c'est troopp durrr :(

1/x + 1/y + 1/z = 5
2/x - 3/y - 1/z = 9
5/x + 7/y - 9/z = - 5

Avec x ≠ 0
         y ≠ 0
         z ≠ 0

On effectue un changement de variable, on pose : 
c = 1/x
b = 1/y
a = 1/z

c + b + a = 5
2c - 3b - a = 9
5c + 7b - 9a = - 5

a = 5 - c - b
2c - 3b - a = 9
5c + 7b - 9a = - 5

a = 5 - c - b
2c - 3b - (5 - c - b) = 9
5c + 7b - 9(5 - c - b) = - 5

a = 5 - c - b
2c - 3b - 5 + c + b = 9
5c + 7b - 9(5 - c - b) = - 5

a = - b - c + 5
- 2b + 3c - 5 = 9
16b + 14c - 45 = - 5

a = - b - c + 5
b = (- 9 + 3c - 5)/2
16b + 14c - 45 = - 5

a = - b - c + 5
b = (- 14 + 3c)/2
16b + 14c - 45 = - 5

a = - (- 14 + 3c)/2 - c + 5
b = (- 14 + 3c)/2
16[(- 14 + 3c)/2] + 14c - 45 = - 5

a = - (- 14 + 3c)/2 - c + 5
b = (- 14 + 3c)/2
(- 224 + 48c)/2 + 14c - 45 = - 5

a = - (- 14 + 3c)/2 - c + 5
b = (- 14 + 3c)/2
c = 4

a = - (- 14 + 3 x 4)/2 - 4 + 5
b = (- 14 + 3 x 4)/2
c = 4

a = - (- 2)/2 + 1
b = (- 2)/2
c = 4

a = 2
b = - 1
c = 4

On remplace les variables : 

1/z = 2
1/y = - 1
1/x = 4

z = 1/2
y = - 1/1
x = 1/4

x = 1/4
y = - 1
z = 1/2