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mathmath333

  • one year ago

Find the number positive integer solutions of 4x+6y=200

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  1. mathmath333
    • one year ago
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    \(\large \color{black}{\begin{align} & \normalsize \text{Find the number positive integer solutions of}\hspace{.33em}\\~\\ & 4x+6y=200 \end{align}}\)

  2. mathmath333
    • one year ago
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    I need to find quick way

  3. Empty
    • one year ago
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    I don't really know but you should look into the Euclidean Algorithm I think that might be a good path to start looking into.

  4. anonymous
    • one year ago
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    as this is Diophantine equation i agree with @Empty

  5. anonymous
    • one year ago
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    so lets try 2x+3y=100

  6. ganeshie8
    • one year ago
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    yeah both equations are same but we don't really need euclid algorithm here as finding a particular solution seems kinda easy by inspection. By inspection, (50, 0) is one particular solution

  7. anonymous
    • one year ago
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    but Euclid would help in finding general solution to show it's infinite solutions right ?

  8. ganeshie8
    • one year ago
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    euclid is for finding particular solution since we already figured that (50, 0) is a particular soltion, we can avoid euclid

  9. anonymous
    • one year ago
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    :3

  10. mathmath333
    • one year ago
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    I found out (50,0),(47,2),.....(2,32) but that took more than 2 min

  11. ganeshie8
    • one year ago
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    finding "one" particular solution is enough

  12. anonymous
    • one year ago
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    anyway i like this way to show its infinite:- |dw:1442748565442:dw|

  13. ganeshie8
    • one year ago
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    after you have one particular solution, try finding the "null" solution : 2x + 3y = 0 Easy to see that (-3, 2) solves above equation. Therefore the complete solution is given by : `(50, 0) + t(-3, 2)`

  14. ganeshie8
    • one year ago
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    since you want just the positive integer solutions, solve : 50 - 3t > 0 0 + 2t > 0

  15. mathmath333
    • one year ago
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    50/3=16.

  16. mathmath333
    • one year ago
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    answer is 17

  17. ganeshie8
    • one year ago
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    right, solve it simultaneously you should get an interval of "t" as solution

  18. anonymous
    • one year ago
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    aha i haven't note positive :O

  19. ganeshie8
    • one year ago
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    I think the answer should be 16

  20. mathmath333
    • one year ago
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    0<t<16

  21. ganeshie8
    • one year ago
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    nope 0 < t < 16.66 there are exactly 16 positive integers in that interval

  22. mathmath333
    • one year ago
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    0<t<17

  23. ganeshie8
    • one year ago
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    0 < t < 50/3 leave it like that

  24. mathmath333
    • one year ago
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    ok u r right (50,0) doesn't count answer is 16

  25. ganeshie8
    • one year ago
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    Yep! lets do one more example maybe ?

  26. ganeshie8
    • one year ago
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    Find the number of positive integer solutions to the equation 7x + 13y = 700

  27. ganeshie8
    • one year ago
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    If you prefer, here are the steps : 1) Find any one `particular` solution by inspection 2) Find the `null` solution 3) Write out the complete solution : `particular` + `null`

  28. mathmath333
    • one year ago
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    (100,0)

  29. ganeshie8
    • one year ago
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    Yep, keep going

  30. mathmath333
    • one year ago
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    how to find null soln

  31. ganeshie8
    • one year ago
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    as the name says, it is the solution to the equation 7x + 13y = 0

  32. ganeshie8
    • one year ago
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    give it a try.. it would feel awesome if you figure out a method to find the null solution on ur own..

  33. mathmath333
    • one year ago
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    (-13,-7)

  34. ganeshie8
    • one year ago
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    Very close, but no. plug them in and see if they really produce 0

  35. mathmath333
    • one year ago
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    (-13,7)

  36. ganeshie8
    • one year ago
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    Excellent! that is one null solution. Notice that any multiple of that also works, so all the null solutions are given by ` t(-13, 7)` where ` t ` belongs to the set of integers

  37. mathmath333
    • one year ago
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    100-13t>0 0+7t>0

  38. ganeshie8
    • one year ago
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    Yes, you have skipped step3 but ok..

  39. ganeshie8
    • one year ago
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    go ahead find the total count

  40. mathmath333
    • one year ago
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    what is step 3 ?

  41. mathmath333
    • one year ago
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    oh this one "3) Write out the complete solution : particular + null"

  42. ganeshie8
    • one year ago
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    Yes, I was refering to that..

  43. mathmath333
    • one year ago
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    7 is answer

  44. ganeshie8
    • one year ago
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    Yep! congratulations! Now you know how to solve any linear diophantine equation of form \(ax+by=c\)

  45. mathmath333
    • one year ago
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    cool!

  46. ganeshie8
    • one year ago
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    so what was your method for finding null solution ?

  47. mathmath333
    • one year ago
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    ax + by = 0 (-b,a)

  48. ganeshie8
    • one year ago
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    thats it!

  49. ganeshie8
    • one year ago
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    that works always! so finding null solution is trivial as you can seethe only hard part is finding a particular solution

  50. anonymous
    • one year ago
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    6y=200-4x 3y=100-2x \[y=\frac{ 100-2x }{ 3 }\] by hit and trial when x=2 \[y=\frac{ 100-4 }{ 3 }=32\] add successively 3 to the value of x x=2+3=5,y=30 x=5+3=8,y=28 x=11,y=26 .... x=47,y=2 x=50,y=0

  51. anonymous
    • one year ago
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    .

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