Wikipedia and other materials typically hide all the intermediate steps, so we have to derive them ourseleves to understand how it works under the hood.
Point addition
$P + Q + R = 0\ where\ P\ne Q\ne R$
Curve equation (Weierstrass form):
$y^2 = x^3 + ax + b$
Our line defined by $P$ and $Q$:
$y = \lambda x + \beta$