Question

Cho a,b,y,x,y,z thỏa mãn \(\hept{\begin{cases}x=by+cz\\y=cz+ax\\z=ax+by\end{cases}}\)

Biết \(a,b,c\ne-1\).Tính giá trị của \(M=\frac{a}{1+a}+\frac{b}{1+b}+\frac{c}{1+c}\)

Answer 1

Có \(x=by+cz\)

=> \(x\left(1+a\right)=ax+x=ax+by+cz\)

=> \(\frac{1}{1+a}=\frac{x}{ax+by+cz}\)

=> \(\frac{a}{1+a}=\frac{ax}{ax+by+cz}\)

Có \(y=cz+ax\)

=> \(y\left(1+b\right)=by+y=by+cz+ax=ax+by+cz\)

=> \(\frac{1}{1+b}=\frac{y}{ax+by+cz}\)

=> \(\frac{b}{1+b}=\frac{by}{ax+by+cz}\)

Có \(z=ax+by\)

=> \(z\left(1+c\right)=cz+z=cz+ax+by=ax+by+cz\)

=> \(\frac{1}{1+c}=\frac{z}{ax+by+cz}\)

=> \(\frac{c}{1+c}=\frac{cz}{ax+by+cz}\)

=> \(M=\frac{a}{1+a}+\frac{b}{1+b}+\frac{c}{1+c}=\frac{ax}{ax+by+cz}+\frac{by}{ax+by+cz}+\frac{cz}{ax+by+cz}\)

\(=\frac{ax+by+cz}{ax+by+cz}=1\)

Vậy giá trị của M là 1