Derandomization of an approximation algorithm for solving a linear system
In your case the method of conditional expectations devolves to a greedy algorithm. At each stage, you choose the value for the variable under consideration that satisfies the most equations (if any; otherwise choose an arbitrary value), and substitute this value.

Why does this work? Suppose that at the beginning of stage $i$, you have $N_i$ remaining equations (i.e., ones in which there is at least one remaining variable with non-zero coefficient). The greedy choice satisfies at least $(N_i-N_{i+1})/p$ equations. In total, the number of equations you satisfy is at least
$$
\sum_{i=1}^n \frac{N_i-N_{i+1}}{p} = \frac{N_1 - N_{n+1}}{p} = \frac{N}{p},
$$
where $N$ is the total number of equations, and $N_{n+1} = 0$ is the number of remaining equations at the end of the algorithm.