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I have three equations that relates five variables {a, b, c, r, s} with a sum and two ratios.

Eq. 1: a = b + c;
Eq. 2: s = b / a;
Eq. 3: r = b / c.

Given two values for any of the five variables I get a solution. But, this is not the automation problem I want to solve.

I can have the solution of variable r by simply knowing s. This is solved by a "human algorithm" as follows.

  1. Substitute a of Eq. 1 in Eq. 2.
  2. Divide the second term of the new Eq. 2 by the variable c.
  3. Replace b/c by the expression of Eq. 3.

That means s = r / (r+1).

The questions is -How can an AI algorithm solve this?, i.e. the machine should recognize that given the variable r she can obtain directly the variable s and do no require another variable.

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What you're asking here is actually not reasoning but rather symbolic computation. Sympy is an example of a library which does it.

Explaining computer algebra systems could be a series of lectures and this is too broad for a single question. To give you an idea: (1) represent equations symbolically (2) bring them in a normal form (3) substitute.

Although it is not elegant, you can simplify the problem by brute Force: try all combinations of solving for a variable and substituting it in the other equations.

Related:

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