Find integer solution [on hold]Solution of a system of linear equationsLinear systems. Please help me solve thisHint: Approach for the solution of the following three equationsSystem of Linear Equations with integer CoefficientsFor what value of $k$ does the linear system have a unique/infinite/no solutionsProblem trying to find kernel of a linear transformationFind the specific values for KHow does one find the solution y(n) from y(n+1)? (System of Difference Equations)System of equations. Find when it has one solution, multiple solutions or no solutionsFind the values of $a$, for which this system of linear equations has one solution, no solution, or infinite solutions

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Find integer solution [on hold]


Solution of a system of linear equationsLinear systems. Please help me solve thisHint: Approach for the solution of the following three equationsSystem of Linear Equations with integer CoefficientsFor what value of $k$ does the linear system have a unique/infinite/no solutionsProblem trying to find kernel of a linear transformationFind the specific values for KHow does one find the solution y(n) from y(n+1)? (System of Difference Equations)System of equations. Find when it has one solution, multiple solutions or no solutionsFind the values of $a$, for which this system of linear equations has one solution, no solution, or infinite solutions













0












$begingroup$


I have given $5$ integers from $1$ to $5$ which are unique. They solve the following equations:



$$beginalign
c+d&=b+e \
b+c&=a+e \
a-d&=e-2
endalign$$



However, I can't find the solution. I have tried various combinations but none seem to work.










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$endgroup$



put on hold as off-topic by TheSimpliFire, user21820, Servaes, Carl Mummert, José Carlos Santos 2 days ago


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – TheSimpliFire, user21820, Servaes, Carl Mummert, José Carlos Santos
If this question can be reworded to fit the rules in the help center, please edit the question.















  • $begingroup$
    I would sort the variables,$$a,b,c,d,e$$
    $endgroup$
    – Dr. Sonnhard Graubner
    2 days ago















0












$begingroup$


I have given $5$ integers from $1$ to $5$ which are unique. They solve the following equations:



$$beginalign
c+d&=b+e \
b+c&=a+e \
a-d&=e-2
endalign$$



However, I can't find the solution. I have tried various combinations but none seem to work.










share|cite|improve this question











$endgroup$



put on hold as off-topic by TheSimpliFire, user21820, Servaes, Carl Mummert, José Carlos Santos 2 days ago


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – TheSimpliFire, user21820, Servaes, Carl Mummert, José Carlos Santos
If this question can be reworded to fit the rules in the help center, please edit the question.















  • $begingroup$
    I would sort the variables,$$a,b,c,d,e$$
    $endgroup$
    – Dr. Sonnhard Graubner
    2 days ago













0












0








0





$begingroup$


I have given $5$ integers from $1$ to $5$ which are unique. They solve the following equations:



$$beginalign
c+d&=b+e \
b+c&=a+e \
a-d&=e-2
endalign$$



However, I can't find the solution. I have tried various combinations but none seem to work.










share|cite|improve this question











$endgroup$




I have given $5$ integers from $1$ to $5$ which are unique. They solve the following equations:



$$beginalign
c+d&=b+e \
b+c&=a+e \
a-d&=e-2
endalign$$



However, I can't find the solution. I have tried various combinations but none seem to work.







linear-algebra systems-of-equations






share|cite|improve this question















share|cite|improve this question













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share|cite|improve this question








edited 2 days ago









Rodrigo de Azevedo

13k41960




13k41960










asked 2 days ago









ToribashToribash

375




375




put on hold as off-topic by TheSimpliFire, user21820, Servaes, Carl Mummert, José Carlos Santos 2 days ago


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – TheSimpliFire, user21820, Servaes, Carl Mummert, José Carlos Santos
If this question can be reworded to fit the rules in the help center, please edit the question.







put on hold as off-topic by TheSimpliFire, user21820, Servaes, Carl Mummert, José Carlos Santos 2 days ago


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – TheSimpliFire, user21820, Servaes, Carl Mummert, José Carlos Santos
If this question can be reworded to fit the rules in the help center, please edit the question.











