Áp dụng dãy tỉ số bằng nhau ta có:

\(\frac{2A+3C}{2B+3D}=\frac{2A-3C}{2B-3D}=\frac{2A+3C+2A-3C}{2B+3D+2B-3D}=\frac{4A}{4B}=\frac{A}{B}\left(1\right)\)\(\frac{2A+3C}{2B+3D}=\frac{2A-3C}{2B-3D}=\frac{2A+3C-2A+3C}{2B+3D-2B+3D}=\frac{6C}{6D}=\frac{C}{D}\left(2\right)\)

Từ (1) và (2) suy ra : \(\frac{A}{B}=\frac{C}{D}\)

Giải :

Từ đảng thức : \(\frac{2a+3c}{2b+3d}=\frac{2a-3c}{2b-3d}\)

\(\Rightarrow\left(2a+3c\right).\left(2b-3d\right)=\left(2b+3d\right).\left(2a-3c\right)\)

\(\Rightarrow4ab-6ad+6bc-9cd=4ab-6bc+6ad-9cd\)

\(\Rightarrow\left(4ab-6ad+6bc-9cd\right)-\left(4ab-6bc+6ad-9cd\right)=0\)

\(\Rightarrow4ab-6ad+6bc-9cd-4ab+6bc-6ad+9cd=0\)

\(\Rightarrow\left(4ab-4ab\right)-\left(6ad+6ad\right)+\left(6bc+6bc\right)-\left(9cd-9cd\right)=0\)

\(\Rightarrow-12ad+12bc=0\)

\(\Rightarrow12bc=12ad\)

\(\Rightarrow bc=ad\)

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

Đặt a/b=c/d=k

=>a=bk; c=dk

a: \(\left(a+c\right)\cdot\left(b-d\right)=\left(bk+dk\right)\left(b-d\right)=k\left(b^2-d^2\right)\)

\(\left(a-c\right)\left(b+d\right)=\left(bk-dk\right)\left(b+d\right)=k\left(b^2-d^2\right)\)

Do đó: \(\left(a+c\right)\left(b-d\right)=\left(a-c\right)\left(b+d\right)\)

b: \(\left(2a+3c\right)\left(2b-3d\right)=\left(2bk+3dk\right)\left(2b-3d\right)=k\left(4b^2-9d^2\right)\)

\(\left(2a-3c\right)\left(2b+3d\right)=\left(2bk-3dk\right)\left(2b+3d\right)=k\left(4b^2-9d^2\right)\)

Do đó: \(\left(2a+3c\right)\left(2b-3d\right)=\left(2a-3c\right)\left(2b+3d\right)\)

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

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

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

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

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

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

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

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

Vì theo định lí sgk thì

\(\frac{a}{b}=\frac{c}{d}\)=>\(\frac{a-c}{b-d}=\frac{a+c}{b+d}\)từ định lí đó suy ra \(\frac{2a-3c}{2b-3d}=\frac{2a+3c}{2b+3d}\)

bạn à viết sai đề rồi nhá

vế phải dưới mẫu là 2b + 3d chứ?

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

Theo tính chất dãy tỉ số bằng nhau, ta có:

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

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

cho \(\frac{a}{b}\)= \(\frac{c}{d}\)

Ta có: \(\frac{a}{b}=\frac{c}{d}\)

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

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

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

Vậy \(\frac{a}{b}=\frac{c}{d}\) thì \(\frac{2a+3c}{2b+3d}=\frac{2a-3c}{2b-3d}\) (đpcm).

_Học tốt_

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

Chúc bạn học tốt!

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

=>\(\frac{2a-3c}{2b-3d}=\frac{2bk-3dk}{2b-3d}=\frac{k.\left(2b-3d\right)}{2b-3d}=k\)

suy ra:\(\frac{a}{b}=\frac{c}{d}=\frac{2a-3c}{2b-3d}\)( vì cùng = k)