Suppose a linear system $Ax=b \tag 1$ is given, $A\in\mathbb{R}^{n\times n}$, $b\in\mathbb{R}^n$, and a solution exists. Now, suppose the system is multiplied from the left by some $C\in\mathbb{R}^{n\times n}$, resulting in $CAx=Cb. \tag 2$ A solution to $(1)$ is also a solution to $(2)$ (solving $(2)$ can be replaced by solving $(1)$), but the system $(2)$ might have more solutions than $(1)$, so solving $(1)$ cannot be replaced by solving $(2)$. Under which conditions on $C$ can solving $(1)$ be replaced by solving $(2)$?
What is with a solution to the system obtained by adding $(1)$ and $(2)$? Does it hold that a solution to the system obtained by summing $Ax=b$ and $Cx=d$, for some $A, C$ and $b, d$ (solutions exist, and it is known that the systems have at least one common solution), would yield a common solution (system (3) $(A+C)x=b+d$. In other words, can it happen that (3) has some solution that does not satisfy both $Ax=b$ and $Cx=d$?