Equation solving?











up vote
0
down vote

favorite












I've got this system of equations.
(a,b,c) ∈ ℤ



$a*b+1 = c$



$a^2 + b^2 +1 = 2c$



$2a + b = c$



I tried to substitute a little bit:



$a^2 + b^2 + 1 -(2a+b) - (a*b+1) = 0$



Ultimately:



$a(a-2-b) + b(b-1) = 0$



Now I'm not sure how to substitute b into a in order to go on.



Maybe I didn't even start well.
Can someone help?



Thx










share|cite|improve this question




























    up vote
    0
    down vote

    favorite












    I've got this system of equations.
    (a,b,c) ∈ ℤ



    $a*b+1 = c$



    $a^2 + b^2 +1 = 2c$



    $2a + b = c$



    I tried to substitute a little bit:



    $a^2 + b^2 + 1 -(2a+b) - (a*b+1) = 0$



    Ultimately:



    $a(a-2-b) + b(b-1) = 0$



    Now I'm not sure how to substitute b into a in order to go on.



    Maybe I didn't even start well.
    Can someone help?



    Thx










    share|cite|improve this question


























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      I've got this system of equations.
      (a,b,c) ∈ ℤ



      $a*b+1 = c$



      $a^2 + b^2 +1 = 2c$



      $2a + b = c$



      I tried to substitute a little bit:



      $a^2 + b^2 + 1 -(2a+b) - (a*b+1) = 0$



      Ultimately:



      $a(a-2-b) + b(b-1) = 0$



      Now I'm not sure how to substitute b into a in order to go on.



      Maybe I didn't even start well.
      Can someone help?



      Thx










      share|cite|improve this question















      I've got this system of equations.
      (a,b,c) ∈ ℤ



      $a*b+1 = c$



      $a^2 + b^2 +1 = 2c$



      $2a + b = c$



      I tried to substitute a little bit:



      $a^2 + b^2 + 1 -(2a+b) - (a*b+1) = 0$



      Ultimately:



      $a(a-2-b) + b(b-1) = 0$



      Now I'm not sure how to substitute b into a in order to go on.



      Maybe I didn't even start well.
      Can someone help?



      Thx







      systems-of-equations quadratics






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Nov 23 at 12:00









      Harry Peter

      5,43111439




      5,43111439










      asked Nov 15 at 11:30









      calculatormathematical

      389




      389






















          3 Answers
          3






          active

          oldest

          votes

















          up vote
          2
          down vote













          You can obtain $|a-b|=1$ by substracting the first equation (where you multiply by 2 both left and right-hand side) from the second equation, so you get $a^2+b^2-2ab -1 = 0$. From this, you can write for instance $b$ and $c$ in function of $a$.






          share|cite|improve this answer




























            up vote
            0
            down vote













            Eliminating $c$ from the first two equations,



            $$(a-b)^2=1$$ or $$a=bpm1.$$



            Then eliminating $a$ and $c$ from the first and third, we get two quadratic,



            $$(b+1)b+1=2(b+1)+b\to b=1pmsqrt2,a=2pmsqrt2,c=5pm3sqrt2,$$



            $$(b-1)b+1=2(b-1)+b\to b=1,3, a=0,2, c=1,7.$$






            share|cite|improve this answer





















            • Ok but what if (a,b,c) ∈ ℤ
              – calculatormathematical
              Nov 15 at 11:54










            • @calculatormathematical: this constraint was added later. Can you tell an irrational from an integer ?
              – Yves Daoust
              Nov 15 at 14:44


















            up vote
            0
            down vote













            Here is a possible way to solve your problem.



            From the first and second equation, we have, $$a^2+b^2+1=2ab+2\or,(a-b)^2=1\or,a-b=pm1$$
            Now, using 3rd equation we get, $$a=frac{1}{3}(cpm 1)$$ So, $$b=amp1=frac{1}{3}(cmp 2)$$
            I have skipped the steps hope you find them. If not then let me know. Further, here I assumed that $a,b,c$ are real numbers and the meaning of $mp=-(pm)$.



            Hope it works.






            share|cite|improve this answer





















              Your Answer





              StackExchange.ifUsing("editor", function () {
              return StackExchange.using("mathjaxEditing", function () {
              StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
              StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
              });
              });
              }, "mathjax-editing");

              StackExchange.ready(function() {
              var channelOptions = {
              tags: "".split(" "),
              id: "69"
              };
              initTagRenderer("".split(" "), "".split(" "), channelOptions);

              StackExchange.using("externalEditor", function() {
              // Have to fire editor after snippets, if snippets enabled
              if (StackExchange.settings.snippets.snippetsEnabled) {
              StackExchange.using("snippets", function() {
              createEditor();
              });
              }
              else {
              createEditor();
              }
              });

              function createEditor() {
              StackExchange.prepareEditor({
              heartbeatType: 'answer',
              convertImagesToLinks: true,
              noModals: true,
              showLowRepImageUploadWarning: true,
              reputationToPostImages: 10,
              bindNavPrevention: true,
              postfix: "",
              imageUploader: {
              brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
              contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
              allowUrls: true
              },
              noCode: true, onDemand: true,
              discardSelector: ".discard-answer"
              ,immediatelyShowMarkdownHelp:true
              });


