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Math Expert V
Joined: 02 Sep 2009
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11 00:00

Difficulty:   75% (hard)

Question Stats: 46% (01:23) correct 54% (01:27) wrong based on 112 sessions

### HideShow timer Statistics 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

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Math Expert V
Joined: 02 Sep 2009
Posts: 56357

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Official Solution:

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

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 $$\frac{\text{diagonal}^2}{2}=\frac{10^2}{2}=50$$.

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Bunuel wrote:
Official Solution:

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

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 $$\frac{\text{diagonal}^2}{2}=\frac{10^2}{2}=50$$.

Another way to find the area of the square, if you don't know how to find it from the diagonal, is to take the area of the four triangles that are made from the x and y axis. You end up with $$(\frac{5*5}{2})*4$$.
=
Additionally, you can find the difference between any two points with the formula $$\sqrt{(X2-X1)^2+(Y2-Y1)^2}$$. Plug in (5,0) and (0,5) into that equation, then square the result to find the area.
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Thanks B

whats the difference between |x+y| + |x-y| = 4 -----(A) and |x|+|y| =5 ------- (B)

For B we find the area through diagonal and for A we simply take one side and square it to get the area.
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Math Expert V
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shrive555 wrote:
Thanks B

whats the difference between |x+y| + |x-y| = 4 -----(A) and |x|+|y| =5 ------- (B)

For B we find the area through diagonal and for A we simply take one side and square it to get the area.

You can find the area of a square in either way: $$area_{square}=side^2=\frac{diagonal^2}{2}$$.

The difference between the square given by |x+y|+|x-y|=4 and the square given by |x|+|y| =5 is that the first one is "horizontal" square and the second one is the square turned by 45 degrees.
|x+y|+|x-y|=4: |x|+|y| =5: Attachment:
MSP86219e41c890f2h195700001928g4634f4e6h4h.gif

Attachment:
MSP232619e41b4bhdg1dh6e00003fi9762d3fb98db7.gif

>> !!!

You do not have the required permissions to view the files attached to this post.

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Bunuel, why cant we assume other values for |x| + |y| = 5

it could also be |-3| + |8|...please advise. i may have a concept wrong
Math Expert V
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ajit257 wrote:
Bunuel, why cant we assume other values for |x| + |y| = 5

it could also be |-3| + |8|...please advise. i may have a concept wrong

We are not assuming the values for x and y. We are expanding |x| + |y| = 5 to get the figure.

If x>0 and y>0 we'll have x+y=5;
If x>0 and y<0 we'll have x-y=5;
If x<0 and y>0 we'll have -x+y=5;
If x<0 and y<0 we'll have -x-y=5;

So we have 4 lines (4 linear equations which will give 4 segments) when drawn these lines give the square shown in my original post. But there is an easier way to get this figure: get X and Y intercepts (which are (0, 5), (0, -5), (5, 0), (-5, 0)) and join them.
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I think this is a high-quality question and I agree with explanation.
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is there a way to know the sides from the information given? thx
Math Expert V
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cocojiz wrote:
is there a way to know the sides from the information given? thx

Yes.

$$5^2 + 5^2 = side^2$$, which gives $$side = 5\sqrt{2}$$.

Or from the area: $$are=side^2=50$$, which gives $$side = 5\sqrt{2}$$.
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I think this is a High quality question and I agree with the explanation.

Are the four points (0, 5), (0, -5), (5, 0), and (-5, 0) the ONLY possible four solutions to this equation? Or are they taken to make the calculation of area simple and direct?

I do understand that even if there are other possible 4 points satisfying the equation in the Coordinate system, the area enclosed will always remain the same. Just curious to know if I am correct in saying this.

Hope, I am able to communicate what I intend to say TIA!
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susheelh wrote:
I think this is a High quality question and I agree with the explanation.

Are the four points (0, 5), (0, -5), (5, 0), and (-5, 0) the ONLY possible four solutions to this equation? Or are they taken to make the calculation of area simple and direct?

I do understand that even if there are other possible 4 points satisfying the equation in the Coordinate system, the area enclosed will always remain the same. Just curious to know if I am correct in saying this.

