How to get single equation equivalent to any conjunction of equations

Question

Let's say we have a system of two equations. These may be any different equations without any constraints.

Is there a simple single rule to get another equation which would be an exact equivalent, which has exactly same solutions such as the system has?

Answer 1

$$\bigwedge_i (a_i = b_i) \iff \sum_i (a_i - b_i)^2=0$$