System of Linear Equation

Question:
If !!x<0!! !!y>0!!
!![x + y +(x/y)]!! !!=!! !!(1/2)!! and !!(x + y)(x/y)!! !!=!! !!(-1/2)!!
Then !!x!! and !!y!! !!?!!

a)!!|x|!! !!=!! !!|y|!!

b)Integers

c)Both !!(a & b)!!

d)None

Solution:
Here let !!P!! !!=!! !!x+y!! and !!Q!! !!=!! !!x/y!! so that
!!P + Q!! !!=!! !!1/2!! and !!PQ!! !!=!! !!-1/2!!

Upon solving we get;
!!P!! !!=!! !!1!! or !!-1/2!!

!!Q!! !!=!! !!-1/2!! or !! 1!!

Case!!-1 :!! When !!P!! !!=!! !!-1/2!! and !!Q!! !!=!! !!1!!
Then,

!!x+y!! !!=!! !!-1/2!! and !!x/y!! !!=!! !!1!!

Solving thses we get;

!!x!! !!=!! !!-1/4!! and !!y!! !!=!! !!-1/4!!.

Here !!x<0!! but !!y!! is negative .So we'll skip this.

Case!!-2:!! When !!P!! !!=!! !!1!! and !!Q!! !!=!! !!-1/2!! so that

!!x+y!! !!=!! !!1!! and !!x/y!! !!=!! !!-1/2!!

Solving these we get ;

!!x!! !!=!! !!-1!! and !!y!! !!=!! !!2!!

Both are integers.

Hence !!(b)!! is the answer.

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