Number of solutions in linear equation with 3 variables











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Is there a way how to determine number of solutions in linear equation like this: $ax + by + cz = d$, where $a,b,c,x,y,z,d$ are non-negative integers and $a,b,c,d$ are known?










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  • 2




    You can use generating functions and look at the coefficient of $X^d$ in the expansion of $(1+X^a+X^{2a}+X^{3a}+dots)(1+X^b+X^{2b}+dots)(1+X^c+X^{2c}+dots)$
    – JMoravitz
    Nov 14 at 19:50










  • See also this question and similar ones.
    – Dietrich Burde
    Nov 14 at 19:57















up vote
1
down vote

favorite












Is there a way how to determine number of solutions in linear equation like this: $ax + by + cz = d$, where $a,b,c,x,y,z,d$ are non-negative integers and $a,b,c,d$ are known?










share|cite|improve this question




















  • 2




    You can use generating functions and look at the coefficient of $X^d$ in the expansion of $(1+X^a+X^{2a}+X^{3a}+dots)(1+X^b+X^{2b}+dots)(1+X^c+X^{2c}+dots)$
    – JMoravitz
    Nov 14 at 19:50










  • See also this question and similar ones.
    – Dietrich Burde
    Nov 14 at 19:57













up vote
1
down vote

favorite









up vote
1
down vote

favorite











Is there a way how to determine number of solutions in linear equation like this: $ax + by + cz = d$, where $a,b,c,x,y,z,d$ are non-negative integers and $a,b,c,d$ are known?










share|cite|improve this question















Is there a way how to determine number of solutions in linear equation like this: $ax + by + cz = d$, where $a,b,c,x,y,z,d$ are non-negative integers and $a,b,c,d$ are known?







diophantine-equations






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edited Nov 14 at 20:10









Alex D

494218




494218










asked Nov 14 at 19:47









Andreisk

61




61








  • 2




    You can use generating functions and look at the coefficient of $X^d$ in the expansion of $(1+X^a+X^{2a}+X^{3a}+dots)(1+X^b+X^{2b}+dots)(1+X^c+X^{2c}+dots)$
    – JMoravitz
    Nov 14 at 19:50










  • See also this question and similar ones.
    – Dietrich Burde
    Nov 14 at 19:57














  • 2




    You can use generating functions and look at the coefficient of $X^d$ in the expansion of $(1+X^a+X^{2a}+X^{3a}+dots)(1+X^b+X^{2b}+dots)(1+X^c+X^{2c}+dots)$
    – JMoravitz
    Nov 14 at 19:50










  • See also this question and similar ones.
    – Dietrich Burde
    Nov 14 at 19:57








2




2




You can use generating functions and look at the coefficient of $X^d$ in the expansion of $(1+X^a+X^{2a}+X^{3a}+dots)(1+X^b+X^{2b}+dots)(1+X^c+X^{2c}+dots)$
– JMoravitz
Nov 14 at 19:50




You can use generating functions and look at the coefficient of $X^d$ in the expansion of $(1+X^a+X^{2a}+X^{3a}+dots)(1+X^b+X^{2b}+dots)(1+X^c+X^{2c}+dots)$
– JMoravitz
Nov 14 at 19:50












See also this question and similar ones.
– Dietrich Burde
Nov 14 at 19:57




See also this question and similar ones.
– Dietrich Burde
Nov 14 at 19:57










1 Answer
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Above equation shown below:



$ax + by + cz = d$ -------$(1)$



For (a, b, c, d)= (3, 2, 5, 32), equation $(1)$ has parametric solution given below:



$x=(1-v)$



$y=(-5u+9v+87)$



$z=(2u-3v-29)$



For $(u,v)$ = $(15, 0)$ we get,



$(x, y, z)$ = $(1, 12, 1)$






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    1 Answer
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    1 Answer
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    Above equation shown below:



    $ax + by + cz = d$ -------$(1)$



    For (a, b, c, d)= (3, 2, 5, 32), equation $(1)$ has parametric solution given below:



    $x=(1-v)$



    $y=(-5u+9v+87)$



    $z=(2u-3v-29)$



    For $(u,v)$ = $(15, 0)$ we get,



    $(x, y, z)$ = $(1, 12, 1)$






    share|cite|improve this answer








    New contributor




    Sam is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.






















      up vote
      0
      down vote













      Above equation shown below:



      $ax + by + cz = d$ -------$(1)$



      For (a, b, c, d)= (3, 2, 5, 32), equation $(1)$ has parametric solution given below:



      $x=(1-v)$



      $y=(-5u+9v+87)$



      $z=(2u-3v-29)$



      For $(u,v)$ = $(15, 0)$ we get,



      $(x, y, z)$ = $(1, 12, 1)$






      share|cite|improve this answer








      New contributor




      Sam is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.




















        up vote
        0
        down vote










        up vote
        0
        down vote









        Above equation shown below:



        $ax + by + cz = d$ -------$(1)$



        For (a, b, c, d)= (3, 2, 5, 32), equation $(1)$ has parametric solution given below:



        $x=(1-v)$



        $y=(-5u+9v+87)$



        $z=(2u-3v-29)$



        For $(u,v)$ = $(15, 0)$ we get,



        $(x, y, z)$ = $(1, 12, 1)$






        share|cite|improve this answer








        New contributor




        Sam is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
        Check out our Code of Conduct.









        Above equation shown below:



        $ax + by + cz = d$ -------$(1)$



        For (a, b, c, d)= (3, 2, 5, 32), equation $(1)$ has parametric solution given below:



        $x=(1-v)$



        $y=(-5u+9v+87)$



        $z=(2u-3v-29)$



        For $(u,v)$ = $(15, 0)$ we get,



        $(x, y, z)$ = $(1, 12, 1)$







        share|cite|improve this answer








        New contributor




        Sam is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
        Check out our Code of Conduct.









        share|cite|improve this answer



        share|cite|improve this answer






        New contributor




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        answered Nov 15 at 16:15









        Sam

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