Two Parameters Families

(iii) Both $m,b$ arbitrary. The full two-parameter family $y=mx+b$ (both constants free) is hard to visualise directly on a graph, but an important sub-case is easy: fixing a point $(x_1,y_1)$ and letting $m$ vary in $y-y_1=m(x-x_1)$ sweeps out every non-vertical line through that one fixed point (the vertical line $x=x_1$ through the same point is the single member this point-slope parametrisation cannot express, since its slope would be undefined).

Family of lines through the intersection of two given lines. Let $L_1\equiv a_1x+b_1y+c_1=0$ and $L_2\equiv a_2x+b_2y+c_2=0$ be two given lines. Then, for a parameter $\lambda$,

is the equation of a family of straight lines, every member of which passes through the (fixed) point of intersection of $L_1=0$ and $L_2=0$ — because at that intersection point both $L_1=0$ and $L_2=0$ simultaneously, so $L_1+\lambda L_2=0+\lambda\cdot0=0$ regardless of $\lambda$. Different real $\lambda$ give different lines through that same fixed point. (This represents every line through the intersection except $L_2=0$ itself, which is the limiting case $\lambda\to\infty$.)

Why this is useful: it lets a line through the intersection of $L_1,L_2$ meeting one further condition be found without first solving for the intersection point — substitute the extra condition (a third point it must pass through, a required slope, etc.) directly into $L_1+\lambda L_2=0$, solve for $\lambda$, and substitute back. …