Let $A, B$ be $n\times n$ matrices and $c, d$ be $n \times 1$ vectors such that the matrix equations $$Ax = c$$
$$Bx = d$$ are consistent, i.e., each equation admits a solution. Can we conclude that $$(A + B)x = (c + d)$$ is also consistent? Prove if true or give a counter example if not true.
i tried hard to find counter example but i coudnt find any
This is just what Michael wrote. Counter example: $A=1$, $B=-1$, $c=1$, $d=\pi$.