Đặt \(\frac{a}{b}=\frac{c}{d}=k\Rightarrow a=bk;c=dk\)

Suy ra : \(\frac{2a+13b}{3a-7b}=\frac{2bk+13b}{3bk-7b}=\frac{b.\left(2k+13\right)}{b.\left(3k-7\right)}=\frac{2k+13}{3k-7}\)

              \(\frac{2c+13d}{3c-7d}=\frac{2dk+13d}{3dk-7d}=\frac{d\left(2k+13\right)}{d\left(3k-7\right)}=\frac{2k+13}{3k-7}\)

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

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

<=> (2a+13b)(3c-7d)=(2c+13d)(7a-7b)

<=>6ac-14ad+39bc-91bd=6c-14bc+39ab-91bd

<=>39bc-14ab=39ab-14bc

<=> bc=ab

<=>\(\frac{a}{b}=\frac{c}{d}\)

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

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

Áp dụng tính chất của dãy tỉ số bằng nhau, ta có:

\(\frac{2a+13b}{2c+13d}=\frac{3a-7b}{3c-7d}=\frac{2a+13b+3a-7b}{2c+13d+3c-7d}=\frac{5a+6b}{5c+6d}\)

\(\Rightarrow\frac{5a+6b}{5c+6d}\Rightarrow\frac{5a}{5c}=\frac{6b}{6d}\)

\(\Rightarrow\frac{a}{c}=\frac{b}{d}\left(đpcm\right)\)

cau hoi tuong tu
 

ko muốn nghĩ thì khỏi làm đi -_-

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

\(\Leftrightarrow6ac-14ad+39bc-91bd=6ac+39ad-14bc-91bd\)

\(\Leftrightarrow-14ad+39bc=39ad-14bc\)\(\Leftrightarrow53bc=53ad\)\(\Leftrightarrow bc=ad\)

\(\Leftrightarrow\frac{a}{b}=\frac{c}{d}\)

Đặt \(\frac{a}{b}=\frac{c}{d}=k\)

\(\Rightarrow a=bk;c=dk\)

\(\frac{2a+13b}{3a-7b}=\frac{2bk+13b}{3bk-7b}=\frac{b\left(2k+13\right)}{b\left(3k-7\right)}=\frac{2k+13}{3k-7}\left(1\right)\)

\(\frac{2c+13d}{3c-7d}=\frac{2dk+13d}{3dk-7d}=\frac{d\left(2k+13\right)}{d\left(3k-7\right)}=\frac{2k+13}{3k-7}\left(2\right)\)

Từ \(\left(1\right)\) và (2) \(\Rightarrow\frac{a}{b}=\frac{c}{d}\)( đpcm ) 

Chúc bạn học tốt !!!

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

\(\Rightarrow\frac{a}{b}=\frac{c}{d}\)