For any chemical system, the goal of breaking or making bonds is to (decrease or increase) the overall potential energy of the system?

When breaking bonds the potential energy of the system increases and when making bonds the potential energy decreases. The overall potential energy of the system increase and then decrease.

Explanation:

The goal of breaking or making bonds is to make chemical reactions possible, a chemical reaction involves two steps, first, the chemical bonds are broken and then new bonds are formed. To make chemical reactions possible we need first to increase the potential energy to break the original bonds to then decrease the potential energy to create new bonds.

Note: the potential energy increase when we break the bonds because the system wants to return to its equilibrium state after the break, an analogy will be the gravity, imagine an apple in the floor 'bonded' with the earth when you separate them you increase the potential energy and when you let the apple fell you decrease the potential energy.

The goal is to decrease the overall potential energy of the system.

Explanation:

The energy stored in chemical bonds is called potential energy. When a bond is formed, energy is released, consenquently, to break a bond we need to provide energy.

Thermodynamics states that the lower the energy of a system, the more stable it is. Therefore, in a chemical reactions occurs in order to decrease the overall potential energy of the system to increase its stability.