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In the xy plane, the area of the region bounded by the graph of x +
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In the xy plane, the area of the region bounded by the graph of x + y + x  y = 4 is A. 8 B. 12 C. 16 D. 20 E. 24
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Originally posted by papillon86 on 08 Nov 2009, 14:11.
Last edited by Bunuel on 26 Oct 2017, 06:46, edited 3 times in total.
Edited the question and added the OA




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Re: In the xy plane, the area of the region bounded by the graph of x +
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08 Nov 2009, 14:34
papillon86 wrote: In xy plane, the area of the region bounded by the graph of x+y + xy = 4 is
a) 8 b) 12 c) 16 d) 20
Need help in solving equations involving Mod...... help? OK, there can be 4 cases: x+y + xy = 4 A. x+y+xy = 4 > x=2 B. x+yx+y = 4 > y=2 C. xy +xy= 4 > y=2 D. xyx+y=4 > x=2 The area bounded by 4 graphs x=2, x=2, y=2, y=2 will be square with the side of 4 so the area will be 4*4=16. Attachment:
MSP17971c13h40gd024h6g10000466ge1e9df941i96.gif [ 1.86 KiB  Viewed 25094 times ]
Answer: C
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Re: In the xy plane, the area of the region bounded by the graph of x +
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08 Nov 2009, 17:41
srini123 wrote: Thanks Bunuel, I used similar method for a similar question and I got wrong answer the question was
what is the area bounded by graph\(x/2 + y/2 = 5\)?
I got hunderd since x=10 x=10 y=10 y=10
isnt the area 400 ? the answer given was 200, please explain I think this one is different. \(\frac{x}{2} + \frac{y}{2} = 5\) After solving you'll get equation of four lines: \(y=10x\) \(y=10+x\) \(y=10x\) \(y=x10\) These four lines will also make a square, BUT in this case the diagonal will be 20 so the \(Area=\frac{20*20}{2}=200\). Or the \(Side= \sqrt{200}\), area=200. If you draw these four lines you'll see that the figure (square) which is bounded by them is turned by 90 degrees and has a center at the origin. So the side will not be 20. Also you made a mistake in solving equation. The red part is not correct. You should have the equations written above. In our original question when we were solving the equation x+y + xy = 4 each time x or y were cancelling out so we get equations of a type x=some value twice and y=some value twice. And these equations give the lines which are parallel to the Y or X axis respectively so the figure bounded by them is a "horizontal" square (in your question it's "diagonal" square). Hope it's clear.
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Re: In the xy plane, the area of the region bounded by the graph of x +
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13 Dec 2012, 04:06
eaakbari wrote: Quote: OK there can be 4 cases:
x+y + xy = 4
A. x+y+xy = 4 > x=2 B. x+yx+y = 4 > y=2 C. xy +xy= 4 > y=2 D. xyx+y=4 > x=2
Any absolute values such as x = 5 could mean that x = 5 or x = 5. Derive both () and (+) possibilities. For the problem: x+y + xy = 4 We could derive two possibilities for x+y could be (x+y) and (x+y) We could derive two possibilities for xy could be (xy) and (xy) This is the reason why we have 4 derived equations. (x+y) + (xy) = 4 (x+y)  (xy) = 4 (x+y) + (xy) = 4 (x+y)  (xy) = 4 Just simplify those... If you want more practice on this question type: http://burnoutorbreathe.blogspot.com/2012/12/absolutevaluessolvingforareaof.html
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Re: In the xy plane, the area of the region bounded by the graph of x +
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21 May 2011, 05:46
xy = xy if xy > 0 xy = (xy) if xy < 0 x+y > 0 => x > y then x !> y x+y + x  y = 4 x = 2 x  y + x  y = 4 (if x < y, then x !< y) y = 2 x + y x + y = 4 => y = 2 xy + x  y = 4 => y = 2 Answer  C
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Re: In the xy plane, the area of the region bounded by the graph of x +
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27 Feb 2012, 23:42
devinawilliam83 wrote: Hi, Can this be solved by graphing. If yes .. how do we graph the equation with 2 mod parts VeritasPrepKarishma wrote: VinuPriyaN wrote: Given xy + x+y = 4
I don't understand why can't xy and x+y be 1 and 3 instead of 2 and 2! (which again equals 4)
Can any one please explain this to me?
