Bonjour,
1.
[tex]\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0<=>\:\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)\left(xyz\right)=0\left(xyz\right)<=>yz+xz+xy=0<=>\frac{yz+xz+xy}{yz}=\frac{0}{yz}<=>1+\frac{x}{y}+\frac{x}{z}=0<=>\frac{x}{y}+\frac{x}{z}=-1<=>x\left(\frac{1}{y}+\frac{1}{z}\right)=-1[/tex]
2.
[tex]\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0<=>yz+xz+xy=0<=>\frac{yz+xz+xy}{xz}=\frac{0}{xz}<=>\frac{y}{x}+\frac{y}{z}+1=0<=>y\left(\frac{1}{x}+\frac{1}{z}\right)=-1[/tex]
On refait de même et on obtient [tex]z\left(\frac{1}{x}+\frac{1}{y}\right)=-1[/tex]
3.
[tex]-3=x\left(\frac{1}{y}+\frac{1}{z}\right)+y\left(\frac{1}{x}+\frac{1}{z}\right)+z\left(\frac{1}{x}+\frac{1}{y}\right)=\frac{x}{y}+\frac{x}{z}+\frac{y}{x}+\frac{y}{z}+\frac{z}{x}+\frac{z}{y}=\frac{x+z}{y}+\frac{x+y}{z}+\frac{y+z}{x}[/tex]
