Question

Cho \(\frac{a+5}{a-5}=\frac{b+6}{b-6}\left(a\ne5;b\ne6\right)\) Chứng minh rằng \(\frac{a}{b}=\frac{5}{6}\)

Answer 1

\(\frac{a+5}{a-5}=\frac{b+6}{b-6}\)

\(\Leftrightarrow\frac{a-5+10}{a-5}=\frac{b-6+12}{b-6}\)

\(\Leftrightarrow1+\frac{10}{a-5}=1+\frac{12}{b-6}\)

\(\Leftrightarrow\frac{10}{a-5}=\frac{12}{b-6}\)

\(\Rightarrow10.\left(b-6\right)=12.\left(a-5\right)\)

\(10b-60=12a-60\)

\(10b=12a\)

\(\frac{a}{b}=\frac{10}{12}=\frac{5}{6}\)

Answer 2

\(\frac{a+5}{a-5}=\frac{b+6}{b-6}\)

\(=>\frac{a+5}{b+6}=\frac{a-5}{b-6}=\frac{a+5-\left(a-5\right)}{b+6-\left(b-6\right)}=\frac{a+5-a+5}{b+6-b+6}=\frac{10}{12}=\frac{5}{6}\)

\(\frac{a+5}{b+6}=\frac{a-5}{b-6}=\frac{a+5+\left(a-5\right)}{b+6+\left(b-5\right)}=\frac{a+5+a-5}{b+6+b-6}=\frac{2a}{2b}=\frac{a}{b}\)

=>\(\frac{a}{b}=\frac{a+5}{b+5}=\frac{5}{6}\)

=>\(\frac{a}{b}=\frac{5}{6}\)