Question

1. For all non-zero numbers x and y such that x= 1/y, what is the value of ( x- 1/x)(y+ 1/y)

Answer 1

if x = 1/y , then xy = 1

(x-1/x)(y+1/y)
= (x^1 -1)/x * (y^2 + 1)/y
= (x^1 - 1)(y^1 + 1)/xy
= (1/y^2 - 1)(y^2 + 1)/1
= 1 + 1/y^2 - y^2 - 1
= 1/y^2 - y^2
or
= ( 1 - y^4)/y^2

or more directly

( x- 1/x)(y+ 1/y)
= xy + x/y - y/x - 1/(xy)
= 1 + (1/y)/y - y/(1/y) - 1
= 1/y^2 - y^2
as above