Consider the family of linear systems $M_tX_t=B_t$ parameterized by $t\in\mathbb R$, where $M_t$ is symmetric. I mean that each entry of the matrix $A_t$ and of the vector $B_t$ is a function of $t$
Assume that for any $t$ the affine space of solutions has dimension $1$. Moreover suppose that we know the asymptotic behavior of $M_t$ and $B_t$: for instance $M_t=O(A_t)$ and $B_t=O(C_t)$ for $t\to+\infty$ (componentwise). Can we somehow relate the asymptotics of $X_t$ with the solutions of $A_tY_t=C_t$?
To relate the asymptotics of the solutions $X_t$ of the system $M_tX_t = B_t$ with the solutions $Y_t$ of the system $A_tY_t = C_t$, we would need some additional assumptions about the relationship between $M_t$ and $A_t$, and between $B_t$ and $C_t$.
In general, if $M_t = O(A_t)$ and $B_t = O(C_t)$ as $t \to +\infty$, this tells us that the magnitude of the entries of $M_t$ and $B_t$ grow no faster than the magnitude of the corresponding entries of $A_t$ and $C_t$ respectively.
However, this doesn't necessarily tell us how the solutions $X_t$ and $Y_t$ relate. For example, consider the case where $M_t = \epsilon A_t$ and $B_t = \epsilon C_t$ for some small $\epsilon > 0$. In this case, $M_t = O(A_t)$ and $B_t = O(C_t)$, but the solutions $X_t$ and $Y_t$ would be quite different (specifically, $X_t = \epsilon Y_t$).
If we knew that $M_t = A_t + O(1/t)$ and $B_t = C_t + O(1/t)$ as $t \to +\infty$, then we could say that the solutions $X_t$ and $Y_t$ are asymptotically equivalent, i.e., $X_t - Y_t = O(1/t)$ as $t \to +\infty$. This would be because the perturbations to the system $A_t Y_t = C_t$ to obtain the system $M_t X_t = B_t$ goes to zero as $t$ goes to infinity.
In summary, to relate the asymptotics of the solutions $X_t$ and $Y_t$, we would need additional assumptions about how the matrices $M_t$ and $A_t$, and the vectors $B_t$ and $C_t$, are related. The condition $M_t = O(A_t)$ and $B_t = O(C_t)$ by itself does not provide enough information to make such a connection.