`1/(2x)-1/y=-1 , 1/x+1/(2y)=8`

Explanation

The given equations are `(1)/(2x) - (1)/(y) = - 1` ….(1)
and `(1)/(x) + (1)/(2y) = 8` …. (2)
Let `(1)/(x) = u` and `(1)/(y) = v` then equation (1) and (2) can be written as
`(u)/(2) - v = - 1`
implies u - 2v = - 2 ….(3)
and `u + (v)/(2) = 8`
implies 2u + v = 16 .... (4)
Multiplying equation (3) by 1 and (4) by 2, we get
u - 2v = - 2. ....(5)
4u + 2v = 32 .... (6)
Adding equations (5) and (6), we get
5u = 30
implies u = 6
Substituting u = 6 in equation (3), we get
6 - 2v = - 2
implies -2v = - 8
implies v = 4
Now, `(1)/(x) = u` implies `(1)/(x) = 6` implies `x = (1)/(6)`
and `(1)/(y) = v` implies `(1)/(y) = 4` implies `y = (1)/(4)`
Hence, the solution is `{:(x = (1)/(6)),(y = (1)/(6)):}}`