Thanks & Regards, Vinu Look at the solution given by Bunuel above. When you solve it, you get four equations. One of them is x = 2 which means that x = 2 and y can take any value. If y = 1, xy = 1 and x+y = 3. For different values of y, xy and x+y will get different values. We are not discounting any of them. Yes, it can be done by graphing. x+y + xy = 4 can expand in four different wasy: A. x+y+xy = 4 > x=2 B. x+yx+y = 4 > y=2 C. xy +xy= 4 > y=2 D. xyx+y=4 > x=2 So you can draw all these four lines x=2, x=2, y=2, y=2 to get a square with the side of 4: Attachment:
Square.gif [ 1.86 KiB  Viewed 29933 times ]
See more examples here: m065absolutevalue108191.htmlgraphsmodulushelp86549.htmlm06q572817.htmlifequationenclosesacertainregion110053.htmlHope it helps.
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Re: In the xy plane, the area of the region bounded by the graph of x +
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15 Apr 2014, 05:01
dheeraj24 wrote: Yeah karishma,
I totally agree with your explanation, but the point is, why couldn't we draw the slant lines for the points (2,0), (2,0), (0,2) and (0,2) instead of horizontal lines and consider the length of diagonal rather than length of side for the original question (x+y + xy = 4).
Thanks in advance.
Because you are asked the area of the region bounded by x+y + xy = 4. This equation gives you ONLY horizontal/vertical lines passing through points (2,0), (2,0), (0,2) and (0,2) such as x = 2, y = 2, x = 2, y = 2. Note that x = 2 is the equation of a line (it is not a coordinate) which passes through point (2, 0) and is parallel to the y axis. Similarly, y = 2 is the equation of a line which is parallel to x axis and passes through the point (0, 2) and so on. I think you are taking x = 2 as a coordinate but that is not the case. A coordinate has a value for y too. x =2 is the equation of a line. It implies that x coordinate is always 2 and y can be anything. So all points lying on a line passing through x = 2 and parallel to y axis satisfy this criteria.
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Re: In the xy plane, the area of the region bounded by the graph of x +
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15 Apr 2014, 09:16
PathFinder007 wrote: Hi Karishma, I am still not clear in question x/2 + y/2 = 5 we are getting following cordinates. x=10 x=10 y=10 y=10and in question x+y + xy = 4. we are having following cordinates x=2 x=2 y=2 y=2 why we are drawing graph differently? Thanks For \(x+y + xy = 4\) the equations are: \(x = 2\); \(y = 2\); \(y = 2\); \(x = 2\). For \(\frac{x}{2} + \frac{y}{2} = 5\) the equations are: \(y=10x\); \(y=10+x\); \(y=10x\); \(y=x10\). You might helpful to reread the thread.
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Re: In the xy plane, the area of the region bounded by the graph of x +
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20 Jun 2015, 03:53
jayanthjanardhan wrote: Bunuel wrote: srini123 wrote: Thanks Bunuel, I used similar method for a similar question and I got wrong answer the question was
what is the area bounded by graph\(x/2 + y/2 = 5\)?
I got hunderd since x=10 x=10 y=10 y=10
isnt the area 400 ? the answer given was 200, please explain I think this one is different. \(\frac{x}{2} + \frac{y}{2} = 5\) After solving you'll get equation of four lines: \(y=10x\) \(y=10+x\) \(y=10x\) \(y=x10\) These four lines will also make a square, BUT in this case the diagonal will be 20 so the \(Area=\frac{20*20}{2}=200\). Or the \(Side= \sqrt{200}\), area=200. If you draw these four lines you'll see that the figure (square) which is bounded by them is turned by 90 degrees and has a center at the origin. So the side will not be 20. Also you made a mistake in solving equation. The red part is not correct. You should have the equations written above. In our original question when we were solving the equation x+y + xy = 4 each time x or y were cancelling out so we get equations of a type x=some value twice and y=some value twice. And these equations give the lines which are parallel to the Y or X axis respectively so the figure bounded by them is a "horizontal" square (in your question it's "diagonal" square). Hope it's clear. Hi bunnel, How did u rhombus for this one and a square for the other one?...I got the limits for both the questions, but could not figure out they turn out to be a square and rhombus!... Even that is a square but never forget that A Square is a specific type of Rhombus onlyI hope, You can understand that the Product of the slopes of the adjacent sides is 1 in that figure which proves the angle between the adjacent sides as 90 degree a Square is a "Rhombus with all angles 90 degrees". So calling it a Rhombus won't be wrong either but you are right about the figure being a Square.
