find $x$, given $\{c_ix = k_i + y_i\}_{i=[1,n]} $
There is no way to recover $x$ in this scenario; there is not even a way to say "this value of $x$ is more likely than that value"

One demonstration is that if we attempt to test a particular value $x'$; we can compute the necessary values:

$$y_i = c_i x - k_i$$

However, if all the values $y_i$ are randomly distributed, and given that we have no further restriction of what the values $y_i$ might be, this distribution is no less probable than any other distribution; hence these relations give us no information about $x$.

Since those relations are the only thing we know about $x$, that means we have no criteria to say which values are more or less probable.

This holds true for any sets of values $c_i, k_i$, and specifically if we have $c_i = c_j$ or $k_i = k_j$.