The common method is to use the determinant
#A(48,7)# #B(93,84)#
The vector formed by #A# and #B# is :
#vec(AB) = (93-48,84-7) = (45,77)#
(which is a vector director to our line)
and now imagine a point #M(x,y)# it can be anything
the vector formed by #A# and #M# is ;
#vec(AM) = (x-48,y-7)#
#vec(AB)# and #vec(AM)# are parallel if and only if #det(vec(AB),vec(AM)) = 0#
in fact they will be parallel and be on the same line, because they share the same point #A#
Why if #det(vec(AB),vec(AM)) = 0# they are parallel ?
because #det(vec(AB),vec(AM)) = AB*AMsin(theta)# where #theta# is the angle formed by the two vectors, since the vectors are not #= vec(0)# the only way #det(vec(AB),vec(AM)) = 0# it is #sin(theta) = 0#
and #sin(theta) = 0# when #theta = pi# or #= 0# if the angle between two line #=0# or #= pi# they are parallel (Euclide definition)
compute the #det# and find
#45(y-7) - 77(x-48) = 0#
And voilà ! You know how to do it geometrically ; )
