We want to formulate the laws of algebra so that

if a = bc, then a/b = c

provided that a/b is defined at all. If we defined 0/0 to be 1, then

0 = (0 times 42) (for instance)

would be true, but

0/0 = 42

would be false! It seems less confusing and dangerous just to forbid division by 0 absolutely, so that equations of the form 0/0 = c never need to be considered.