Product property [closed]

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What is the name of the following property of two equal products?




If $ab = cd$, then $a(b-d)=(c-a)d$











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closed as off-topic by Saad, Brahadeesh, Frpzzd, Widawensen, user593746 Dec 20 '18 at 15:03


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." – Saad, Brahadeesh, Frpzzd, Widawensen, Community

If this question can be reworded to fit the rules in the help center, please edit the question.












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    $begingroup$
    I don't suppose this has a name. There are many, many algebraic identities. Most of them lack names.
    $endgroup$
    – lulu
    Dec 20 '18 at 11:58
















0












$begingroup$


What is the name of the following property of two equal products?




If $ab = cd$, then $a(b-d)=(c-a)d$











share|cite|improve this question









$endgroup$



closed as off-topic by Saad, Brahadeesh, Frpzzd, Widawensen, user593746 Dec 20 '18 at 15:03


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." – Saad, Brahadeesh, Frpzzd, Widawensen, Community

If this question can be reworded to fit the rules in the help center, please edit the question.












  • 2




    $begingroup$
    I don't suppose this has a name. There are many, many algebraic identities. Most of them lack names.
    $endgroup$
    – lulu
    Dec 20 '18 at 11:58














0












0








0





$begingroup$


What is the name of the following property of two equal products?




If $ab = cd$, then $a(b-d)=(c-a)d$











share|cite|improve this question









$endgroup$




What is the name of the following property of two equal products?




If $ab = cd$, then $a(b-d)=(c-a)d$








universal-property






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asked Dec 20 '18 at 11:56









Michael MuntaMichael Munta

97111




97111




closed as off-topic by Saad, Brahadeesh, Frpzzd, Widawensen, user593746 Dec 20 '18 at 15:03


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." – Saad, Brahadeesh, Frpzzd, Widawensen, Community

If this question can be reworded to fit the rules in the help center, please edit the question.







closed as off-topic by Saad, Brahadeesh, Frpzzd, Widawensen, user593746 Dec 20 '18 at 15:03


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." – Saad, Brahadeesh, Frpzzd, Widawensen, Community

If this question can be reworded to fit the rules in the help center, please edit the question.








  • 2




    $begingroup$
    I don't suppose this has a name. There are many, many algebraic identities. Most of them lack names.
    $endgroup$
    – lulu
    Dec 20 '18 at 11:58














  • 2




    $begingroup$
    I don't suppose this has a name. There are many, many algebraic identities. Most of them lack names.
    $endgroup$
    – lulu
    Dec 20 '18 at 11:58








2




2




$begingroup$
I don't suppose this has a name. There are many, many algebraic identities. Most of them lack names.
$endgroup$
– lulu
Dec 20 '18 at 11:58




$begingroup$
I don't suppose this has a name. There are many, many algebraic identities. Most of them lack names.
$endgroup$
– lulu
Dec 20 '18 at 11:58










2 Answers
2






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$a(b-d)=(c-a)d iff ab-ad=cd-ad iff ab=cd$.



The reason ist the distributive property of addition and multiplication in $ mathbb R$.






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    1












    $begingroup$

    It is the multilinearity of the determinant (substracting the first column from the second one does not change the determinant), i.e.,
    $$
    ab-cd=det begin{pmatrix} a & c cr d & b end{pmatrix}=det
    begin{pmatrix} a & c-a cr d & b-d end{pmatrix}=a(b-d)-(c-a)d.
    $$






    share|cite|improve this answer









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      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      0












      $begingroup$

      $a(b-d)=(c-a)d iff ab-ad=cd-ad iff ab=cd$.



      The reason ist the distributive property of addition and multiplication in $ mathbb R$.






      share|cite|improve this answer









      $endgroup$


















        0












        $begingroup$

        $a(b-d)=(c-a)d iff ab-ad=cd-ad iff ab=cd$.



        The reason ist the distributive property of addition and multiplication in $ mathbb R$.






        share|cite|improve this answer









        $endgroup$
















          0












          0








          0





          $begingroup$

          $a(b-d)=(c-a)d iff ab-ad=cd-ad iff ab=cd$.



          The reason ist the distributive property of addition and multiplication in $ mathbb R$.






          share|cite|improve this answer









          $endgroup$



          $a(b-d)=(c-a)d iff ab-ad=cd-ad iff ab=cd$.



          The reason ist the distributive property of addition and multiplication in $ mathbb R$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 20 '18 at 12:04









          FredFred

          48.7k11849




          48.7k11849























              1












              $begingroup$

              It is the multilinearity of the determinant (substracting the first column from the second one does not change the determinant), i.e.,
              $$
              ab-cd=det begin{pmatrix} a & c cr d & b end{pmatrix}=det
              begin{pmatrix} a & c-a cr d & b-d end{pmatrix}=a(b-d)-(c-a)d.
              $$






              share|cite|improve this answer









              $endgroup$


















                1












                $begingroup$

                It is the multilinearity of the determinant (substracting the first column from the second one does not change the determinant), i.e.,
                $$
                ab-cd=det begin{pmatrix} a & c cr d & b end{pmatrix}=det
                begin{pmatrix} a & c-a cr d & b-d end{pmatrix}=a(b-d)-(c-a)d.
                $$






                share|cite|improve this answer









                $endgroup$
















                  1












                  1








                  1





                  $begingroup$

                  It is the multilinearity of the determinant (substracting the first column from the second one does not change the determinant), i.e.,
                  $$
                  ab-cd=det begin{pmatrix} a & c cr d & b end{pmatrix}=det
                  begin{pmatrix} a & c-a cr d & b-d end{pmatrix}=a(b-d)-(c-a)d.
                  $$






                  share|cite|improve this answer









                  $endgroup$



                  It is the multilinearity of the determinant (substracting the first column from the second one does not change the determinant), i.e.,
                  $$
                  ab-cd=det begin{pmatrix} a & c cr d & b end{pmatrix}=det
                  begin{pmatrix} a & c-a cr d & b-d end{pmatrix}=a(b-d)-(c-a)d.
                  $$







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Dec 20 '18 at 12:08









                  Dietrich BurdeDietrich Burde

                  81.2k648106




                  81.2k648106















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                      SVrEG vfpgdDr vwsY6AGM9NT HMlbb CngB8mWX2EXdOs6ZXpZDbLd9z3,UM2ImPiciH wZeBr3OGiB 9QTEo83X1lQOcwerfyF,C5

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