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Re: In the xy plane, the area of the region bounded by the graph of x +
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08 Nov 2009, 15:27
Bunuel wrote: papillon86 wrote: In xy plane, the area of the region bounded by the graph of x+y + xy = 4 is
a) 8 b) 12 c) 16 d) 20
Need help in solving equations involving Mod...... help? I've never seen such kind of question in GMAT before. OK there can be 4 cases: x+y + xy = 4 A. x+y+xy = 4 > x=2 B. x+yx+y = 4 > y=2 C. xy +xy= 4 > y=2 D. xyx+y=4 > x=2 The area bounded by 4 graphs x=2, x=2, y=2, y=2 will be square with the side of 4 so the area will be 4*4=16. Answer: C Why cant we consider (4,0) and (0,4) as points on graph ? then area would be different... , right?
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Re: In the xy plane, the area of the region bounded by the graph of x +
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08 Nov 2009, 15:39
srini123 wrote: Why cant we consider (4,0) and (0,4) as points on graph ? then area would be different... , right? First of all we are not considering points separately, as we have XY plane and roots of equation will represent lines, we'll get the figure bounded by this 4 lines. The equations for the lines are: x=2 x=2 y=2 y=2 This lines will make a square with the side 4, hence area 4*4=16. Second: points (4,0) or (0,4) doesn't work for x+y + xy = 4.
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Re: In the xy plane, the area of the region bounded by the graph of x +
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08 Nov 2009, 16:58
Bunuel wrote: srini123 wrote: Why cant we consider (4,0) and (0,4) as points on graph ? then area would be different... , right? First of all we are not considering points separately, as we have XY plane and roots of equation will represent lines, we'll get the figure bounded by this 4 lines. The equations for the lines are: x=2 x=2 y=2 y=2 This lines will make a square with the side 4, hence area 4*4=16. Second: points (4,0) or (0,4) doesn't work for x+y + xy = 4. Thanks Bunuel, I used similar method for a similar question and I got wrong answer the question was what is the area bounded by graph\(x/2 + y/2 = 5\)? I got hunderd since x=10 x=10 y=10 y=10 isnt the area 400 ? the answer given was 200, please explain
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Re: In the xy plane, the area of the region bounded by the graph of x +
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08 Nov 2009, 20:23
Thanks Bunuel , once again wonderful explanation +1 Kudos.. have a good day...
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Re: In the xy plane, the area of the region bounded by the graph of x +
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22 May 2011, 07:21
Given xy + x+y = 4
I don't understand why can't xy and x+y be 1 and 3 instead of 2 and 2! (which again equals 4)
Can any one please explain this to me?
Thanks & Regards, Vinu



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Re: In the xy plane, the area of the region bounded by the graph of x +
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22 May 2011, 08:38
VinuPriyaN wrote: Given xy + x+y = 4
I don't understand why can't xy and x+y be 1 and 3 instead of 2 and 2! (which again equals 4)
Can any one please explain this to me?
Thanks & Regards, Vinu Look at the solution given by Bunuel above. When you solve it, you get four equations. One of them is x = 2 which means that x = 2 and y can take any value. If y = 1, xy = 1 and x+y = 3. For different values of y, xy and x+y will get different values. We are not discounting any of them.
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Re: In the xy plane, the area of the region bounded by the graph of x +
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15 Feb 2012, 00:25
prashantbacchewar wrote: In the XY plane, the area of the region bounded by the graph of x + y + x – y = 4 is (1) 8 (2) 12 (3) 16 (4) 20 (5) 24 Merging similar topics. Please ask if anything remains unclear. Some questions on the same subject to practice: m065absolutevalue108191.htmlgraphsmodulushelp86549.htmlm06q572817.htmlifequationenclosesacertainregion110053.htmlHope it helps.
