Why does heat at constant pressure equal enthalpy?

[Nisarg Shah 1C]

Why does the heat exchanged by a system and its environment at constant pressure equal ? What happens when pressure changes, and how would that mathematically change how you calculate enthalpy?

[Jingyi Li 2C]

By definition, H = U + PV, where U is internal energy, P is pressure of the system, V is volume.
Also, ΔU = w + q, where q is heat and w is work.
As a result, ΔH= ΔU + PΔV = w + q + PΔV, and w = -P(external) ΔV
when you combine them together, ΔH = q - PexΔV + PΔV
if pressure is constant, Pex = P, ΔH = q