In Einstein's book "Relativity" there is a wonderful derivation of the Lorentz transformation, requiring no more than high school algebra (pp. 117 - 121). It is quite clear but I do not understand one early step.
Equation (1) is $$x - ct = 0$$ Equation (2) is $$x' - ct' = 0$$
I don't see how (1) and (2) imply
Equation (3) $$(x - ct) = \lambda (x' - ct')$$
This seems to be saying $0 = \lambda 0$ mathematically, which makes no sense.
Other questions in this forum have dealt with this, and one commenter said that (3) follows because the transformation between the two coordinate systems is linear.
Linear transformations do take straight lines through the origin of one coordinate system to straight lines through the origin of another, but are (1) and (2) enough to imply that the transformation is linear, and if so does that make them imply (3)?