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Math Expert V
Joined: 02 Sep 2009
Posts: 58381
If equation |x| + |y| = 5 encloses a certain region on the graph, what  [#permalink]

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Difficulty:   65% (hard)

Question Stats: 50% (01:26) correct 50% (01:23) wrong based on 148 sessions

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If equation $$|x| + |y| = 5$$ encloses a certain region on the graph, what is the area of that region?

A. 5
B. 10
C. 25
D. 50
E. 100

M06-05

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Math Expert V
Joined: 02 Sep 2009
Posts: 58381
Re: If equation |x| + |y| = 5 encloses a certain region on the graph, what  [#permalink]

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Bunuel wrote:
If equation $$|x| + |y| = 5$$ encloses a certain region on the graph, what is the area of that region?

A. 5
B. 10
C. 25
D. 50
E. 100

M06-05

X and Y intercepts are (0, 5), (0, -5), (5, 0), and (-5, 0) (just make x equal to zero and find y and then make y equal to zero and find x). Now if we join these points we'll get the following region: The diagonals of the quadrilateral are equal (10 and 10), and also are perpendicular bisectors of each other (as they are on X and Y axis), so the figure must be a square. Area of a square equals to $$\frac{\text{diagonal}^2}{2}=\frac{10^2}{2}=50$$.

Attachment: m06-05.gif [ 2.86 KiB | Viewed 1575 times ]

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Intern  B
Joined: 07 Mar 2019
Posts: 29
Re: If equation |x| + |y| = 5 encloses a certain region on the graph, what  [#permalink]

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1
What about coordinate points (-2, 3), (2, 3), (-2, -3), (2, -3)? Those satisfy the equation and yield a different area

Area: (2 - (-2)) * (3 - (-3)) = 24

I don't understand why this isn't valid?
Senior Manager  G
Joined: 25 Feb 2019
Posts: 336
Re: If equation |x| + |y| = 5 encloses a certain region on the graph, what  [#permalink]

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You may have combination of 1,4 also which will satisfy the given equation .

BUT THIS IS NOT CORRECT

Actually given equation is a collection of 4 straight lines which are caluclated on below logic

when x and y both negative
x negative and y positive
x positive and y negative
x positive and y positive

this way we get four lines amd then draw them on the xy plane

and calculate the enclosed area .

award kudos if helpful

Posted from my mobile device
Intern  B
Joined: 07 Mar 2019
Posts: 29
If equation |x| + |y| = 5 encloses a certain region on the graph, what  [#permalink]

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1
m1033512 wrote:
You may have combination of 1,4 also which will satisfy the given equation .

BUT THIS IS NOT CORRECT

Actually given equation is a collection of 4 straight lines which are caluclated on below logic

when x and y both negative
x negative and y positive
x positive and y negative
x positive and y positive

this way we get four lines amd then draw them on the xy plane

and calculate the enclosed area .

award kudos if helpful

Posted from my mobile device
I'm still confused. Possible combinations of (5,0), (4,1), (3,2) all yield inconsistent results, wouldn't this be an invalid problem? What is wrong about using (1,4) or (2,3)? They both satisfy the equation and yield straight lines on the graph. I'm guessing this is supposed to be wrong because only (0,5) doesn't plot vertical lines, but I don't see anything in the problem that prevents this. The problem seems to state plotting the coordinates and interpreting them visually.
Manager  S
Joined: 11 Feb 2018
Posts: 80
Re: If equation |x| + |y| = 5 encloses a certain region on the graph, what  [#permalink]

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1
georgethomps wrote:
m1033512 wrote:
You may have combination of 1,4 also which will satisfy the given equation .

BUT THIS IS NOT CORRECT

Actually given equation is a collection of 4 straight lines which are caluclated on below logic

when x and y both negative
x negative and y positive
x positive and y negative
x positive and y positive

this way we get four lines amd then draw them on the xy plane

and calculate the enclosed area .

award kudos if helpful

Posted from my mobile device
I'm still confused. Possible combinations of (5,0), (4,1), (3,2) all yield inconsistent results, wouldn't this be an invalid problem? What is wrong about using (1,4) or (2,3)? They both satisfy the equation and yield straight lines on the graph. I'm guessing this is supposed to be wrong because only (0,5) doesn't plot vertical lines, but I don't see anything in the problem that prevents this. The problem seems to state plotting the coordinates and interpreting them visually.

Note that the question has asked for the area enclosed by the equation. In other words it is asking for the area where all these lines (4 in this case) intersect. From the equation we can see that the lines will intersect at (5,0), (0,5), (-5,0) and (0,-5). While your coordinates may satisfy the equation, they are not the points at which these lines intersect. To make it simpler, when asked to compute area under the graph in questions of this type, look for the coordinates at which the lines intersect and compute the area enclosed by them.

Hope its clear now. Re: If equation |x| + |y| = 5 encloses a certain region on the graph, what   [#permalink] 19 Jun 2019, 02:43
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