              }
              });














              draft saved

              draft discarded


















              StackExchange.ready(
              function () {
              StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2999578%2fequation-solving%23new-answer', 'question_page');
              }
              );

              Post as a guest















              Required, but never shown

























              3 Answers
              3






              active

              oldest

              votes








              3 Answers
              3






              active

              oldest

              votes









              active

              oldest

              votes






              active

              oldest

              votes








              up vote
              2
              down vote













              You can obtain $|a-b|=1$ by substracting the first equation (where you multiply by 2 both left and right-hand side) from the second equation, so you get $a^2+b^2-2ab -1 = 0$. From this, you can write for instance $b$ and $c$ in function of $a$.






              share|cite|improve this answer

























                up vote
                2
                down vote













                You can obtain $|a-b|=1$ by substracting the first equation (where you multiply by 2 both left and right-hand side) from the second equation, so you get $a^2+b^2-2ab -1 = 0$. From this, you can write for instance $b$ and $c$ in function of $a$.






                share|cite|improve this answer























                  up vote
                  2
                  down vote










                  up vote
                  2
                  down vote









                  You can obtain $|a-b|=1$ by substracting the first equation (where you multiply by 2 both left and right-hand side) from the second equation, so you get $a^2+b^2-2ab -1 = 0$. From this, you can write for instance $b$ and $c$ in function of $a$.






                  share|cite|improve this answer












                  You can obtain $|a-b|=1$ by substracting the first equation (where you multiply by 2 both left and right-hand side) from the second equation, so you get $a^2+b^2-2ab -1 = 0$. From this, you can write for instance $b$ and $c$ in function of $a$.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Nov 15 at 11:41









                  Lacramioara

                  32126




                  32126






















                      up vote
                      0
                      down vote













                      Eliminating $c$ from the first two equations,



                      $$(a-b)^2=1$$ or $$a=bpm1.$$



                      Then eliminating $a$ and $c$ from the first and third, we get two quadratic,



                      $$(b+1)b+1=2(b+1)+b\to b=1pmsqrt2,a=2pmsqrt2,c=5pm3sqrt2,$$



                      $$(b-1)b+1=2(b-1)+b\to b=1,3, a=0,2, c=1,7.$$






                      share|cite|improve this answer





















                      • Ok but what if (a,b,c) ∈ ℤ
                        – calculatormathematical
                        Nov 15 at 11:54










                      • @calculatormathematical: this constraint was added later. Can you tell an irrational from an integer ?
                        – Yves Daoust
                        Nov 15 at 14:44















                      up vote
                      0
                      down vote













                      Eliminating $c$ from the first two equations,



                      $$(a-b)^2=1$$ or $$a=bpm1.$$



                      Then eliminating $a$ and $c$ from the first and third, we get two quadratic,



                      $$(b+1)b+1=2(b+1)+b\to b=1pmsqrt2,a=2pmsqrt2,c=5pm3sqrt2,$$



                      $$(b-1)b+1=2(b-1)+b\to b=1,3, a=0,2, c=1,7.$$






                      share|cite|improve this answer





















                      • Ok but what if (a,b,c) ∈ ℤ
                        – calculatormathematical
                        Nov 15 at 11:54










                      • @calculatormathematical: this constraint was added later. Can you tell an irrational from an integer ?
                        – Yves Daoust
                        Nov 15 at 14:44













                      up vote
                      0
                      down vote










                      up vote
                      0
                      down vote









                      Eliminating $c$ from the first two equations,



                      $$(a-b)^2=1$$ or $$a=bpm1.$$



                      Then eliminating $a$ and $c$ from the first and third, we get two quadratic,



                      $$(b+1)b+1=2(b+1)+b\to b=1pmsqrt2,a=2pmsqrt2,c=5pm3sqrt2,$$



                      $$(b-1)b+1=2(b-1)+b\to b=1,3, a=0,2, c=1,7.$$






                      share|cite|improve this answer












                      Eliminating $c$ from the first two equations,



                      $$(a-b)^2=1$$ or $$a=bpm1.$$



                      Then eliminating $a$ and $c$ from the first and third, we get two quadratic,



                      $$(b+1)b+1=2(b+1)+b\to b=1pmsqrt2,a=2pmsqrt2,c=5pm3sqrt2,$$



                      $$(b-1)b+1=2(b-1)+b\to b=1,3, a=0,2, c=1,7.$$







                      share|cite|improve this answer












                      share|cite|improve this answer



                      share|cite|improve this answer










                      answered Nov 15 at 11:38









                      Yves Daoust

                      122k668217




                      122k668217












                      • Ok but what if (a,b,c) ∈ ℤ
                        – calculatormathematical
                        Nov 15 at 11:54










