Last visit was: 23 Jul 2024, 19:23 It is currently 23 Jul 2024, 19:23
Close
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Close
Request Expert Reply
Confirm Cancel
SORT BY:
Date
Tags:
Difficulty: 605-655 Level,   Absolute Values,   Algebra,                              
Show Tags
Hide Tags
Math Expert
Joined: 02 Sep 2009
Posts: 94589
Own Kudos [?]: 643386 [347]
Given Kudos: 86728
Send PM
Most Helpful Reply
Math Expert
Joined: 02 Sep 2009
Posts: 94589
Own Kudos [?]: 643386 [105]
Given Kudos: 86728
Send PM
Retired Moderator
Joined: 29 Oct 2013
Posts: 220
Own Kudos [?]: 2069 [36]
Given Kudos: 204
Concentration: Finance
GPA: 3.7
WE:Corporate Finance (Retail Banking)
Send PM
General Discussion
Current Student
Joined: 10 Mar 2013
Posts: 359
Own Kudos [?]: 2741 [7]
Given Kudos: 200
Location: Germany
Concentration: Finance, Entrepreneurship
GMAT 1: 580 Q46 V24
GPA: 3.7
WE:Marketing (Telecommunications)
Send PM
Re: For how many ordered pairs (x, y) that are solutions of the system abo [#permalink]
5
Kudos
2
Bookmarks
|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).
Target Test Prep Representative
Joined: 04 Mar 2011
Status:Head GMAT Instructor
Affiliations: Target Test Prep
Posts: 3036
Own Kudos [?]: 6624 [13]
Given Kudos: 1646
Send PM
Re: For how many ordered pairs (x , y) that are solutions of the [#permalink]
9
Kudos
4
Bookmarks
Expert Reply
tonebeeze wrote:
\(2x + y = 12\)
\(|y| \leq 12\)

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


From the inequality |y| ≤ 12, we see that -12 ≤ y ≤ 12. If we want y to be an integer, then y is any integer from -12 to 12 inclusive. Since we want x to be an integer also, let’s isolate x in terms of y:

2x + y = 12

2x = 12 - y

x = 6 - y/2

We see that in order for x to be an integer, y must be even so that y/2 is an integer. Thus, y is any of the integers in the following list:

-12, -10, -8, -6, -4, -2, 0, 2, 4, 6, 8, 10, 12

For any of the 13 integers in the above list, x will be also an integer. Thus, there are 13 ordered pairs that satisfy the system with both x and y being integers.

Answer: D
Tutor
Joined: 16 Oct 2010
Posts: 15140
Own Kudos [?]: 66815 [4]
Given Kudos: 436
Location: Pune, India
Send PM
For how many ordered pairs (x, y) that are solutions of the system abo [#permalink]
3
Kudos
1
Bookmarks
Expert Reply
Bunuel wrote:
\(2x + y = 12\)
\(|y| \leq 12\)

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


The solution of \(|y| \leq 12\) is straight forward.
\(-12 \leq y \leq 12\)
(If you are not comfortable with this, check out my video: https://youtu.be/oqVfKQBcnrs )


If both x and y have to be integers, y should be an integer and hence can take any value from the set {-12, -11, -10 ... 10, 11, 12} i.e. any one of 25 values (these are 25 values -12 to -1 (12 values), 0, 1 to 12 (another 12 values)) 13 of them are even and 12 of them are odd.

\(2x + y = 12\)
Every time y is even, x will be integer. e.g. y = 12, x = 0 (because x = (12 - even)/2 will be an integer)
Every time y is odd, x will be non-integer e.g. y = 1, x = 5.5 (because x = (12 - odd)/2 will not be an integer)

Therefore, for 13 values, x and y both will be integers.

