With a vector equation r=p+td can you use any point the vector passes through as position vector p?
For instance if one has line 4y + 3x = 0 (and needs to make a vector equation for it) then d, the direction vector, is calculated from the gradient:
m = -3/4
direction vector = 4 i - 3 j.
But what is p? I could put infinite numbers into 4y + 3x = 0, for instance when y=3 then x=-4, the position vector for this is -4 i + 3 j but is that vector p?
How do I decide what number to put into the function to find p?
As you noticed, there are infinitely many choices for $p$, as it must any point on the line, i.e. in your case any solution of the equation $$4y+3x=0.$$ An easy choice could be $p=(0,0)$.
Remark: note that also $d$ is not unique as any vector which is a (non-zero) multiple of $d$ does the same job. For example $d=(-8,6)$.