#p794049
Attachment:

graph.png [ 8.83 KiB | Viewed 42643 times ]

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Re: In the xy-plane, region R... [#permalink]

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02 Sep 2012, 13:41
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2x +3y <= 6
3y <= 6 - 2x
y <= 2 - 2x/3

Statement (1):
3r + 2s = 6
2s = 6 - 3r
s = 3 - 2r

Since this line is not parralel to the edge of the region (when 2x+3y=6), it will at some point be above, and at some point be below it. So we don't have enough information.

Statement (2):
Let's calculate point (3, 2): 2*3+3*2=12>6, so not inside region R
But since we can go as low as we want with this point, it will eventually be in region R.

Combining (1) + (2):
Let's see, where region R intersects with the line of statement (1).
y=2-2x/3
s=3-2r --> y=3-2x
2-2x/3=3-2x
4x/3 = 1
x = 3/4
y=3-2*(3/4)=1.5

Intersection at (0.75, 1.5).
So our line from statement (1) will intersect with region R within the region of statement (2). So there will be points on that line that also fullfill statement (2), but are not in region R. And there will be points that are in region R. --> not sufficient
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Re: In the xy-plane, region R consists of all the points (x, y) [#permalink]

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17 Dec 2012, 12:10
look, the question is about the point (r,5) not the point (r,s).
if the question was about the point (r,s), I think the (2) is sufficient, the correct answer is B
However, if the question is about the point (r,5), (1) and (2) together are sufficient but none of them alone is sufficient. The corrent answer is C
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Re: In the xy-plane, region R consists of all the points (x, y) [#permalink]

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27 Feb 2013, 00:44
I want to follow this posting.

this question test out ability to react to numbers, to plug in number. This process is done by Bruno already. og explanation is the same. this process of plugging takes me 2 minutes.

drawing or solving the equation is too time consuming and is not the correct the method.
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Re: In the xy-plane, region R consists of all the points (x, y) [#permalink]

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05 Jul 2013, 02:44
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27 Sep 2013, 11:26
Bunuel wrote:
shreya717 wrote:

This is how I solved the question and im arriving at the wrong answer. Please could someone correct me.

Given:- equation of the line is 2x+3y<6
Thus, y<2/3x+2
the y and x intercepts of the given line are y<2 and x <3 respectively.

Statement 2 gives us that r (x) < 2 and s (y) <3. Therefore i concluded statment 2 is suff to answer the question. Am i going wrong in the calculation?

Thanks.

First of all $$2x+3y\leq{6}$$ can be rewritten as $$y\leq{2-\frac{2}{3}*x}$$, not as $$y\leq{2+\frac{2}{3}*x}$$. Next, it's not equation of a line, it gives the region which is below line $$y={2-\frac{2}{3}*x}$$, so all your farther conclusions are wrong.

Thanks for the explanation. I used the same incorrect approach to get to B and I missed that the equation defines a region and not a line.

If the equation was a line, then would "B" be the correct answer?
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30 Sep 2013, 23:37
Bunuel wrote:
metallicafan 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 $$s=< 2$$

Though the solution provided by shrouded1 above is perfectly OK, it's doubtful that can be done in 2-3 minutes.

So 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.

Hope it's clear.

Hi Bunuel,
Just a quick clarification-
Combining two statements, to get NO as answer, we can also try the first values from Stat.1 - If r=0 and s=3 then 2r+3s=9>6, so the answer is NO..

Please correct me if I'm wrong.
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01 Oct 2013, 00:13
bagdbmba wrote:
Bunuel wrote:
metallicafan 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 $$s=< 2$$

Though the solution provided by shrouded1 above is perfectly OK, it's doubtful that can be done in 2-3 minutes.

So 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.

Hope it's clear.

Hi Bunuel,
Just a quick clarification-
Combining two statements, to get NO as answer, we can also try the first values from Stat.1 - If r=0 and s=3 then 2r+3s=9>6, so the answer is NO..

Please correct me if I'm wrong.

When combining the values must satisfy both statements: r=0 and s=3 does not satisfy the second one.
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Re: In the xy-plane, region R consists of all the points (x, y) [#permalink]

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01 Oct 2013, 00:19
Oops! My bad...I missed that($$s\leq{2}$$).

