How to solve this? #{((x+y)/(xy)+(xy)/(x+y)=a+1/a),((x-y)/(xy)+(xy)/(x-y)=b+1/b):}#
1 Answer
May 29, 2017
At least four solutions...
Explanation:
Given:
#{ ((x+y)/(xy)+(xy)/(x+y) = a+1/a), ((x-y)/(xy)+(xy)/(x-y) = b+1/b) :}#
Note that any solutions of the following system will be solutions of the given system:
#{ ((x+y)/(xy) = a), ((x-y)/(xy) = b) :}#
Adding these, we find:
#a+b = (x+y)/(xy) + (x-y)/(xy) = (2x)/(xy) = 2/y#
Hence:
#y = 2/(a+b)#
Subtracting them, we find:
#a-b = (x+y)/(xy) - (x-y)/(xy) = (2y)/(xy) = 2/x#
Hence:
#x = 2/(a-b)#
So one solution is:
#(x, y) = (2/(a-b), 2/(a+b))#
Three other solutions can be found by replacing one or both of
#(x, y) = (2/(1/a-b), 2/(1/a+b))#
#(x, y) = (2/(a-1/b), 2/(a+1/b))#
#(x, y) = (2/(1/a-1/b), 2/(1/a+1/b))#