  • $begingroup$
    I would sort the variables,$$a,b,c,d,e$$
    $endgroup$
    – Dr. Sonnhard Graubner
    2 days ago
















  • $begingroup$
    I would sort the variables,$$a,b,c,d,e$$
    $endgroup$
    – Dr. Sonnhard Graubner
    2 days ago















$begingroup$
I would sort the variables,$$a,b,c,d,e$$
$endgroup$
– Dr. Sonnhard Graubner
2 days ago




$begingroup$
I would sort the variables,$$a,b,c,d,e$$
$endgroup$
– Dr. Sonnhard Graubner
2 days ago










4 Answers
4






active

oldest

votes


















2












$begingroup$

Since $d+e = a + 2$, we have:



$$2b + e = b + c + d = a + e + d = 2a + 2$$



so that $e = 2a + 2 - 2b$, and $d = -a + 2b$. We thus have:



$$c = a + e - b = 3a + 2 - 3b$$



so that we have a 2-parameter set of solutions.



In summary, we have $a = a, b = b, c = 3a - 3b + 2, d = -a + 2b$, and $e = 2a - 2b + 2$.



Now in order to find the required answer, we run through the possible values for $a, b$ in order to ensure that $c, d, e$ are in the required range and that $a, b, c, d, e$ are all distinct. One sees that $a = 3, b = 2$ works.



As a sanity check:



$$a = 3$$
$$b = 2$$
$$c = 3^*3 - 3^*2 + 2 = 5$$
$$d = - 3 + 2^*2 = 1$$
$$e = 2^*3 - 2^*2 + 2 = 4$$



and so $c + d = 6 = b + e$, $b + c = 7 = a + e$, and $a - d = 2 = e - 2$, as required.






share|cite|improve this answer









$endgroup$




















    1












    $begingroup$

    Using the three equations in the order they are stated you get
    $$c=b+e-d=(a+e-c)+e-d=((d+e-2)+e-c)+e-d$$
    and, hence,
    $$2(c+1)=3e.$$
    This proves that $c+1$ must be divisible by 3 and $e$ must be divisible by 2. If $c+1$ were equal to 3, then $c$ would have to be equal to 2 and, hence, $e$ would have to be equal to 2. Since $c=e$ is not allowed, we must have $c=5$ and $e=4$. It is now easy to derive the values for $a$, $b$, and $d$ from the three equations: $a=3$, $b=2$, and $d=1$.






    share|cite|improve this answer









    $endgroup$




















      1












      $begingroup$

      I have got $$a=C_1,b=C_2,c=2+3C_1,d=-C_1+2C_2,e=2+2C_1-2C_2$$ where $$C_1,C_2$$ are integer numbers.






      share|cite|improve this answer









      $endgroup$




















        0












        $begingroup$

        Brute-forcing in Haskell:



        λ> filter ((a,b,c,d,e) -> c+d==b+e && b+c==a+e && a-d==e-2) [ (a,b,c,d,e) | a <- [1..5], b <- [1..5], c <- [1..5], d <- [1..5], e <- [1..5] ]
        [(1,1,2,1,2),(2,2,2,2,2),(3,2,5,1,4),(3,3,2,3,2),(4,3,5,2,4),(4,4,2,4,2),(5,4,5,3,4),(5,5,2,5,2)]


        Of the eight solutions, the only $5$-tuple that is admissible (no repetitions) is $(3,2,5,1,4)$, which is the solution found by Nethesis and Gerhard.






        share|cite|improve this answer









        $endgroup$



















          4 Answers
          4






          active

          oldest

          votes








          4 Answers
          4






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          2












          $begingroup$

          Since $d+e = a + 2$, we have:



          $$2b + e = b + c + d = a + e + d = 2a + 2$$



          so that $e = 2a + 2 - 2b$, and $d = -a + 2b$. We thus have:



          $$c = a + e - b = 3a + 2 - 3b$$



          so that we have a 2-parameter set of solutions.