Originally posted by KarishmaB on 26 Sep 2017, 10:32.
Last edited by KarishmaB on 19 Sep 2023, 04:39, edited 1 time in total.
VP
VP
Joined: 09 Mar 2016
Posts: 1142
Own Kudos [?]: 1028 [0]
Given Kudos: 3851
Send PM
For how many ordered pairs (x, y) that are solutions of the system abo [#permalink]
Bunuel 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
 

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
Intern
Joined: 19 Jun 2017
Posts: 5
Own Kudos [?]: 11 [6]
Given Kudos: 166
Send PM
For how many ordered pairs (x, y) that are solutions of the system abo [#permalink]
6
Kudos
dave13 wrote:
Bunuel 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
 

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
 
Target Test Prep Representative
Joined: 04 Mar 2011
Status:Head GMAT Instructor
Affiliations: Target Test Prep
Posts: 3036
Own Kudos [?]: 6624 [1]
Given Kudos: 1646
Send PM
For how many ordered pairs (x, y) that are solutions of the system abo [#permalink]
1
Bookmarks
Expert Reply
Bunuel 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

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
GMAT Club Legend
GMAT Club Legend
Joined: 03 Jun 2019
Posts: 5312
Own Kudos [?]: 4245 [0]
Given Kudos: 161
Location: India
GMAT 1: 690 Q50 V34
WE:Engineering (Transportation)
Send PM
For how many ordered pairs (x, y) that are solutions of the system abo [#permalink]
Bunuel 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


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
Intern
Intern
Joined: 01 Jul 2017
Posts: 13
Own Kudos [?]: 13 [2]
Given Kudos: 47
WE:Other (Other)
Send PM
Re: For how many ordered pairs (x, y) that are solutions of the system abo [#permalink]
2
Kudos
y=12-2x
y=2(6-x)

also |y| <= 12

so |2(6-x)| <= 12

i.e. -12 <= 2 (6-x) <= 12
i.e. -6 <= 6-x <= 6
i.e. -12 <= -x <= 0
i.e. 0<= x<= 12

So x can be any interger among integers from 0 to 12 .

Answer 13
Intern
Intern
Joined: 22 Jan 2021
Posts: 15
Own Kudos [?]: 10 [0]
Given Kudos: 41
Send PM
Re: For how many ordered pairs (x, y) that are solutions of the system abo [#permalink]
SHUBHAM GAUTAM wrote:
y=12-2x
y=2(6-x)

also |y| <= 12

so |2(6-x)| <= 12

i.e. -12 <= 2 (6-x) <= 12
i.e. -6 <= 6-x <= 6
i.e. -12 <= -x <= 0
i.e. 0<= x<= 12

So x can be any interger among integers from 0 to 12 .

Answer 13


VeritasKarishma JeffTargetTestPrep
I used this same process above ^^. I substituted x into |y| <= 12 to find that there are 13 values of x because the range was [0,12]
Is this a correct way to solve this problem? I see that it is different from your solutions...
Tutor
Joined: 16 Oct 2010
Posts: 15140
Own Kudos [?]: 66815 [0]
Given Kudos: 436
Location: Pune, India
Send PM
Re: For how many ordered pairs (x, y) that are solutions of the system abo [#permalink]
Expert Reply
dc2880 wrote:
SHUBHAM GAUTAM wrote:
y=12-2x
y=2(6-x)

also |y| <= 12

so |2(6-x)| <= 12

i.e. -12 <= 2 (6-x) <= 12
i.e. -6 <= 6-x <= 6
i.e. -12 <= -x <= 0
i.e. 0<= x<= 12

So x can be any interger among integers from 0 to 12 .

Answer 13


VeritasKarishma JeffTargetTestPrep
I used this same process above ^^. I substituted x into |y| <= 12 to find that there are 13 values of x because the range was [0,12]
Is this a correct way to solve this problem? I see that it is different from your solutions...


