a) \(\frac{a+b-x}{c}+\frac{b+c-x}{a}+\frac{c+a-x}{b}+\frac{4x}{a+b+c}=1\)
\(\Leftrightarrow\frac{a+b-x}{c}+1+\frac{b+c-x}{a}+1+\frac{c+a-x}{b}+1+\frac{4x}{a+b+c}-4=0\)
\(\Leftrightarrow\frac{a+b+c-x}{c}+\frac{a+b+c-x}{a}+\frac{a+b+c-x}{b}+\frac{4x-4\left(a+b+c\right)}{a+b+c}=0\)
\(\Leftrightarrow\left(x-a-b-x\right)\left(\frac{1}{bc}+\frac{1}{ca}+\frac{1}{ab}\right)=0\)
b)đề bài như trên
\(\Leftrightarrow\left(\frac{x-a-b-c}{bc}\right)+\left(\frac{x-b}{ca}-\frac{1}{a}-\frac{1}{c}\right)+\left(\frac{x-c}{ab}-\frac{1}{a}-\frac{1}{b}\right)=0\)
\(\Leftrightarrow\left(x-a-b-c\right)\left(\frac{1}{bc}+\frac{1}{ca}+\frac{1}{ab}\right)=0\)
\(a,\frac{a+b-x}{c}+\frac{b+c-x}{a}+\frac{c+a-x}{b}+\frac{4x}{a+b+c}=1\)
\(a,\frac{a+b-x}{c}+1+\frac{b+c-x}{a}+1+\frac{c+a-x}{b}+1+\frac{4x}{a+b+c}-4=0\)
\(\Leftrightarrow\frac{a+b+c-x}{c}+\frac{a+b+c-x}{a}+\frac{a+b+c-x}{b}-\frac{4a+4b+4c-4x}{a+b+c}=0\)
\(\Leftrightarrow\left(a+b+c-x\right)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}-\frac{4}{a+b+c}\right)=0\)
\(\Leftrightarrow a+b+c-x=0\)Do \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}-\frac{4}{a+b+c}\ne0\)
\(\Leftrightarrow x=a+b+c\)
Vậy phương trình có nghiệm \(x=a+b+c\)
\(b,\frac{1}{2}\left(\frac{x-a}{bc}+\frac{x-b}{ac}+\frac{x-c}{ab}\right)=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\)
\(\Leftrightarrow\frac{x-a}{bc}+\frac{x-b}{ac}+\frac{x-c}{ab}-\frac{2}{a}-\frac{2}{b}-\frac{2}{c}=0\)
\(\Leftrightarrow\left(\frac{x-a}{bc}-\frac{1}{b}-\frac{1}{c}\right)+\left(\frac{x-b}{ac}-\frac{1}{a}-\frac{1}{c}\right)+\left(\frac{x-c}{ab}-\frac{1}{a}-\frac{1}{b}\right)=0\)
\(\Leftrightarrow\frac{x-a-b-c}{bc}+\frac{x-a-b-c}{ac}+\frac{x-a-b-c}{ab}=0\)
\(\Leftrightarrow\left(x-a-b-c\right)\left(\frac{1}{bc}+\frac{1}{ac}+\frac{1}{ab}\right)=0\)
\(\Leftrightarrow x-a-b-c=0\)Vì \(\frac{1}{bc}+\frac{1}{ac}+\frac{1}{ab}\ne0\)
\(\Leftrightarrow x=a+b+c\)
Vậy phương trình có nghiệm \(x=a+b+c\)
