This is from A.P. French Special Relativity book, Chapter 3 (page 78)
Setup of the proof: $S$ and $S'$ be inertial reference frame. $S'$ move to the right with respect to $S$ at velocity $v$.
Let co-ordinates in $S$ be $(x,t)$ and co-ordinates in $S'$ be $(x',t')$
Equation (3-8) in the book, he writes that transformation will be of the form:
$x = ax' + bt'$ and by symmetry of the reference frames as implied by relativity principle, $x' = ax - bt$
How does symmetry of reference frame argument lead to the above conclusion? For e.g. why can't the second equation above be of the form $x' = -ax - bt$ or maybe $x' = -ax + bt$. These equations look as symmetric to me (mathematically!) as the one author uses. (I know I'm wrong but want to understand more clearly why am I wrong)