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Why is the answer (output) from the Solve expression shown below {}? Can someone please tell me what's wrong with the expression. I want to know σ1 and σ2}.

Solve[{S - N1 == 0, S - N2 == 0, σ1 == N1/A1, σ2 == N2/A2}, {σ1, σ2}]
m_goldberg
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marcus
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    From Solve documentation page; {} is returned when there are no solutions. – Edmund Jun 11 '17 at 10:38
  • Solving for {A1, A2, N1, N2, S, \[Sigma]1, \[Sigma]2} instead of just {\[Sigma]1, \Sigma[2]} gives {N1 -> S, N2 -> S, \[Sigma]1 -> S/A1, \[Sigma]2 -> S/A2} – I should change my Username Jun 11 '17 at 10:45
  • when i do as you write a get:

    Solve[{S - N1 == 0, S - N2 == 0, [Sigma]1 == N1/A1, [Sigma]2 == N2/A2}, {A1, A2, N1, N2, S, [Sigma]1, [Sigma]2}] I get this {{N1->A1 [Sigma]1,N2->A1 [Sigma]1,S->A1 [Sigma]1,[Sigma]2->(A1 [Sigma]1)/A2}}

     – marcus Jun 11 '17 at 10:58
  • dont know whats wrong – marcus Jun 11 '17 at 10:59
  • I was editing my answers too much, I accidentally deleted the part where I did Reduce first. But going directly with Solve[{S - N1 == 0, S - N2 == 0, \[Sigma]1 == N1/A1, \[Sigma]2 == N2/A2}, {N2, S, \[Sigma]1, \[Sigma]2}] solves to {{N2 -> N1, S -> N1, \[Sigma]1 -> N1/A1, \[Sigma]2 -> N1/A2}}. But mathematically I have no idea if its right, I just brute forced it until I got an output with \[Sigma1] and \[Sigma]2 – I should change my Username Jun 11 '17 at 11:09
  • Thank you very much! i solved it whit your help. =) – marcus Jun 11 '17 at 11:22
  • Note (emphasis mine): "Solve gives generic solutions only. Solutions that are valid only when continuous parameters satisfy equations are removed. Other solutions that are only conditionally valid are expressed as ConditionalExpression objects." (Docs.) If you replace Solve by Reduce, you get, as part of the reduce system, two equations N2 == S && N1 == S, and so the solution is removed because N1 and N2 are treated as continuous parameters. There are two ways to solve it that treat N1 and N2 as variables instead of parameters.... – Michael E2 Jun 11 '17 at 16:37
  • I feel the preceding comment might be worth posting as an answer...unless there's a duplicate of this question on site already. (I may have voted to put this Q on hold too hastily.) – Michael E2 Jun 11 '17 at 16:38
  • ...[Continuing with my answer]...Method 1: Make N1, N2 variables to be solved for: Solve[system, {\[Sigma]1, \[Sigma]2, N1, N2}]. Method 2: Make N1, N2 variables to be eliminated: Solve[system, {\[Sigma]1, \[Sigma]2}, {N1, N2}]. In both methods, the symbols N1, N2 are treated as variables and not as parameters. (And these methods assume that is the intention. If not, identify which symbols are variables and which are parameters and proceed accordingly.) – Michael E2 Jun 11 '17 at 16:56

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