regexp_matrix_solver.rb

require 'matrix'
module RegexpMatrixSolver
  refine Array do
    def ~
      keys = map(&:first).flat_map(&:keys).uniq
      matrix = []
      binding = first.last
      vec = map{_2}
      each do |values, rhs|
        matrix << keys.map { values[_1] || 0 }
      end
      keys.zip Matrix[*matrix].lup.solve(vec) do
        binding.local_variable_set _1, _2
      end
    end
  end
  refine Regexp do 
    def =~(binding)
      segments = source.scan(/(^|[+-])(?:(?:(\d+)\*)?\(\?<([^<>]+)>\)(?:\*(\d+))?|(\d+))/)
      values = Hash.new(0)
      lhs = 0
      segments.map do |sgn, n, name, m, c|
        sign = sgn == '-' ? -1 : 1
        if c
          lhs += sign * c.to_i
        else
          values[name] += sign * (n || 1).to_i * (m || 1).to_i
        end
      end
      [values, $rhs - lhs, binding]
    end
  end
  refine Integer do
    def ~
      [$rhs = self].tap{
        _1.singleton_class.class_eval{
          alias ~ binding
          public :~
        }
      }
    end
  end
end

using RegexpMatrixSolver
~[
  /2*(?<x>)+3*(?<y>)-3/       =~~~ 5,
  /2*(?<y>)-(?<z>)+3/         =~~~ 4,
  /-(?<x>)+5*(?<y>)-2*(?<z>)/ =~~~ 3,
]
p [x, y, z] #=> [(1/1), (2/1), (3/1)]