                      • @calculatormathematical: this constraint was added later. Can you tell an irrational from an integer ?
                        – Yves Daoust
                        Nov 15 at 14:44


















                      • Ok but what if (a,b,c) ∈ ℤ
                        – calculatormathematical
                        Nov 15 at 11:54










                      • @calculatormathematical: this constraint was added later. Can you tell an irrational from an integer ?
                        – Yves Daoust
                        Nov 15 at 14:44
















                      Ok but what if (a,b,c) ∈ ℤ
                      – calculatormathematical
                      Nov 15 at 11:54




                      Ok but what if (a,b,c) ∈ ℤ
                      – calculatormathematical
                      Nov 15 at 11:54












                      @calculatormathematical: this constraint was added later. Can you tell an irrational from an integer ?
                      – Yves Daoust
                      Nov 15 at 14:44




                      @calculatormathematical: this constraint was added later. Can you tell an irrational from an integer ?
                      – Yves Daoust
                      Nov 15 at 14:44










                      up vote
                      0
                      down vote













                      Here is a possible way to solve your problem.



                      From the first and second equation, we have, $$a^2+b^2+1=2ab+2\or,(a-b)^2=1\or,a-b=pm1$$
                      Now, using 3rd equation we get, $$a=frac{1}{3}(cpm 1)$$ So, $$b=amp1=frac{1}{3}(cmp 2)$$
                      I have skipped the steps hope you find them. If not then let me know. Further, here I assumed that $a,b,c$ are real numbers and the meaning of $mp=-(pm)$.



                      Hope it works.






                      share|cite|improve this answer

























                        up vote
                        0
                        down vote













                        Here is a possible way to solve your problem.



                        From the first and second equation, we have, $$a^2+b^2+1=2ab+2\or,(a-b)^2=1\or,a-b=pm1$$
                        Now, using 3rd equation we get, $$a=frac{1}{3}(cpm 1)$$ So, $$b=amp1=frac{1}{3}(cmp 2)$$
                        I have skipped the steps hope you find them. If not then let me know. Further, here I assumed that $a,b,c$ are real numbers and the meaning of $mp=-(pm)$.



                        Hope it works.






                        share|cite|improve this answer























                          up vote
                          0
                          down vote










                          up vote
                          0
                          down vote









                          Here is a possible way to solve your problem.



                          From the first and second equation, we have, $$a^2+b^2+1=2ab+2\or,(a-b)^2=1\or,a-b=pm1$$
                          Now, using 3rd equation we get, $$a=frac{1}{3}(cpm 1)$$ So, $$b=amp1=frac{1}{3}(cmp 2)$$
                          I have skipped the steps hope you find them. If not then let me know. Further, here I assumed that $a,b,c$ are real numbers and the meaning of $mp=-(pm)$.



                          Hope it works.






                          share|cite|improve this answer












                          Here is a possible way to solve your problem.



                          From the first and second equation, we have, $$a^2+b^2+1=2ab+2\or,(a-b)^2=1\or,a-b=pm1$$
                          Now, using 3rd equation we get, $$a=frac{1}{3}(cpm 1)$$ So, $$b=amp1=frac{1}{3}(cmp 2)$$
                          I have skipped the steps hope you find them. If not then let me know. Further, here I assumed that $a,b,c$ are real numbers and the meaning of $mp=-(pm)$.



                          Hope it works.







                          share|cite|improve this answer












                          share|cite|improve this answer



                          share|cite|improve this answer










                          answered Nov 15 at 11:40









                          Sujit Bhattacharyya

                          883316




                          883316






























                              draft saved

                              draft discarded




















































                              Thanks for contributing an answer to Mathematics Stack Exchange!


                              • Please be sure to answer the question. Provide details and share your research!

                              But avoid



                              • Asking for help, clarification, or responding to other answers.

                              • Making statements based on opinion; back them up with references or personal experience.


                              Use MathJax to format equations. MathJax reference.


                              To learn more, see our tips on writing great answers.





                              Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


                              Please pay close attention to the following guidance:


                              • Please be sure to answer the question. Provide details and share your research!

                              But avoid



                              • Asking for help, clarification, or responding to other answers.

                              • Making statements based on opinion; back them up with references or personal experience.


                              To learn more, see our tips on writing great answers.




                              draft saved


                              draft discarded














                              StackExchange.ready(
                              function () {
                              StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2999578%2fequation-solving%23new-answer', 'question_page');
                              }
                              );

                              Post as a guest















                              Required, but never shown





















































                              Required, but never shown














                              Required, but never shown












                              Required, but never shown







                              Required, but never shown

































                              Required, but never shown














                              Required, but never shown












                              Required, but never shown







                              Required, but never shown







                              Popular posts from this blog

                              Berounka

                              Sphinx de Gizeh

                              Different font size/position of beamer's navigation symbols template's content depending on regular/plain...