Assume $A \in \mathbb{R}^{m \times n}, \ m > n$ is full rank, and the linear system $Ax = b$ is inconsistent. The solution to $\min_x \|Ax-b\|^2$ is given by the least-square, $A^+b.$
Similarly, assume $C \in \mathbb{R}^{m \times n}, \ m > n$ is full rank, and the linear system $Cx = d$ is inconsistent. The solution to $\min_x \|Cx-d\|^2$ is given by the least-square, $C^+d.$
Now consider $\min_x \|Ax-b\|^2 + \|Cx-d\|^2.$ The solution to this problem is the least-square solution $E^+f,$ where $E$ and $f$ corresponds to the augmented linear system.
Is there a closed-form solution for $E^+f$ as a function (among others) of $C^+d$ and $A^+b$?
Similarly, can we relate the residuals?