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 ; )