          In summary, we have $a = a, b = b, c = 3a - 3b + 2, d = -a + 2b$, and $e = 2a - 2b + 2$.



          Now in order to find the required answer, we run through the possible values for $a, b$ in order to ensure that $c, d, e$ are in the required range and that $a, b, c, d, e$ are all distinct. One sees that $a = 3, b = 2$ works.



          As a sanity check:



          $$a = 3$$
          $$b = 2$$
          $$c = 3^*3 - 3^*2 + 2 = 5$$
          $$d = - 3 + 2^*2 = 1$$
          $$e = 2^*3 - 2^*2 + 2 = 4$$



          and so $c + d = 6 = b + e$, $b + c = 7 = a + e$, and $a - d = 2 = e - 2$, as required.






          share|cite|improve this answer









          $endgroup$

















            2












            $begingroup$

            Since $d+e = a + 2$, we have:



            $$2b + e = b + c + d = a + e + d = 2a + 2$$



            so that $e = 2a + 2 - 2b$, and $d = -a + 2b$. We thus have:



            $$c = a + e - b = 3a + 2 - 3b$$



            so that we have a 2-parameter set of solutions.



            In summary, we have $a = a, b = b, c = 3a - 3b + 2, d = -a + 2b$, and $e = 2a - 2b + 2$.



            Now in order to find the required answer, we run through the possible values for $a, b$ in order to ensure that $c, d, e$ are in the required range and that $a, b, c, d, e$ are all distinct. One sees that $a = 3, b = 2$ works.



            As a sanity check:



            $$a = 3$$
            $$b = 2$$
            $$c = 3^*3 - 3^*2 + 2 = 5$$
            $$d = - 3 + 2^*2 = 1$$
            $$e = 2^*3 - 2^*2 + 2 = 4$$



            and so $c + d = 6 = b + e$, $b + c = 7 = a + e$, and $a - d = 2 = e - 2$, as required.






            share|cite|improve this answer









            $endgroup$















              2












              2








              2





              $begingroup$

              Since $d+e = a + 2$, we have:



              $$2b + e = b + c + d = a + e + d = 2a + 2$$



              so that $e = 2a + 2 - 2b$, and $d = -a + 2b$. We thus have:



              $$c = a + e - b = 3a + 2 - 3b$$



              so that we have a 2-parameter set of solutions.



              In summary, we have $a = a, b = b, c = 3a - 3b + 2, d = -a + 2b$, and $e = 2a - 2b + 2$.



              Now in order to find the required answer, we run through the possible values for $a, b$ in order to ensure that $c, d, e$ are in the required range and that $a, b, c, d, e$ are all distinct. One sees that $a = 3, b = 2$ works.



              As a sanity check:



              $$a = 3$$
              $$b = 2$$
              $$c = 3^*3 - 3^*2 + 2 = 5$$
              $$d = - 3 + 2^*2 = 1$$
              $$e = 2^*3 - 2^*2 + 2 = 4$$



              and so $c + d = 6 = b + e$, $b + c = 7 = a + e$, and $a - d = 2 = e - 2$, as required.






              share|cite|improve this answer









              $endgroup$



              Since $d+e = a + 2$, we have:



              $$2b + e = b + c + d = a + e + d = 2a + 2$$



              so that $e = 2a + 2 - 2b$, and $d = -a + 2b$. We thus have:



              $$c = a + e - b = 3a + 2 - 3b$$



              so that we have a 2-parameter set of solutions.



              In summary, we have $a = a, b = b, c = 3a - 3b + 2, d = -a + 2b$, and $e = 2a - 2b + 2$.



              Now in order to find the required answer, we run through the possible values for $a, b$ in order to ensure that $c, d, e$ are in the required range and that $a, b, c, d, e$ are all distinct. One sees that $a = 3, b = 2$ works.



