Poser
X = 1/x et Y = 1/y et Z = 1/z

Le système devient
X - 2Y + 3Z = 1
2X + 3Y - Z = 19/6
-3X + Y - 2Z = 19/6

En ajoutant les 3 équations ->
2Y = 1 + (19/3)
2Y = 22/3
Y = 11/3

le systeme devient alors:

X - (22/3) + 3Z = 1
2X + 11 - Z = 19/6

X + 3Z = 25/3
2X - Z = -47/6

X = (25/3) - 3Z
(50/3) - 6Z - Z = -47/6
-7Z = -(47/6) - (50/3)
7Z = 147/6
Z = 7/2

X = (25/3) - 3*(7/2)
X = (25/3) - (21/2)
X = -13/6
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X = 1/x = -13/6  -> x = -6/13
Y = 1/y = 11/3  -> y = 3/11
Z = 1/z = 2/7   -> z = 7/2
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Les solutions sont donc:
x = -6/13
y = 3/11
z = 7/2
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Sauf distraction, refais les calculs.