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# For how many ordered pairs (x , y) that are solutions of the

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Manager
Joined: 20 Mar 2005
Posts: 169
For how many ordered pairs (x , y) that are solutions of the [#permalink]

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24 May 2007, 06:07
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Question Stats:

62% (02:10) correct 38% (01:35) wrong based on 410 sessions

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2x + y = 12
|y| <= 12

For how many ordered pairs (x , y) that are solutions of the system above are x and y both integers?

A. 7
B. 10
C. 12
D. 13
E. 14

OPEN DISCUSSION OF THIS QUESTION IS HERE: for-how-many-ordered-pairs-x-y-that-are-solutions-of-the-110687.html
[Reveal] Spoiler: OA

Last edited by Bunuel on 09 Mar 2014, 23:57, edited 2 times in total.
Renamed the topic, edited the question and added the OA.
Intern
Joined: 20 Apr 2007
Posts: 12

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24 May 2007, 09:28
Can you explain why the Y values go from -12 to 12? I a little new at this forum. I originally got 12 total possibilities.
Senior Manager
Joined: 11 Jun 2006
Posts: 254

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24 May 2007, 10:17
1
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2x + y = 12

|y| <= 12 means that y can be anything between 12 and -12 inclusive. Absolute values always indicates a range of numbers... this is the easy way to think about abs. values.

Ok, now you've narrowed down the answer choices to 25 possible numbers... which doesn't help you with the answers given. Next you need to find a way of eliminating more answer choices...

Simplify 2x + y = 12 to

x + y/2 = 3

Now looking at that, you know y has to be an even number to yield an integer... so the initial pool of 25 numbers is now narrowed down to 13, hence the answer.
Manager
Joined: 18 Sep 2006
Posts: 58

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24 May 2007, 19:57
This is a good question, coz I did not pay attention to the "(x,y) that will yield x and y to be integers" part of the question, so I was stuck with the answer being 25 and was stumped by the choices.
Good job you guys....i guess I should read the question clearly
Math Forum Moderator
Joined: 20 Dec 2010
Posts: 2010
Re: PS-ordered pairs (x , y) [#permalink]

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07 Sep 2011, 00:04
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Balvinder wrote:
2x + y = 12
|y| <= 12

For how many ordered pairs (x , y) that are solutions of the system above are x and y both integers?
A. 7
B. 10
C. 12
D. 13
E. 14

|y| <= 12
-12<=y<=12

2x + y = 12
x=(12-y)/2

To find number of integer pairs, we just need to find even number of y's, because even y will make "(12-y)" even as well and only even numbers divide by 2 evenly to give an integer.

e.g.
x=(12-y)/2; for y=1; x=(12-1)/2=11/2=5.5(Not an integer because y is odd)
x=(12-y)/2; for y=0; x=(12-0)/2=12/2=6(An integer because y is even)

Thus, if we find the number of even y's, we should be good.

-12<=y<=12
What is the first even number greater than or equal to -12?
-12
What is the last even number smaller than or equal to +12?
+12

$$Count=\frac{Last Even-First Even}{2}+1$$

$$Count=\frac{12-(-12)}{2}+1=12+1=13$$

Ans: "D"
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Joined: 06 Jun 2011
Posts: 147
Re: PS-ordered pairs (x , y) [#permalink]

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07 Sep 2011, 02:02
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Fluke ,
The approach was awesome
+1
abhijit
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Joined: 24 Jul 2011
Posts: 1254
GMAT 1: 780 Q51 V48
GRE 1: 1540 Q800 V740
Re: PS-ordered pairs (x , y) [#permalink]

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07 Sep 2011, 05:34
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Number of ordered pairs = number of integers between 12 and 0 (both inclusive)
= 13
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Re: 2x + y = 12 |y| <= 12 For how many ordered pairs (x [#permalink]

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09 Mar 2014, 17:11
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Posts: 39720
Re: For how many ordered pairs (x , y) that are solutions of the [#permalink]

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09 Mar 2014, 23:58
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Expert's post
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2x + y = 12
|y| <= 12

For how many ordered pairs (x , y) that are solutions of the system above are x and y both integers?

A. 7
B. 10
C. 12
D. 13
E. 14

Given: $$-12\leq{y}\leq{12}$$ and $$2x+y=12$$ --> $$y=12-2x=2(6-x)=even$$, (as $$x$$ must be an integer). Now, there are 13 even numbers in the range from -12 to 12, inclusive each of which will give an integer value of $$x$$.

OPEN DISCUSSION OF THIS QUESTION IS HERE: for-how-many-ordered-pairs-x-y-that-are-solutions-of-the-110687.html
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Re: For how many ordered pairs (x , y) that are solutions of the   [#permalink] 09 Mar 2014, 23:58
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