Consider $Ax=b$ where $A$ is invertible, so we have $x=A^{-1}b$.
Now, let's consider a low-rank approximation of $A$, say $\bar{A}$ such that $rank(\bar{A})\leq r$ and $||A-\bar{A}||_F\leq \epsilon$ where $||\cdot||_F$ denotes the Frobenius norm, and we set $\bar{x}= \bar{A}^+ b$, where $A^+$ denotes the pseudo-inverse of $A$.
The question is, can we bound the error $||x-\bar{x}||$ (say, by 2-norm). Moreover, computationally, is computing $\bar{A}^+ b$ easier since the rank of $\bar{A}$ is low?