First question, go easy on me. =)
From text: Show that only those points satisfying $y-y_1=m(x-x_1)$ can lie on the line $L$ passing through $P_1(x_1,y_1)$ with slope $m$.
I can evaluate $m$ from two points: $P_1(x_1,y_1)$ and $P(x,y)$, and plug all these values into the point-slope form to give me the equation for a line ($L_1$).
Furthermore, I can demonstrate that a second line ($L_2$) has a slope equal to the slope of $L_1$, but at different $Ps$ in the plane. $L_1$ || $L_2$ if and only if $L_1$ and $L_2$ have equal slopes and the set of $Ps$ defining the lines $L_1$ and $L_2$ are unique from each other.
Though I've demonstrated that the points on $L_1$ are unique from the points on $L_2$, I'm not sure that I am answering this rigorously enough or proving the statement completely true. What am I missing? Or do I have this completely backwards?
TYIA.