Thanks for correcting it.
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Re: In the xy-plane, region R consists of all the points (x, y) [#permalink]

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06 Jul 2014, 16:45
I can't see why B is insufficient. Since r<= 3 means that x<=3 and s<=2 means y<=2 this covers the region under the line. which is f(x) <0 .
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Re: In the xy-plane, region R consists of all the points (x, y) [#permalink]

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06 Jul 2014, 16:54
amz14 wrote:
I can't see why B is insufficient. Since r<= 3 means that x<=3 and s<=2 means y<=2 this covers the region under the line. which is f(x) <0 .

Have you checked Graphic Approach HERE and Number Plugging approach HERE and HERE?
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Re: In the xy-plane, region R consists of all the points (x, y) [#permalink]

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08 Jul 2014, 13:58
gmat1220 wrote:
Bunuel,
Thanks for the solution ! Please correct me.

If (r,s) is inside the region R on the XY plane, as per equation 2r + 3s <=6 is true.
That also means that r is bounded (when s=0) i.e. r<=3
s is also bounded (when r=0) is s<=2

Hence r<=3 and s<=2 should be sufficient. Am I missing something? Please clarify.

I made the same mistake as you and marked B.
But the region enclosed by the second statement is a rectangular region where r < = 3 and s<= 2. Only some part of this rectangle lies below the original line 2x + 3y <= 6.
So this option is not sufficient to say that all points (r,s) lie below the original line.
Others have shown this via algebra.
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Re: In the xy-plane, region R consists of all the points (x, y) [#permalink]

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29 Jun 2015, 05:08
Hi Bunuel,

Any link to similar problems like this ?

Thanks.
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In the xy-plane, region R consists of all the points (x, y) [#permalink]

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05 Jul 2015, 18:35
shrouded1 wrote:
metallicafan 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 $$s=< 2$$

Ok, the easiest way to solve this is to visualize the graph with the lines plotted on it.
2x+3y<=6, is the region below the line with X-intercept 3 and Y-intercept 2. We know it is that region, because (0,0) lies below the line and it satisfies the inequality. And all points on one side of the line satisfy the same sign of inequality. (BLUE LINE)

(1) : The line 3r+2s=6 (PURPLE LINE) represents the second line shown in the figure. It can be above or below the other line. So insufficient.
(2) : r<=3 & s<=2. Again easy to see from the graph even with that constraint, the point (r,s) may lie above or below the line in question

(1+2) : the two conditions together, only take a section of the 3r+2s=6 line as a solution, but again even with r<=3, s<=2, its not sufficient to keep solutions below the 2x + 3y =< 6 line

In questions like these, once you are comfortable with graphs, you can solve in less than 30 seconds fairly easily.
Let me know if the method isn't clear

I did it a little differently.

I solved for y to get $$y <=\frac{-2}{3}x + 2$$

(1) I solved for s, so $$s = \frac{-3}{2}r + 3$$

Both x-y and r-s lines are downward sloping and have different y-intercepts. We know that they deviate in the -x region if you draw or visualize a crude graph.

Plug in an arbitrary negative x-value for r. Say, -10. Then solve for s, which is 18.

Plug the x-value of r into the x-y equation. $$y <=\frac{-2}{3}*-10$$+ 2; y <= $$\frac{26}{3}$$ or 8$$\frac{2}{3}$$

If (r,s) is within the x-y region, then the y-value (s) must be <= 8$$\frac{2}{3}$$. 18 <= 8$$\frac{2}{3}$$ does not hold.

(2) Plug given r and s values into given 2r+3s≤6 equation. 12 ≤ 6 does not hold.

Insufficient.

Both (1) and (2) together neither confirms nor eliminates the large negative x-value concern that we have, and it also neither confirms nor eliminates the positive x-values. So both together are insufficient.
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In the xy-plane, region R consists of all the points (x, y) [#permalink]

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29 Nov 2015, 14:08
Given Info: A region R consists of all the points (x,y) such that $$2x + 3y\leq{6}$$.

Interpreting the Question Statement: We have to check whether the point(r,s) is in Region R, according to the 2 statements given to us.

