Linear system's AX=b solution with small variation with vector b.

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Let's say that I have a linear system $Ax=b$, where $A$ is square matrix $nxn$, $x$ - searching variable, and $b$ - vector $nx1$.

I solved this equation and got a solution.

Then I changed some of the vectors $b$ values with small variation (added/subtracted a number close to zero). And then also solved that system.

Depending on linear systems matrix $A$ and vector $b$ initial values the $x$ variables might change a lot or have a small different. I tried to figure out on what that depends. Is it somehow related with eigenvalues of matrix $A$?

Any ideas? Thanks in advance.