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

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Math Expert
Joined: 02 Sep 2009
Posts: 47015
For how many ordered pairs (x , y) that are solutions of the [#permalink]

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11 Mar 2014, 23:54
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36
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Difficulty:

55% (hard)

Question Stats:

67% (01:32) correct 33% (01:39) wrong based on 1349 sessions

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The Official Guide For GMAT® Quantitative Review, 2ND Edition

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

Problem Solving
Question: 152
Category: Algebra Absolute value
Page: 82
Difficulty: 600

GMAT Club is introducing a new project: The Official Guide For GMAT® Quantitative Review, 2ND Edition - Quantitative Questions Project

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Joined: 02 Sep 2009
Posts: 47015
For how many ordered pairs (x , y) that are solutions of the [#permalink]

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15 Mar 2014, 10:33
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SOLUTION

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 $$2x+y=12$$

Rearrange $$2x+y=12$$ to get $$y=12-2x=2(6-x)=even$$, (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 $$x$$.

P.S. The ordered pairs of (x, y)would be:
(12, -12)
(11, -10)
(10, -8)
(9, -6)
(8, -4)
(7, -2)
(6, 0)
(5, 2)
(4, 4)
(3, 6)
(2, 8)
(1, 10)
(0, 12)
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Re: For how many ordered pairs (x , y) that are solutions of the [#permalink]

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01 Jun 2014, 23:31
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2
2x + y = 12 --> x= (12 - y)/2 = 6 - y/2. Thus every even value of y will yield integer value of x too.

|y| <= 12 --> There are 13 even values of y: 12 - (-12) = 24/2 + 1 = 13

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

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18 Dec 2014, 14:58
1
|y| <= 12 means the range of Y is -12<=Y<=12
Let'ssimplify the first equation X=(12-y)/2 -> So in order both x and y to be an integer 12-y must be even.
We have 13 even numbers in the range of -12<=Y<=12: These are -12,-10,-8,-6,-4,-2,0,2,4,6,8,10,12 (don't forget to count 0 and 12)

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

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26 Sep 2017, 10:32
1
Given that |y| <= 12 -----> y<=12 and y>= -12
Given that 2x + y = 12 and x,y are integers
In order for x to be integer, y would always take even value
Hence, no of even integers between -12 and 12(both inclusive) are 13.
Option D.

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Director
Joined: 09 Mar 2016
Posts: 648
Re: For how many ordered pairs (x , y) that are solutions of the [#permalink]

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03 Apr 2018, 10:11
Bunuel wrote:
The Official Guide For GMAT® Quantitative Review, 2ND Edition

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

Problem Solving
Question: 152
Category: Algebra Absolute value
Page: 82
Difficulty: 600

GMAT Club is introducing a new project: The Official Guide For GMAT® Quantitative Review, 2ND Edition - Quantitative Questions Project

Each week we'll be posting several questions from The Official Guide For GMAT® Quantitative Review, 2ND Edition and then after couple of days we'll provide Official Answer (OA) to them along with a slution.

We'll be glad if you participate in development of this project:
2. Please vote for the best solutions by pressing Kudos button;
3. Please vote for the questions themselves by pressing Kudos button;
4. Please share your views on difficulty level of the questions, so that we have most precise evaluation.

Thank you!

hello, my quant session continues

guys what does "many ordered pairs" mean ? i didnt understand the question itself. i thought it was coordinate geometry question

why are we looking into ODD and EVEN integers ?

can someone explain this please ?
Intern
Joined: 19 Jun 2017
Posts: 2
Re: For how many ordered pairs (x , y) that are solutions of the [#permalink]

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03 Apr 2018, 23:22
1
dave13 wrote:
Bunuel wrote:
The Official Guide For GMAT Quantitative Review, 2ND Edition

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

Problem Solving
Question: 152
Category: Algebra Absolute value
Page: 82
Difficulty: 600

GMAT Club is introducing a new project: The Official Guide For GMAT Quantitative Review, 2ND Edition - Quantitative Questions Project

Each week we&#39;ll be posting several questions from The Official Guide For GMAT Quantitative Review, 2ND Edition and then after couple of days we&#39;ll provide Official Answer (OA) to them along with a slution.

We&#39;ll be glad if you participate in development of this project:
2. Please vote for the best solutions by pressing Kudos button;
3. Please vote for the questions themselves by pressing Kudos button;
4. Please share your views on difficulty level of the questions, so that we have most precise evaluation.

Thank you!

hello, my quant session continues

guys what does "many ordered pairs" mean ? i didnt understand the question itself. i thought it was coordinate geometry question

why are we looking into ODD and EVEN integers ?

can someone explain this please ?
Hi Dave,

Ordered pair means for what values of x and y the given condition satisfy.
Here we are discussing about the odd and even because from the first equation after simplifying further we can get x= 6-y/2.
So we have figure out for what values of y x is an integer.And from equation 2 we can get the values of y as -12<= y<=12.
So for x to be integer y has to an even integer( as only even integers are divisible by 2).so here our answer is to find how even integers are present between -12 and 12 i.e 13 .(don't forget to include 0).hope it helps

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Affiliations: Target Test Prep
Joined: 04 Mar 2011
Posts: 2669
Re: For how many ordered pairs (x , y) that are solutions of the [#permalink]

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05 Apr 2018, 17:01
Bunuel wrote:
The Official Guide For GMAT® Quantitative Review, 2ND Edition

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

For the inequality |y| ≤ 12, we see that -12 ≤ y ≤ 12

For the equation 2x + y = 12, we see that x = (12 - y)/2. If x has to be an integer, then y has to be an even integer; thus, y can be any of the even integers from -12 to 12, inclusive. Since there are

(12 - (-12))/2 + 1 = 24/2 + 1 = 13

even integers for y, there will be 13 corresponding integers for x. Hence, there are 13 ordered pairs (x, y) that are solutions to the system and where x and y are both integers.

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Re: For how many ordered pairs (x , y) that are solutions of the   [#permalink] 05 Apr 2018, 17:01
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