Solution: Here we will use the brute force of plugging in some numbers and checking for the solution to the problem

Statement 1:
Putting r=0 in the first statement, we get s=3

Checking for Region R in equation $$2x + 3y\leq{6}$$

We get, $$2(0)+3(3)=9>6$$.
So the coordinates (r,s)=(0,3) does not lie in region R

Putting s=0 in the first statement, we get r=2

Checking for Region R in equation $$2x + 3y\leq{6}$$

We get,$$2(2)+3(0)=4<6$$.
So the coordinates (r,s)=(2,0) lies in region R

For one value, (r,s) lies in region R and for other it does not lie in region R. So we cannot determine with a surety from Statement 1 whether point (r,s) will lie in Region R or not.

So Statement 1 alone is not sufficient to answer the question.

Statement 2:
Putting r=3, s=2 (It satisfies statement 2)

Checking for region R in equation $$2x + 3y\leq{6}$$

We get,$$2(3)+3(2)=12>6$$
So the coordinates (r,s)=(3,2) does not lie in region R

Putting r=1,s=1 (It satisfies staement 2)

Checking for region R in equation $$2x + 3y\leq{6}$$

We get,$$2(1)+3(1)=5<6$$
So the coordinates (r,s)=(1,1) lies in region R

For one value, (r,s) lies in region R and for other it does not lie in region R. So we cannot determine with a surety from Statement 2 whether point (r,s) will lie in Region R or not.

So Statement 2 alone is not sufficient to answer the question.

Combining statements 1 and 2

Putting r=2,s=0 (It satisfies both statements 1 and 2)
Checking for region R in equation $$2x + 3y\leq{6}$$

We get,$$2(2)+3(0)=4<6$$
So the coordinates (r,s)=(2,0) lies in region R

Putting s=2 in equation 1, we get r=2/3

These values satisfy statement 2 as well.
Checking for region R in equation $$2x + 3y\leq{6}$$

We get,$$2(2/3)+3(2)=22/3>6$$
So the coordinates (r,s)=(2/3,2) does not lie in region R

For one value, (r,s) lies in region R and for other it does not lie in region R. So we cannot determine with a surety from both Statement 1 and Statement 2 whether point (r,s) will lie in Region R or not.

So Statement 1 and Statement 2 both together are also not sufficient to answer the question.

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Re: In the xy-plane, region R consists of all the points (x, y) [#permalink]

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30 Nov 2015, 10:43
Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

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

There are 2 variables (x,y) and one equations (2x+3y<=6) and 2 more equations are given by the 2 conditions
This makes (D) a likely answer.
For condition 1, the answer to the question becomes 'yes' for (r,s)=(2,0), but 'no' for (0,3). This is therefore insufficient.
For condition 2, the answer to the question becomes 'yes' for (r,s)=(2,0), but 'no' for (2/3,2). This is also insufficient and the answer therefore becomes (E).

For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously there may be cases where the answer is A, B, C or E.
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Re: In the xy-plane, region R consists of all the points (x, y) [#permalink]

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11 Dec 2015, 01:44
I like the graphical approach. I think also whats helpful to realize here is - 3r+2s=6 will be the mirror image of 2x+3y=6 about x=y
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Re: In the xy-plane, region R consists of all the points (x, y) [#permalink]

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13 Dec 2015, 04:20
Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

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

-> When you modify the original condition and question, it becomes 2r+3s<=6? and there are 2 variables(r, s), which should match with the number equations. So, you need 2 equations. For 1) 1 equation, for 2) 1 equation, which is most likely to make C the answer. In 1) & 2), (r,s)=(2,0) yes, (r,s)=(2/3,2), which is no and therefore not sufficient. Thus, the answer is E.

-> For cases where we need 3 more equations, such as original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 80% chance that E is the answer (especially about 90% of 2 by 2 questions where there are more than 3 variables), while C has 15% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since E is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, C or D.
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Re: In the xy-plane, region R consists of all the points (x, y) [#permalink]

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22 Jan 2016, 09:57
metallicafan wrote:
In the xy-plane, region R consists of all the points (x,y) such that $$2x + 3y\leq{6}$$. Is the point (r,s) in region R?

(1) $$3r + 2s = 6$$
(2) $$r\leq{3}$$ and $$s\leq{2}$$

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Re: In the xy-plane, region R consists of all the points (x, y)   [#permalink] 22 Jan 2016, 09:57

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