When we have a system of linear equations we are trying to find the solution of the system (if exists). Are we doing the same when we use system of linear equations to describe something (make an abstraction from natural language)?
For example suppose that we are given that John has $x$ dollars, Mark has $M=x+2$ dollars and George has $G=M+3=x+5$ dollars and $x=2$. Then according to the above $M=4$ dollars and $G=7$ dollars. Now what if we found out that $G=4$ dollars? That contradicts the above statements and also contradicts what we were given. That means there is no number we can assign to the amount of money of John and make true the other equations but this doesn't mean that John has $0$ dollars right? Is this equivalent to say that at least one of our assumptions (what we were give were wrong) because we can assign a number to John's money?
I want to understand the difference when we solve a system of linear equations in mathematics and when we solve it as an everyday problem. From a mathematical point of view we just trying to find the unknowns that make every equation of the system true. From an everydays problem it seems we use some initial information (system of linear equations) to find more information. So could we say that from this point of view inconsistent system in mathematics translates to contradictions about initial information in everyday's problem?