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Math Expert V
Joined: 02 Sep 2009
Posts: 65785
For how many ordered pairs (x , y) that are solutions of the  [#permalink]

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103 00:00

Difficulty:   55% (hard)

Question Stats: 68% (02:03) correct 32% (02:14) wrong based on 2503 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

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.

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Math Expert V
Joined: 02 Sep 2009
Posts: 65785
For how many ordered pairs (x , y) that are solutions of the  [#permalink]

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16
22
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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5
8
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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2
2
|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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Joined: 24 Jul 2017
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Re: For how many ordered pairs (x , y) that are solutions of the  [#permalink]

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1
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.

Kudos if it helps VP  D
Joined: 09 Mar 2016
Posts: 1261
Re: For how many ordered pairs (x , y) that are solutions of the  [#permalink]

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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  B
Joined: 19 Jun 2017
Posts: 10
Re: For how many ordered pairs (x , y) that are solutions of the  [#permalink]

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3
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

Sent from my XT1663 using GMAT Club Forum mobile app
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Re: For how many ordered pairs (x , y) that are solutions of the  [#permalink]

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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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Posts: 440
Re: For how many ordered pairs (x , y) that are solutions of the  [#permalink]

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1
|y|≤12
-12 ≤ y ≤ 12
&
2x+y = 12
x = (12-y)/2

Now, for both x, y to be ints we need an even y value because x must be (even# - even#)/2 to be an int.
So we need all the even numbers in the range of y: -12,-10,-8,-6,-4,-2,0,2,4,6,8,10,12
That's 13 numbers (alternatively: max y - min y = 12 - (-12) = 24 / 2 = 12 numbers + 1 for 0 = 13 numbers)
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Concentration: Entrepreneurship, Marketing
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Re: For how many ordered pairs (x , y) that are solutions of the  [#permalink]

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1
[quote="Bunuel"]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

lyl ⩽ 12
—> -12 ⩽ y ⩽ 12

2x + y = 12
—> x = (12 - y)/2
We can see that x is an integer for all even values of y
—> Number of solutions (x, y) = Number of even values of y from -12 to 12
—> -12, -10, -8, . . . . . 12
AP Series, Use last term
—> 12 = a + (n - 1)d
—> 12 = -12 + (n - 1)2
—> 24 = (n - 1)2
—> n - 1 = 12
—> n = 13

IMO Option D

Pls Hit kudos if you like the solution

Posted from my mobile device
Manager  S
Joined: 21 Jun 2019
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Concentration: Finance, Accounting
GMAT 1: 670 Q48 V34
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Re: For how many ordered pairs (x , y) that are solutions of the  [#permalink]

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hello,

given 2x+y=12 ===> x= Y-12/2. we can conclude that x will be an integer if and only if y is even.

and -12 ≤ y ≤ 12.

if we combine both information we can eliminate all odd numbers in the range -12 ≤ y ≤ 12

the even numbers remaining are -12,-10,-8,-6,-4,-2,0,2,4,6,8,10,12

n=13
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Re: For how many ordered pairs (x , y) that are solutions of the  [#permalink]

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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!

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

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

2x + y = 12 => y is even since both 2x and 12 are even
|y| = {0,2,4,6,8,10,12}

y={0,2,-2,4,-4,6,-6,8,-8,10,-10,12,-12}
x={6,5,7,4,8,3,9,2,10,1,11,0,12}
There are 12 (x,y) ordered pairs.

The ordered pairs of (x, y) are:
(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)

IMO D
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Director  V
Joined: 24 Oct 2016
Posts: 714
GMAT 1: 670 Q46 V36 GMAT 2: 690 Q47 V38 GMAT 3: 690 Q48 V37 GMAT 4: 710 Q49 V38 (Online) For how many ordered pairs (x , y) that are solutions of the  [#permalink]

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Bunuel wrote:
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)

# of Possible values for y = 25

-12 <= y <= 12
-12 <= 12 - 2x <= 12
-24 <= -2x <= 0
24 >= 2x >= 0
12 >= x >= 0

# of Possible values for x = 13

# of Possible values for (x, y) = Min (# of Possible values for x, # of Possible values for y) = Min (13, 25) = 13

Intern  B
Joined: 28 Jan 2020
Posts: 4
For how many ordered pairs (x , y) that are solutions of the  [#permalink]

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2x + y = 12
y = 12 - 2x

|y| <= 12
-12 <= 12 - 2x <= 12 (bring 12 - 2x from equation above)

0 <= x <= 12

Since x can take any number from 0 to 12 included, then y = 12 - 2x has 13 solutions that yield an integer
Manager  B
Joined: 11 Oct 2019
Posts: 103
Re: For how many ordered pairs (x , y) that are solutions of the  [#permalink]

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2x + y = 12
x = 6 - y/2
Therefore x is an integer of y=0 and every even value of y.

|y| <= 12
-12 < y < 12
There are 12 even values of y and 0 in this region.
Hence there are 13 integer pairs Re: For how many ordered pairs (x , y) that are solutions of the   [#permalink] 01 Jul 2020, 11:29

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