y is sum of two variables , one of which varies directly with x and another varies inversely with x . When x=1, then y =-1 and when x =3 ,then y =5 . Find the relation between x and y .

Explanation

Let the variables are p and q , where p `prop` x and q `peop 1/x`
`rArr p =k_(1) x and q =(k^(2))/(x) , [k_(1),k_(2)ne0]`
As per question , y =p+q `rArr y =k_(1)x +(k_(2))/(x)…………(1)`
Now if x =1 ,then y =-1
`therefore` by (1) we get , -1=`k_(1).1 +(k_(2))/(1) or, k_(1) +k_(2) =-1...........(2)`
Also , if x =3 ,then y=5 ,
`therefore` from (1) we get , 5 =`k_(1) .3 +(k_(2))/(3)` or , 15 =`9k_(1)+k_(2) or , 9k_(1)+k_(2) =15 ..............(3) `
Then subtracting (2) from (3) we get , 8`k_(1)=16rArr k_(1)=2`
From (2) we get ,`k_(1) =-1-2=-3` .
`therefore` (1) becomes , y =2x-`3/x`.
Hence the required relation between x and y is y =2x`-3/x` .