In the Finite element, we take a function with free parameters and put it into an equation, but it doesn't solve this equation, so why don't we get $0=1$?
Why does this method give an approximate solution?
If I have the equation $2*f(x)^2=x$, and put in $f(x)=ax+b$ into it, I get $0=1$, not a good approximation, so why does this method give a good approximation?
Your description of FEM looks much simpler than I can remember. :)
A characteristic property is that the domain of the sought solution is divided into parts, the finite elements, and that on each element an approximation is sought.
So I see that you intend to you use linear functions as approximations, but what is the finite element you are working on? I would expect at least one $I \subset \mathbb{R}$.
Then out of the infinite many linear functions characterized by $(a, b) \in \mathbb{R}^2$, according to what criteria will you select the best fit?
Here are two continuous functions which would solve your equation:
You see that the choice of the best linear approximation depends on the interval you want to approximate on.
