Question 984771

If  x = y+z,  y = z+x,  z = x+y,  then what is the value of {{{(1/(x+1))}}} + {{{(1/(y+1))}}} + {{{(1/(z+1))}}} ?
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You have the system of three linear equations with three unknowns


{{{system(x - y - z = 0,
-x + y - z = 0,
-x - y + z = 0)}}}.


Add all three equations.  You will get


-x - y - z = 0,     or


x + y + z = 0.      (*)


Now,  add the last equation and the first one of the system.  You will get


2x = 0,   and,   hence,   x = 0.


Next,  add the equation  (*)  and the second equation of the system.  You will get


2y = 0,   and,   hence,   y = 0.


Finally,  add the equation  (*)  and the third equation of the system.  You will get


2z = 0,   and,   hence,   z = 0.


Thus your system has a unique solution   x = y = z = 0.


Hence,  the expression under the question is 


{{{(1/(x+1))}}} + {{{(1/(y+1))}}} + {{{(1/(z+1))}}} = 3.