Question

solve for the value of x and y.

X+y=15

x^y+y^x=30

Answer 1

The graph of x^y + y^x =30
looks something like this
http://www.wolframalpha.com/input/?i=plot+x%5Ey%2By%5Ex%3D30

is there an x-intercept?
let y = 0 to get
x^0 + 0^x = 30
1 + 0^x = 30
1 + 0 = 30 , false, there is no x-intercept
same reason ---> no y-intercept

but x+y = 15, has
x-intercept of (15,0) and y-intercept of (0,15)

but x+y = 15 will definitely cross the curve.
it must be very close to (15,0)

I then plugged in some values close to x=15 into
http://www.wolframalpha.com/input/?i=solve+15%5Ey+%2B+y%5E15%3D30

to get :
(13, 1.18513)
(14, 1.16423)
(15, 1.14602)
the curve is getting closer to the x-axis, but becomes an asymptote
e.g. (20, 1.07952)

so in x+y = 15
we know that y is appr 1.164
then x = 15-1.164 = 13.846

let's set up an iteration using Wolfram

now plug in x = 13.846 into Wolfram
y = 1.16726
x = 15 - 1.16726 = 13.83274

13.83274 --->1.16752
x = 15-1.16752 = 13.83248

13.83248 ---> 1.16752 , same as before

<b>Wow, how about (13.83248, 1.16752)
and (1.16752, 13.83248) </b>

check:
for x+y = 15
1.16752 + 13.83248 = 15 , yeahhh
in the curve:
1.16752^13.83248 + 13.83248^1.16752 = 29.9993..

That's about as good as it gets.