              As a sanity check:



              $$a = 3$$
              $$b = 2$$
              $$c = 3^*3 - 3^*2 + 2 = 5$$
              $$d = - 3 + 2^*2 = 1$$
              $$e = 2^*3 - 2^*2 + 2 = 4$$



              and so $c + d = 6 = b + e$, $b + c = 7 = a + e$, and $a - d = 2 = e - 2$, as required.







              share|cite|improve this answer












              share|cite|improve this answer



              share|cite|improve this answer










              answered 2 days ago









              NethesisNethesis

              1,9121823




              1,9121823





















                  1












                  $begingroup$

                  Using the three equations in the order they are stated you get
                  $$c=b+e-d=(a+e-c)+e-d=((d+e-2)+e-c)+e-d$$
                  and, hence,
                  $$2(c+1)=3e.$$
                  This proves that $c+1$ must be divisible by 3 and $e$ must be divisible by 2. If $c+1$ were equal to 3, then $c$ would have to be equal to 2 and, hence, $e$ would have to be equal to 2. Since $c=e$ is not allowed, we must have $c=5$ and $e=4$. It is now easy to derive the values for $a$, $b$, and $d$ from the three equations: $a=3$, $b=2$, and $d=1$.






                  share|cite|improve this answer









                  $endgroup$

















                    1












                    $begingroup$

                    Using the three equations in the order they are stated you get
                    $$c=b+e-d=(a+e-c)+e-d=((d+e-2)+e-c)+e-d$$
                    and, hence,
                    $$2(c+1)=3e.$$
                    This proves that $c+1$ must be divisible by 3 and $e$ must be divisible by 2. If $c+1$ were equal to 3, then $c$ would have to be equal to 2 and, hence, $e$ would have to be equal to 2. Since $c=e$ is not allowed, we must have $c=5$ and $e=4$. It is now easy to derive the values for $a$, $b$, and $d$ from the three equations: $a=3$, $b=2$, and $d=1$.






                    share|cite|improve this answer









                    $endgroup$















                      1












                      1








                      1





                      $begingroup$

                      Using the three equations in the order they are stated you get
                      $$c=b+e-d=(a+e-c)+e-d=((d+e-2)+e-c)+e-d$$
                      and, hence,
                      $$2(c+1)=3e.$$
                      This proves that $c+1$ must be divisible by 3 and $e$ must be divisible by 2. If $c+1$ were equal to 3, then $c$ would have to be equal to 2 and, hence, $e$ would have to be equal to 2. Since $c=e$ is not allowed, we must have $c=5$ and $e=4$. It is now easy to derive the values for $a$, $b$, and $d$ from the three equations: $a=3$, $b=2$, and $d=1$.






                      share|cite|improve this answer









                      $endgroup$



                      Using the three equations in the order they are stated you get
                      $$c=b+e-d=(a+e-c)+e-d=((d+e-2)+e-c)+e-d$$
                      and, hence,
                      $$2(c+1)=3e.$$
                      This proves that $c+1$ must be divisible by 3 and $e$ must be divisible by 2. If $c+1$ were equal to 3, then $c$ would have to be equal to 2 and, hence, $e$ would have to be equal to 2. Since $c=e$ is not allowed, we must have $c=5$ and $e=4$. It is now easy to derive the values for $a$, $b$, and $d$ from the three equations: $a=3$, $b=2$, and $d=1$.







                      share|cite|improve this answer












                      share|cite|improve this answer



                      share|cite|improve this answer










                      answered 2 days ago









                      Gerhard S.Gerhard S.

                      1,07529




                      1,07529





















                          1












                          $begingroup$

                          I have got $$a=C_1,b=C_2,c=2+3C_1,d=-C_1+2C_2,e=2+2C_1-2C_2$$ where $$C_1,C_2$$ are integer numbers.






                          share|cite|improve this answer









                          $endgroup$

















                            1












                            $begingroup$

                            I have got $$a=C_1,b=C_2,c=2+3C_1,d=-C_1+2C_2,e=2+2C_1-2C_2$$ where $$C_1,C_2$$ are integer numbers.






                            share|cite|improve this answer









                            $endgroup$















                              1












                              1








                              1





                              $begingroup$

                              I have got $$a=C_1,b=C_2,c=2+3C_1,d=-C_1+2C_2,e=2+2C_1-2C_2$$ where $$C_1,C_2$$ are integer numbers.






