Consider a case where a theoretically crafted truss problem has more variables than number of equations, for example by simply adding a straight rod between any two points of a truss system that was initially solvable. Will it possible to make that structure in reality? I think there should not be any problem in that. Then, experimentally we would find some value of compression or tension in each two-force member. But then why when we try to solve the problem theoretically, we declare the system to be insolvable?
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I suspect the issue causing your confusion is to confuse the mathematical model with the reality. In physics we only produce models of reality. The mathematics does not necessarily match the reality.
Physics is not mathematics. Mathematics does not define reality. We can construct (and do) lots of theories which are (more or less) discarded by physics, as they do not match reality. We keep the best matching theoretical frameworks.
What you have done is create two different systems.
The different systems of equations that describe each system will, by definition, have a "solution" as that's a given if you actually build the system.
A system of mathematical equations not having a closed form analytical solution, allowing only numerical solutions, does not invalidate the theory.
If the model does not match the experiment to within expected error margins, them the model is wrong.
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