How do you know when a system of equations is inconsistent?
When you try to solve the system, you get an impossibility.
You get something like
If you're working in the real numbers with nonlinear systems, you might instead get an imaginary solution.
A system is inconsistent if, being a solution to one equation is inconsistent with being a solution of another equation in the system.
Being "inconsistent with" mean they can't both happen.
For example: being negative is inconsistent with being positive.
Being less than 4 is inconsistent with being greater than 9.
Being a solution to