                              share|cite|improve this answer









                              $endgroup$



                              I have got $$a=C_1,b=C_2,c=2+3C_1,d=-C_1+2C_2,e=2+2C_1-2C_2$$ where $$C_1,C_2$$ are integer numbers.







                              share|cite|improve this answer












                              share|cite|improve this answer



                              share|cite|improve this answer










                              answered 2 days ago









                              Dr. Sonnhard GraubnerDr. Sonnhard Graubner

                              77.7k42866




                              77.7k42866





















                                  0












                                  $begingroup$

                                  Brute-forcing in Haskell:



                                  λ> filter ((a,b,c,d,e) -> c+d==b+e && b+c==a+e && a-d==e-2) [ (a,b,c,d,e) | a <- [1..5], b <- [1..5], c <- [1..5], d <- [1..5], e <- [1..5] ]
                                  [(1,1,2,1,2),(2,2,2,2,2),(3,2,5,1,4),(3,3,2,3,2),(4,3,5,2,4),(4,4,2,4,2),(5,4,5,3,4),(5,5,2,5,2)]


                                  Of the eight solutions, the only $5$-tuple that is admissible (no repetitions) is $(3,2,5,1,4)$, which is the solution found by Nethesis and Gerhard.






                                  share|cite|improve this answer









                                  $endgroup$

















                                    0












                                    $begingroup$

                                    Brute-forcing in Haskell:



                                    λ> filter ((a,b,c,d,e) -> c+d==b+e && b+c==a+e && a-d==e-2) [ (a,b,c,d,e) | a <- [1..5], b <- [1..5], c <- [1..5], d <- [1..5], e <- [1..5] ]
                                    [(1,1,2,1,2),(2,2,2,2,2),(3,2,5,1,4),(3,3,2,3,2),(4,3,5,2,4),(4,4,2,4,2),(5,4,5,3,4),(5,5,2,5,2)]


                                    Of the eight solutions, the only $5$-tuple that is admissible (no repetitions) is $(3,2,5,1,4)$, which is the solution found by Nethesis and Gerhard.






                                    share|cite|improve this answer









                                    $endgroup$















                                      0












                                      0








                                      0





                                      $begingroup$

                                      Brute-forcing in Haskell:



                                      λ> filter ((a,b,c,d,e) -> c+d==b+e && b+c==a+e && a-d==e-2) [ (a,b,c,d,e) | a <- [1..5], b <- [1..5], c <- [1..5], d <- [1..5], e <- [1..5] ]
                                      [(1,1,2,1,2),(2,2,2,2,2),(3,2,5,1,4),(3,3,2,3,2),(4,3,5,2,4),(4,4,2,4,2),(5,4,5,3,4),(5,5,2,5,2)]


                                      Of the eight solutions, the only $5$-tuple that is admissible (no repetitions) is $(3,2,5,1,4)$, which is the solution found by Nethesis and Gerhard.






                                      share|cite|improve this answer









                                      $endgroup$



                                      Brute-forcing in Haskell:



                                      λ> filter ((a,b,c,d,e) -> c+d==b+e && b+c==a+e && a-d==e-2) [ (a,b,c,d,e) | a <- [1..5], b <- [1..5], c <- [1..5], d <- [1..5], e <- [1..5] ]
                                      [(1,1,2,1,2),(2,2,2,2,2),(3,2,5,1,4),(3,3,2,3,2),(4,3,5,2,4),(4,4,2,4,2),(5,4,5,3,4),(5,5,2,5,2)]


                                      Of the eight solutions, the only $5$-tuple that is admissible (no repetitions) is $(3,2,5,1,4)$, which is the solution found by Nethesis and Gerhard.







                                      share|cite|improve this answer












                                      share|cite|improve this answer



                                      share|cite|improve this answer










                                      answered 2 days ago









                                      Rodrigo de AzevedoRodrigo de Azevedo

                                      13k41960




                                      13k41960













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