To determine how far a small piston must be pushed to lift a larger piston by 20 cm, the principle of hydraulic systems is applied, which states that the volume displaced by the small piston equals the volume displaced by the larger piston. The equation used involves the relationship between the diameters and the distances moved by the pistons, considering the incompressibility of the liquid. The relevant equations include those related to pressure and volume, specifically focusing on the area of the pistons and the distances they move. The calculations will yield the necessary distance the smaller piston must be pushed down. Understanding these principles is crucial for solving hydraulic lift problems effectively.