Many problems requiring numerical solutions can only be solved by setting up some equations and then some detailed analysis. Often the equations have some symmetry, which may involve some manipulation.
Example: Solve the simultaneous equations
(1)
(2)
(3)
Multiply (1) by c, (2) by b and (3) by a to give the three equations
(4)
(5)
(6)
From these we obviously obtain
hence
If
then from (1)
so
and then
and
The solutions are
or
If
(so that
) then from (1)
The solutionis found above. The other possibility is
and these also satisfy all three equations.
and
are symmetrical in the equations (1), (2) and (3) so that if any two are interchanged, the same equations result, in a different order. This means we can swap values ofandto obtain different solution.
Swapping
and
gives the solution
Swappingandgives the solution
Example: Solve the simultaneous equations
(1)
(2)
(3)
Addto (1), addto (2) and addto (30 to give
and
are all equal to a+b+c, they are all equal to each other, so
and
The solutions are
for any number
Example: Solve the simultaneous equations:
(1)
(2)
(3)
Adding these equations gives
then
(4)
(5)
(6)
(1)+(4) gives
(2)+(5) gives
(1)+(4) gives