Least Squares Regression $Ax=b$ when $A$ is fixed and $b$ is varied
In least squares regression, the problem that you solve is:

$X^T X \beta = X^T y$

where $\beta$ are the parameters, $X$ the regressors and $y$ the predicted variable. If you want, in your notation $A$ to be constant and $b$ to change, you need to have constant regressors and change the predicted variable $y$. I don't think there are applications that the regressors are not changed but the predictors do change, but if you want to find one it will probably be in online learning.