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

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

\(\Rightarrow\frac{3a-2b}{5a+2b}=\frac{3c-2d}{2c+2d}\) ( đpcm )

a) \(\hept{\begin{cases}\frac{a}{b}=\frac{c}{d}\Rightarrow\frac{a}{c}=\frac{b}{d}=\frac{5a}{5c}=\frac{3b}{3d}=\frac{5a-3b}{5c-3d}\\\frac{a}{c}=\frac{b}{d}\Rightarrow\frac{3a}{3c}=\frac{2b}{2d}=\frac{3a+2b}{3c+2d}\end{cases}}\)

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

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

b) Chứng minh tương tự 

ko biet nghen

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

Áp dụng dãy tỉ số bằng nhau ta có:

\(\frac{a}{c}=\frac{b}{d}=\frac{5a}{5c}=\frac{3b}{3d}=\frac{5a-3b}{5c-3d}\)

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

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

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

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

\(\frac{a}{b}=\frac{c}{d}\Rightarrow\frac{a}{c}=\frac{b}{d}=\frac{5a}{5c}=\frac{3b}{3d}=\frac{5a-3b}{5c-3d}\)

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

=> \(\frac{5a-3b}{5c-3d}=\frac{3a+2b}{3c+2d}\) ( Vì cùng bằng \(\frac{a}{c}\))

\(\frac{a}{b}=\frac{c}{d}\Rightarrow\)\(\frac{a}{c}=\frac{b}{d}\)\(\Rightarrow\)\(\frac{5a}{5c}=\frac{3b}{3d}=\frac{3a}{3c}=\frac{2b}{2d}\)

=> \(\frac{5a-3b}{5c-3d}=\frac{3a+2b}{3c+2d}\Rightarrow\)\(\frac{5a-3b}{3a+2b}=\frac{5c-3d}{3c+2d}\) (đpcm)

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

\(\Rightarrow\frac{5a}{5c}=\frac{2b}{2d}\)

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

\(\frac{5a}{5c}=\frac{2b}{2d}=\frac{5a+2b}{5c+2d}=\frac{5a-2b}{5c-2d}\)

                 \(\Rightarrow\frac{5a+2b}{5a-2b}=\frac{5c+2d}{5c-2d}\left(đpcm\right)\)

ta có:

\(\frac{5a+2b}{5a-2b}=\frac{5c+2d}{5c-2d}\Rightarrow\frac{5a+2b}{5c+2d}=\frac{5a-2b}{5c-2d}\)

\(\frac{a}{b}=\frac{c}{d}\Rightarrow\frac{5a}{5c}=\frac{2b}{2d}=\frac{5a-2b}{5c-2d}=\frac{5a+2b}{5c+2d}\)(đpcm)

Đặt \(\frac{a}{b}=\frac{c}{d}=k\)

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

\(\Rightarrow\frac{5a+2b}{5a-2b}=\frac{5bk+2b}{5bk-2b}=\frac{b\left(5k+2\right)}{b\left(5k-2\right)}=\frac{5k+2}{5k-2}\left(1\right)\)

\(\Rightarrow\frac{5c+2d}{5c-2d}=\frac{5dk+2d}{5dk-2d}=\frac{d\left(5k+2\right)}{d\left(5k-2\right)}=\frac{5k+2}{5k-2}\left(2\right)\)

Từ (1) và (2) 

\(\Rightarrow\frac{5a+2b}{5a-2b}=\frac{5c+2d}{5c-2d}\left(\text{đpcm}\right)\)

\(\frac{a}{b}=\frac{c}{d}\Rightarrow\frac{a}{c}=\frac{b}{d}=\frac{4a}{4c}=\frac{2b}{2d}=\frac{4a-2b}{4c-2d}=\frac{5a}{5c}=\frac{2b}{2d}=\frac{5a+2b}{5c+2d}\)

Suy ra \(\frac{4a-2b}{4c-2d}=\frac{5a+2b}{5c+2d}\)Suy ra điều phải chứng minh: \(\frac{4a-2b}{5a+2b}=\frac{4c-2d}{5c+2d}\)

Ta có:

a/b =c/d

⟹a/c=b/d

Áp dụng tính chất dãy tỉ số bằng nhau ta có

a/c=b/d=5a+2b/5c+2b=5a-2b/5c-2d

Vì 5a=2b/5c=2d=5a-2b/5c-2d

⟹5a+2b/5a-2b=5c+2d/5c-2d

Ta có: \(\frac{a}{b}=\frac{c}{d}\Rightarrow\frac{a}{c}=\frac{b}{d}.\)

\(\Rightarrow\frac{5a}{5c}=\frac{2b}{2d}.\)

Áp dụng tính chất dãy tỉ số bằng nhau ta được:

\(\frac{5a}{5c}=\frac{2b}{2d}=\frac{5a+2b}{5c+2d}\) (1)

\(\frac{5a}{5c}=\frac{2b}{2d}=\frac{5a-2b}{5c-2d}\) (2)

Từ (1) và (2) \(\Rightarrow\frac{5a+2b}{5c+2d}=\frac{5a-2b}{5c-2d}\)

\(\Rightarrow\frac{5a+2b}{5a-2b}=\frac{5c+2d}{5c-2d}\left(đpcm\right).\)

Chúc bạn học tốt!

Đặt a/b=c/d=k

=>a=bk; c=dk

a: \(\dfrac{3a+2b}{a}=\dfrac{3bk+2b}{bk}=\dfrac{3k+2}{k}\)

\(\dfrac{3c+2d}{c}=\dfrac{3dk+2d}{dk}=\dfrac{3k+2}{k}\)

Do đó: \(\dfrac{3a+2b}{a}=\dfrac{3c+2d}{c}\)

b: \(\dfrac{2a-3b}{b}=\dfrac{2bk-3b}{b}=2k-3\)

\(\dfrac{2c-3d}{d}=\dfrac{2dk-3d}{d}=2k-3\)

Do đó: \(\dfrac{2a-3b}{b}=\dfrac{2c-3d}{d}\)

c: \(\dfrac{a}{a-2b}=\dfrac{bk}{bk-2b}=\dfrac{k}{k-2}\)

\(\dfrac{c}{c-2d}=\dfrac{dk}{dk-2d}=\dfrac{k}{k-2}\)

Do đó: \(\dfrac{a}{a-2b}=\dfrac{c}{c-2d}\)