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In the xy plane, the area of the region bounded by the graph of x + [#permalink]
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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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Last edited by Bunuel on 26 Oct 2017, 05:46, edited 3 times in total.
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Re: In the xy plane, the area of the region bounded by the graph of x + [#permalink]
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08 Nov 2009, 14:27
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 + [#permalink]
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08 Nov 2009, 14:39
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 + [#permalink]
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08 Nov 2009, 15:58
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 + [#permalink]
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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 + [#permalink]
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08 Nov 2009, 19:23
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 + [#permalink]
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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 + [#permalink]
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22 May 2011, 06:21
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 + [#permalink]
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22 May 2011, 07:38
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 + [#permalink]
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Re: In the xy plane, the area of the region bounded by the graph of x + [#permalink]
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27 Feb 2012, 22:33
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 + [#permalink]
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27 Feb 2012, 22:42
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 20425 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 + [#permalink]
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05 Dec 2012, 22:17
(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 + [#permalink]
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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 + [#permalink]
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13 Dec 2012, 03:26
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 + [#permalink]
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13 Dec 2012, 03:30
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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Re: In the xy plane, the area of the region bounded by the graph of x + [#permalink]
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21 Aug 2013, 20:51
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. The side of the square can't be 4, instead its sqrt(8)
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Re: In the xy plane, the area of the region bounded by the graph of x + [#permalink]
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22 Aug 2013, 02:21
honchos wrote: 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. The side of the square can't be 4, instead its sqrt(8)The side of the square IS 4: Attachment:
MSP39361d6ehgde6ie87a8800003827f7f92a367c60.gif [ 1.86 KiB  Viewed 3156 times ]
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Re: In the xy plane, the area of the region bounded by the graph of x + [#permalink]
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27 Jan 2014, 02:00
Bunuel,
wouldn't it be sufficient to look at only two cases?
(x+y) + (xy) = 4 ==> x=2 (x+y)  (xy) = 4 ==> y=2
Which would give us 2*2 * 4 = 16?




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