Yes, this method is correct too. It uses algebraic substitution to solve.
Intern
Intern
Joined: 22 May 2021
Posts: 48
Own Kudos [?]: 18 [0]
Given Kudos: 107
Send PM
Re: For how many ordered pairs (x, y) that are solutions of the system abo [#permalink]
Here :
2x+y=12
|y|≤12

implies -12 ≤ y ≤ 12 y can take, since it's given x n y are integers then , 2x = 12 -y so 12 -y needs to be multiple of 2.
If we subtract odd values from 12 it will give odd value, so need to avoid odd values of y between -12 and 12 which give us 12 even values.
Need to consider 0 as well.
So total values 13.
Ans D
Intern
Intern
Joined: 22 Sep 2018
Posts: 12
Own Kudos [?]: 4 [3]
Given Kudos: 281
GMAT 1: 620 Q45 V30
Send PM
Re: For how many ordered pairs (x, y) that are solutions of the system abo [#permalink]
3
Kudos
1) 2 x + y = 12, from this equation we can find y = 12 - 2x
2) |y|≤12, from this inequality we can deduce following: -12 ≤ y ≤12
Take equation from 1st step and put into inequality in 2nd step and get following inequality: -12 ≤ 12-2x ≤12. Solve it as follow:
-12 ≤ 12-2x ≤12 |subtract 12 each side
-24≤ - 2x ≤ 0 |divide by -2 each side
12 ≥ x ≥ 0, so x can be anything from 0 to 12 including, and that is 13 numbers which is our answer choice D.
Tutor
Joined: 26 Jun 2014
Status:Mentor & Coach | GMAT Q51 | CAT 99.98
Posts: 450
Own Kudos [?]: 810 [0]
Given Kudos: 8
Send PM
Re: For how many ordered pairs (x, y) that are solutions of the system abo [#permalink]
Expert Reply
Bunuel wrote:
\(2x + y = 12\)
\(|y| \leq 12\)

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


2x+y=12
|y|≤12

x and y are integers
2x+y=12
Here 2x and 12 are even
So y must be even
For every even y, you will definitely get a value of x
For example: y = -6, x = 9 | y = 6, x = 3, etc.
Total possible values of y are:
-12, -11, ... 0, ... 11, 12
Of these, the even ones are:

-12, -10, ... 0, ... 6, 8, 10, 12

Thus, there are 13 solution sets

Answer D

Posted from my mobile device
Intern
Intern
Joined: 26 Apr 2024
Posts: 39
Own Kudos [?]: 15 [0]
Given Kudos: 19
Location: India
Concentration: Entrepreneurship, Strategy
GPA: 9.07
WE:Engineering (Manufacturing)
Send PM
Re: For how many ordered pairs (x, y) that are solutions of the system abo [#permalink]
Hello Bunuel,

What is the difference between ordered and unordered pairs and how this question will have been different if it has asked about unordered pairs?
Math Expert
Joined: 02 Sep 2009
Posts: 94589
Own Kudos [?]: 643386 [0]
Given Kudos: 86728
Send PM
For how many ordered pairs (x, y) that are solutions of the system abo [#permalink]
Expert Reply
1616 wrote:
Hello Bunuel,

What is the difference between ordered and unordered pairs and how this question will have been different if it has asked about unordered pairs?

­The difference between ordered and unordered pairs is in the sequence and significance of the elements:

  • Ordered Pairs: The sequence of elements matters. For example, (x=1, y=2) is different from (x=2, y=1) unless x equals y. Each specific pair is counted as unique based on the order of elements.
  • Unordered Pairs: The sequence of elements does not matter. For example, (x=1, y=2) is considered the same as (x=2, y=1).

For this particular question, the number of unordered pairs would not be different from the number of ordered pairs. But if there were a solution such as (4, 6) and (6, 4) satisfying the conditions, those would be counted separately as ordered pairs because (x=4, y=6) is different from (x=6, y=4).

However, if the question were to find the number of ordered pairs of (x, y) where both x and y are positive integers such that x + y = 4, the answer would be:

  • Ordered pairs: (1, 3), (2, 2), and (3, 1). This gives us 3 ordered pairs.
  • Unordered pairs: (1, 3) and (2, 2). Since (1, 3) and (3, 1) are considered the same pair, the correct unordered pairs are (1, 3) and (2, 2). This gives us 2 unordered pairs.
­
GMAT Club Bot
For how many ordered pairs (x, y) that are solutions of the system abo [#permalink]
Moderator:
Math Expert
94589 posts