Given
$$c_1x = k_1 + y_1 $$ $$c_2x = k_2 + y_2 $$ $$\vdots $$ $$c_nx = k_n + y_n $$
where the values of $\{c_1 \ldots c_n \}$ and $\{ k_1 \ldots k_n \}$ are known, and $x, \{y_1 \ldots y_n \}$ are unknown. $y_i$ is chosen uniformly at random.
We are working in a group of $\mathbb{Z}_p^*$, so all $c_i, k_i, x, y_i ∈ \mathbb{Z}_p^*$.
From here, how can we recover the value of $x$ ? (suppose $c_i = c_j$ for some $i,j$ that we know)