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The number of ordered pairs of positive integers (x, y) satisfying the

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Bunuel
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The number of ordered pairs of positive integers (x, y) satisfying the [#permalink] New post   28 Feb 2020, 02:33
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The number of ordered pairs of positive integers (x, y) satisfying the equation 7x + 3y = 123 is:

A. 3
B. 5
C. 12
D. 13
E. Infinite



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B
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The number of ordered pairs of positive integers (x, y) satisfying the [#permalink] New post   28 Feb 2020, 03:47
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Bunuel wrote:
The number of ordered pairs of positive integers (x, y) satisfying the equation 7x + 3y = 123 is:

A. 3
B. 5
C 12
D. 13
E. Infinite


Are You Up For the Challenge: 700 Level Questions


Solution:


    • x and y are positive integers.
    • \(7x + 3y = 123\)
      o \(3y = (123-7x)\)
    • We need to find the number of positive values of \(x\) for which \(123 -7x\) is a multiple of \(3\).
    • \(123 – 7x\) is a multiple of 3 when x is also a multiple of 3.
      o \(123-7x > 0\)
      o \(123/7 > x\)
      o \(17.57>x\)
    • Now the possible values of \(x\) are \(3, 6, 9, 12, 15\), i.e. there are \(5\) possible values of \(x\) and for each \(x\) there will be a unique \(y\).
    • There are \(5\) ordered pairs of \((x, y)\), for which the given equation is true.
Hence, the correct answer is Option B.
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Re: The number of ordered pairs of positive integers (x, y) satisfying the [#permalink] New post   28 Feb 2020, 03:42
straight line/Linear equations have infinite ordered pair

please add/correct your views
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The number of ordered pairs of positive integers (x, y) satisfying the [#permalink] New post   28 Feb 2020, 03:50
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Bunuel wrote:
The number of ordered pairs of positive integers (x, y) satisfying the equation 7x + 3y = 123 is:

A. 3
B. 5
C 12
D. 13
E. Infinite


Are You Up For the Challenge: 700 Level Questions


7x + 3y = 123

123 is divisible by 3 i.e. 7x should also be divisible by 3
so first pair of (x, y) = (3, 34)

Now, next value of x will differ by co-efficient of y which is 3 and next value of y will differ by co-efficient of x which is 7

Now, if x increased by 3 then y will decrease by 7

i.e. 7*4 = 28 is maximum that can be subtracted from 34 to keep it positive

So total Solutions = 1+4 = 5

Answer: Option B
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Re: The number of ordered pairs of positive integers (x, y) satisfying the [#permalink] New post   28 Feb 2020, 03:52
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mdsaddamforgmat wrote:
straight line/Linear equations have infinite ordered pair

please add/correct your views



Question mentioned positive integer solution.

I hope this helps!!! :)
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Re: The number of ordered pairs of positive integers (x, y) satisfying the [#permalink] New post   28 Feb 2020, 04:04
Oh... Silly mistake

Posted from my mobile device
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The number of ordered pairs of positive integers (x, y) satisfying the [#permalink] New post   28 Feb 2020, 05:51
Bunuel wrote:
The number of ordered pairs of positive integers (x, y) satisfying the equation 7x + 3y = 123 is:

A. 3
B. 5
C 12
D. 13
E. Infinite


Are You Up For the Challenge: 700 Level Questions


N0te that 123 is divisible by 3. so Any 7x where x> 0, x is multiple of 3, and 7x < 123 will satisfy the given equation. Max value of such 7x = 105 where x =5.
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Re: The number of ordered pairs of positive integers (x, y) satisfying the [#permalink] New post   01 Mar 2020, 02:26
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The number of ordered pairs of positive integers (x, y) satisfying the equation 7x + 3y = 123 is:

A. 3
B. 5
C. 12
D. 13
E. Infinite

Solution: 7x + 3y = 123
3y = 123-7x
Y = 41-7x/3 ……………….(1)
To be integer the value of y, 41-7x/3 must be integer. So the value of x must be multiple of 3.
When, x = 3,6,9,12,15, then, y = positive integer.
When x = 18, y = 41 – 7/3(18)
• y = 41 – 42 = -1, Not acceptable.
Therefore, there are 5 ordered pair of positive integers.
Answer: B
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The number of ordered pairs of positive integers (x, y) satisfying the [#permalink] New post   28 Mar 2020, 06:25
Bunuel wrote:
The number of ordered pairs of positive integers (x, y) satisfying the equation 7x + 3y = 123 is:

A. 3
B. 5
C. 12
D. 13
E. Infinite



Are You Up For the Challenge: 700 Level Questions


Asked: The number of ordered pairs of positive integers (x, y) satisfying the equation 7x + 3y = 123 is:

7x + 3y = 123
x = (123-3y)/7

(x,y) = (9,20) is a solution
x will change by 3 and y will change in opposite sign by 7 for each solution

(x,y) = {(3,34), (6,27), (9,20), (12,13),(15,6)}

IMO B
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Re: The number of ordered pairs of positive integers (x, y) satisfying the [#permalink] New post   28 Mar 2020, 06:26
mdsaddamforgmat wrote:
straight line/Linear equations have infinite ordered pair

please add/correct your views


But it is mentioned that solution is valid for only positive integers.
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Re: The number of ordered pairs of positive integers (x, y) satisfying the [#permalink]
28 Mar 2020, 06:26
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