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Re: In the xy plane, the area of the region bounded by the graph of x +
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27 Feb 2012, 23:33
Hi, Can this be solved by graphing. If yes .. how do we graph the equation with 2 mod parts VeritasPrepKarishma wrote: VinuPriyaN wrote: Given xy + x+y = 4
I don't understand why can't xy and x+y be 1 and 3 instead of 2 and 2! (which again equals 4)
Can any one please explain this to me?
Thanks & Regards, Vinu Look at the solution given by Bunuel above. When you solve it, you get four equations. One of them is x = 2 which means that x = 2 and y can take any value. If y = 1, xy = 1 and x+y = 3. For different values of y, xy and x+y will get different values. We are not discounting any of them.



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Re: In the xy plane, the area of the region bounded by the graph of x +
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05 Dec 2012, 23:17
(1) derive all equations from x+y + xy = 4 x+y+xy =4 ==> x=2 x+yx+y =4 ==> y=2 xy+xy =4 ==> y=2 xyx+y =4 ==> x=2 (2) Plot your four lines (3) Notice you have formed a square region bounded by x=2, y=2, y=2 and x=2 lines (4) Area = 4*4 = 16 Answer: C For more detailed solutions for similar question types:
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Re: In the xy plane, the area of the region bounded by the graph of x +
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13 Dec 2012, 04:26
Bunuel wrote: srini123 wrote: Thanks Bunuel, I used similar method for a similar question and I got wrong answer the question was
what is the area bounded by graph\(x/2 + y/2 = 5\)?
I got hunderd since x=10 x=10 y=10 y=10
isnt the area 400 ? the answer given was 200, please explain I think this one is different. \(\frac{x}{2} + \frac{y}{2} = 5\) After solving you'll get equation of four lines: \(y=10x\) \(y=10+x\) \(y=10x\) \(y=x10\) These four lines will also make a square, BUT in this case the diagonal will be 20 so the \(Area=\frac{20*20}{2}=200\). Or the \(Side= \sqrt{200}\), area=200. If you draw these four lines you'll see that the figure (square) which is bounded by them is turned by 90 degrees and has a center at the origin. So the side will not be 20. Also you made a mistake in solving equation. The red part is not correct. You should have the equations written above. In our original question when we were solving the equation x+y + xy = 4 each time x or y were cancelling out so we get equations of a type x=some value twice and y=some value twice. And these equations give the lines which are parallel to the Y or X axis respectively so the figure bounded by them is a "horizontal" square (in your question it's "diagonal" square). Hope it's clear. Hii Bunuel. What is the best approach of finding the points of intersection in order to make the square.
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Re: In the xy plane, the area of the region bounded by the graph of x +
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13 Dec 2012, 04:30
Marcab wrote: Bunuel wrote: srini123 wrote: Thanks Bunuel, I used similar method for a similar question and I got wrong answer the question was
what is the area bounded by graph\(x/2 + y/2 = 5\)?
I got hunderd since x=10 x=10 y=10 y=10
isnt the area 400 ? the answer given was 200, please explain I think this one is different. \(\frac{x}{2} + \frac{y}{2} = 5\) After solving you'll get equation of four lines: \(y=10x\) \(y=10+x\) \(y=10x\) \(y=x10\) These four lines will also make a square, BUT in this case the diagonal will be 20 so the \(Area=\frac{20*20}{2}=200\). Or the \(Side= \sqrt{200}\), area=200. If you draw these four lines you'll see that the figure (square) which is bounded by them is turned by 90 degrees and has a center at the origin. So the side will not be 20. Also you made a mistake in solving equation. The red part is not correct. You should have the equations written above. In our original question when we were solving the equation x+y + xy = 4 each time x or y were cancelling out so we get equations of a type x=some value twice and y=some value twice. And these equations give the lines which are parallel to the Y or X axis respectively so the figure bounded by them is a "horizontal" square (in your question it's "diagonal" square). Hope it's clear. Hii Bunuel. What is the best approach of finding the points of intersection in order to make the square. I'd say substituting x=0 and y=0 in the equations of lines and making a drawing.
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