Question:Do we have to worry about dividing by zero in the
Lagrange method if we get rid of the lambda parameter
by dividing the first equation by the second

Answer:
When dividing out lambda by dividing the first equation by the second,
you actually do nothing dangerous if you afterwards multiply out again.
In two dimensions, you will get from
f_x = lambda g_x
f_y = lambda g_y
that f_x/f_y = g_x/g_y which means f_x g_y  f_y g_x = 0
This just means that the cross product between the two vectors is zero
that they are parallel. The Lagrange equations could also have been written as
nabla f x nabla g = 0
g = c
which is in some sense better since the case nabla g=0 is covered by this.
The reason that this is not replacing the Lagrange equations is that the
Lagrange equations hold in any dimensions and with arbitrarily many
constraints. It is a powerful general tool even so in the cases we usually
look at to practice, other methods would work too.
