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I am new to Mathematica and struggling with its syntax. I have a set of equations, which I would like to solve for my unknowns. I would like to solve the first equation for the first unknown and substitute this unknown in a second equation with the result I have obtained from Solve. Sounds easy, but for me this seems to be undoable.

Here my MWE:

g1 = a + b + 2*c + 5 == 0
g2 = a + e + c - 5 == 0
sol = Solve[g1, c]
g2 = g2 /. c -> sol

The result is:

{{-5 + a + e + (c -> 1/2 (-5 - a - b))}} == 0

which is clearly not what I wanted. In the second step I want to solve g2 for a, assuming b and e are known.

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    $\begingroup$ Try g2/.sol and don't overwrite g2=g2... $\endgroup$
    – Ulrich Neumann
    Commented Dec 4, 2020 at 9:39

2 Answers 2

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Clear["Global`*"]

g1 = a + b + 2*c + 5 == 0;
g2 = a + e + c - 5 == 0;

With two equations you can solve for two unknowns.

sol = Solve[{g1, g2}, {a, c}]

(* {{a -> 15 + b - 2 e, c -> -10 - b + e}} *)

Or, if you only want to solve for a while eliminating c

sola = Solve[{g1, g2}, a, {c}]

(* {{a -> 15 + b - 2 e}} *)

For your larger system of equations, try structuring it as a minimization problem.

min = Minimize[Total[(Subtract @@@ {g1, g2})^2], {a, c}] // Simplify

(* {0, {a -> 15 + b - 2 e, c -> -10 - b + e}} *)

If an exact solution exists, the minimum is zero and the approaches are equivalent.

sol[[1]] === min[[2]]

(* True *)
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  • $\begingroup$ Thank you I will definitely try it. Is the chance for solving my larger problem with minimize bigger? $\endgroup$
    – Zorg
    Commented Dec 6, 2020 at 12:28
  • $\begingroup$ I have no way of knowing or even guessing. Since the approach that you are using doesn't seem to be working, it is at least an alternative. $\endgroup$
    – Bob Hanlon
    Commented Dec 6, 2020 at 15:55
  • $\begingroup$ okay, I will try it. The three @ are strange to me, so I did some research. According to Wolfram this means "Verify that their differences are integers:", but since I am seeking the symbolic solutions is thisusefull or necessary? $\endgroup$
    – Zorg
    Commented Dec 7, 2020 at 10:55
  • $\begingroup$ Okay I have to admit that the longer I look at our minimize code the less I understand it. You subtract g2 from g1 and square the result, okay I understand this. But why 'Total', where does the sum come from? $\endgroup$
    – Zorg
    Commented Dec 7, 2020 at 11:50
  • $\begingroup$ In general the equations could be of the form lhs == rhs, i.e., Equal[lhs, rhs] where rhs is not necessarily zero. Highlight @@@ and press F1 for help. You will go to a link for Apply. The documentation there states "f@@@expr or Apply[f, expr, {1}] replaces heads at level 1 of expr by f." So Subtract @@@ eqns results in {lhs1-rhs1, lhs2-rhs2, ...}. To satisfy the equations, all of these numbers must be zero. A standard way of doing this is to minimize the sum of the squares (positive numbers), i.e., minimize Total[(Subtract @@@ eqns)^2] $\endgroup$
    – Bob Hanlon
    Commented Dec 7, 2020 at 16:10
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You may use your approach after applying the correction proposed by @Ulrich Neumann. There is, however, a function specifically for such an operation:

g1 = a + b + 2*c + 5 == 0
g2 = a + e + c - 5 == 0

Eliminate[{g1, g2}, c]

(*    15 + b - 2 e == a  *)

The both approaches yield the same final equation.

Have fun!

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  • $\begingroup$ Thank you, my real problem is a little bit more complictaed than my MWE. I have 9 non-linear equations which I would like to solve for 9 unknowns. Numerically this is no problem but I need the symbolic solutions, if they exist. Simply Solve[{equations}{unkowns}] does not work, it runs for days without a result. $\endgroup$
    – Zorg
    Commented Dec 4, 2020 at 9:57
  • $\begingroup$ My question is, is it worth trying to use Eliminate? $\endgroup$
    – Zorg
    Commented Dec 4, 2020 at 10:05
  • $\begingroup$ My experience tells me that exactly in the situation you described in some cases I could not get a solution with Solve, but succeeded first using Eliminate and after that Solve. In other cases it did not help. Of course, 9 nonlinear equations is a lot, and I would not be very optimistic about finding the exact solution. However, everything depends on the form of equations. I wish you to be lucky. $\endgroup$
    – Alexei Boulbitch
    Commented Dec 4, 2020 at 11:17
  • $\begingroup$ Nine nonlinear equations... If they are polynomials, Solve is most likely to be hanging either in computing a Groebner basis, or in unravelling the roots. The complexity with depend on degrees and on presence/absence of symbolic parameters. If they are polynomials and the input is not mongo-large, I will suggest posting as a new MSE question. $\endgroup$
    – Daniel Lichtblau
    Commented Dec 4, 2020 at 14:48

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