Solving x^y=y^x & x+y=6 with Deduction

[arka.sharma]

Hi All,

I was given two equations back in my school days to solve for both x and y it is as follows

x^y = y^x & x+y = 6

Now it can be seen that following would be possible solutions (x=4,y=2),(x=3,y=3),(x=2,y=4).
But is there any deductive way to solve this ?

Regards,
Arka

 

[maajdl]

Solving (6-y)^y == y^(6-y)
Which can be represented graphically.

 

[arka.sharma]

Thanks for your reply. But how to solve the equation (6-y)^y == y^(6-y) ?

 

[maajdl]

Put the RHS and the LHS of the equation on a graphic and see where these curve cut each other.
You will find the root that you already know.
If there are other roots, you will likely find out.
Afterward, you might find out arguments to "prove" you found all the roots.
The graphical representation will help a lot.

 

[HallsofIvy]

Thanks for your reply. But how to solve the equation (6-y)^y == y^(6-y) ?

You were told to solve it graphically. It can also be solved using a numerical iteration method. If you are looking for an elementary algebraic way of solving it, there is none.