Пускай 1/х = a

1/y = b

1/z = c

a + b = 1/2

b + c = 1/4  из него b = 1/4 - c

a + c = 1/8  из него  a = 1/8 - c

Подставим a, b в первое уравнение a + b = 1/2

1/8 - c + 1/4 - c = 1/2

-2c = 1/2 - 1/4 - 1/8

- 2c = 4/8 - 2/8 - 1/8

-2c = 1/8

c = - 1/16

b = 1/4 - c      b = 1/4 - (- 1/16) = 4/16 + 1/16 = 5/16

a = 1/8 - c      a = 1/8 - (-1/16) = 2/16 + 1/16 = 3/16

a = 3/16 = 1 / (16/3)    x = 16/3

b = 5/16 = 1 / (16/5) = 1/3,2    b = 3,2

c = -1/16      z = -16  

Ответ проверил