Question

[lequangdung]

(2a+13b)/(3a-7b)=(2c+13d)/(3c-7d) cm a/b=c/d

Answer 1

[Nguyễn Linh Chi]

\(\frac{2a+13b}{3a-7b}=\frac{2c+13d}{3c-7d}\)

\(\Rightarrow\frac{2a+13b}{2c+13d}=\frac{3a-7b}{3c-7d}=\frac{3\left(2a+13b\right)}{3\left(2c+13d\right)}=\frac{2\left(3a-7b\right)}{2\left(3c-7d\right)}\)

\(=\frac{3\left(2a+13b\right)-2\left(3a-7b\right)}{3\left(2c+13d\right)-2\left(3c-7d\right)}=\frac{53b}{53d}=\frac{b}{d}\)(1)

\(\Rightarrow\frac{2a+13b}{2c+13d}=\frac{3a-7b}{3c-7d}=\frac{7\left(2a+13b\right)}{7\left(2a+13d\right)}=\frac{13\left(3a-7b\right)}{13\left(3c-7d\right)}\)

\(=\frac{7\left(2a+13b\right)+13\left(3a-7b\right)}{7\left(2c+13d\right)+13\left(3c-7d\right)}=\frac{53a}{53c}=\frac{a}{c}\)(2)

Từ (1) (2) => \(\frac{b}{d}=\frac{a}{c}\Rightarrow\frac{c}{d}=\frac{a}{b}\)

Answer 2

[Nguyệt]

\(\frac{2a+13b}{3a-7b}=\frac{2c+13d}{3c-7d}\Leftrightarrow\left(2a+13b\right).\left(3c-7d\right)=\left(2c+13d\right).\left(3a-7b\right)\)

\(\Rightarrow6ac-14ad+39bc-91bd=6ac-14cb+39ad-91bd\)

\(\Rightarrow-14ad+39bc=-14cb+39ad\)

\(\Rightarrow-53ad=-53bc\Rightarrow ad=bc\Rightarrow\frac{a}{b}=\frac{c}{d}\left(đpcm\right)\)