Hope, I am able to communicate what I intend to say TIA!

|x|+|y|=5 has infinitely may solutions. For any value of x from -5 to 5 there will exist some value of y which will satisfy |x|+|y|=5. Since there are infinitely many real numbers from -5 to 5, then there are infinitely many pairs of (x, y) which satisfy |x|+|y|=5.

(0, 5), (0, -5), (5, 0), and (-5, 0) are x and y-intercepts of |x|+|y|=5. x-intercept is the value of x when y = 0 and y-intercept is the value of y when x = 0. If you plug x = 0, you'll get two value of y and if you plug y = 0, you'll get two value of x. Thus 4 points.

Hope it's clear.
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A million thanks to you Bunuel for the quick answer. I fully understand this now. Awesome solution to an equally awesome question!

Bunuel wrote:
susheelh wrote:
I think this is a High quality question and I agree with the explanation.

Are the four points (0, 5), (0, -5), (5, 0), and (-5, 0) the ONLY possible four solutions to this equation? Or are they taken to make the calculation of area simple and direct?

I do understand that even if there are other possible 4 points satisfying the equation in the Coordinate system, the area enclosed will always remain the same. Just curious to know if I am correct in saying this.

Hope, I am able to communicate what I intend to say TIA!

|x|+|y|=5 has infinitely may solutions. For any value of x from -5 to 5 there will exist some value of y which will satisfy |x|+|y|=5. Since there are infinitely many real numbers from -5 to 5, then there are infinitely many pairs of (x, y) which satisfy |x|+|y|=5.

(0, 5), (0, -5), (5, 0), and (-5, 0) are x and y-intercepts of |x|+|y|=5. x-intercept is the value of x when y = 0 and y-intercept is the value of y when x = 0. If you plug x = 0, you'll get two value of y and if you plug y = 0, you'll get two value of x. Thus 4 points.

Hope it's clear.

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Hi,
I got the points as (5,5), (5,-5), (-5,5) and (-5,-5). This makes each side 10 units and is a square. So the area is 100. Where am I going wrong? I got the points by substituting x=0 and y=0 for each equation of the line |x| + |y| = 5.
Math Expert V
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Hi,
I got the points as (5,5), (5,-5), (-5,5) and (-5,-5). This makes each side 10 units and is a square. So the area is 100. Where am I going wrong? I got the points by substituting x=0 and y=0 for each equation of the line |x| + |y| = 5.

How did you get those values if you substituted x = 0 and y = 0?

|x| + |y| = 5:

If x = 0, then y is 5 or -5 --> two y-intercepts (0, 5), and (0, -5).
If y = 0, then x is 5 or -5 --> two x-intercepts (5, 0), and (-5, 0).

So, X and Y intercepts are (0, 5), (0, -5), (5, 0), and (-5, 0)
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Bunuel niks18 chetan2u generis pushpitkc KarishmaB EMPOWERgmatRichC GMATPrepNow

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

It is interesting to see jump from absolute modulus to intercepts in this problem.

I believe it is very difficult to solve |x| + |y| = 5 on its own. Let me know your thoughts.
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Bunuel niks18 chetan2u generis pushpitkc KarishmaB EMPOWERgmatRichC GMATPrepNow

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

It is interesting to see jump from absolute modulus to intercepts in this problem.

I believe it is very difficult to solve |x| + |y| = 5 on its own. Let me know your thoughts.

It is a co-ordinate geometry question (keywords are graph, area of the region). Just that the equations of the lines are given in a more compact format involving absolute values.
If you know your absolute values, you know the equations you will get are x + y = 5, x - y = 5 etc depending on the signs of x and y.
Also, it is very easy to visualise these lines by putting each variable equal to 0 in turn.
x + y = 5
If x = 0, y = 5 and if y = 0, x = 5 so line will intersect the axis at (0, 5) and (5, 0) and so on.
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If there was an answer choice $$\frac{25}{2}$$. I would have selected that without taking double look. Re: M06-05   [#permalink] 10 Jul 2019, 21:26
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# M06-05

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