#x+y=5# and #x^y+y^x=17# Find the value of #x and y#?

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Answer 1 · 2017-06-19T14:44:17

#x=2, y=3#
Assuming #{x,y} in NN#

Here we have two equations:

#x+y =5#

#x^y+y^x=17#

{Although there is no resaon to assume that #x# and #y# are naturals, it seemed like a sensible place to start.}

Assume #{x,y} in NN#

Since #x+y = 5#

#x in { 1, 2, 3, 4}# and #y in {4, 3, 2, 1}#

Testing each #(x, y)# pair in turn we notice that:

#2^3+3^2 = 8 + 9 =17#

Hence a solution to this system is #x=2, y=3#

NB: I have not worked through the cases where #(x, y) in RR#, Hence, I have not proved that there are no other real solutions to this system.