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

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New post 11 Mar 2014, 23:54
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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


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

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New post 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 \(x\).

Answer: D.

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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New post 01 Jun 2014, 23:31
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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

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

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New post 18 Dec 2014, 14:58
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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)

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

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New post 26 Sep 2017, 10:32
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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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Re: For how many ordered pairs (x , y) that are solutions of the  [#permalink]

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New post 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:
1. Please provide your solutions to the questions;
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 ? :)
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Re: For how many ordered pairs (x , y) that are solutions of the  [#permalink]

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New post 03 Apr 2018, 23:22
2
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:
1. Please provide your solutions to the questions;
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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Re: For how many ordered pairs (x , y) that are solutions of the  [#permalink]

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New post 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.

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

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New post 16 Jun 2019, 12:41
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|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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Re: For how many ordered pairs (x , y) that are solutions of the  [#permalink]

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New post 16 Jun 2019, 13:52
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[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

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

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New post 25 Jun 2019, 02:50
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] 25 Jun 2019, 02:50
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