Lời giải:
\(\frac{a}{b}=\frac{c}{d}=k\Rightarrow a=bk, c=dk \)

Khi đó:

\(\frac{2002a+2003b}{2002a-2003b}=\frac{2002bk+2003b}{2002bk-2003b}=\frac{b(2002k+2003)}{b(2002k-2003)}=\frac{2002k+2003}{2002k-2003}(1)\)

\(\frac{2002c+2003d}{2002c-2003d}=\frac{2002dk+2002d}{2002dk-2003d}=\frac{d(2002k+2003)}{d(2002k-2003)}=\frac{2002k+2003}{2002k-2003}(2)\)

Từ \((1);(2)\Rightarrow \frac{2002a+2003b}{2002a-2003b}=\frac{2002c+2003d}{2002c-2003d}\)

Ta có đpcm.

Xét tỉ lệ thức \(\dfrac{a}{b}=\dfrac{c}{d}\) . Gọi giá trị chung của các tỉ số đó là k, ta có:

\(\dfrac{a}{b}=\dfrac{c}{d}=k\)

=> \(a=k.b,c=k.d\)

Ta có :

( 1 )

= \(\dfrac{2002a+2003b}{2002a-2003b}=\dfrac{2002kb+2003b}{2002kb-2003b}\)

= \(\dfrac{b.\left(2002k+2003\right)}{b.\left(2002k-2003\right)}=\dfrac{2002k+2003}{2002k-2003}\)

( 2 ) \(\dfrac{2002c+2003d}{2002c-2003d}=\dfrac{2002kd+2003d}{2002kd-2003d}\)

= \(\dfrac{d.\left(2002k+2003\right)}{d.\left(2002k-2003\right)}=\dfrac{2002k+2003}{2002k-2003}\)

Từ ( 1 ) và ( 2 ) => \(\dfrac{2002a+2003b}{2002a-2003b}=\dfrac{2002c+2003d}{2002c-2003d}\)

Từ \(\dfrac{a}{b}=\dfrac{c}{d}\Rightarrow\dfrac{a}{c}=\dfrac{b}{d}=\dfrac{2002a}{2002c}=\dfrac{2003b}{2003d}\)

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

\(\dfrac{2002a}{2002c}=\dfrac{2003b}{2003d}=\dfrac{2002a+2003b}{2002c+2003d}=\dfrac{2002a-2003b}{2002c-2003d}\)

Từ \(\dfrac{2002a+2003b}{2002c+2003d}=\dfrac{2002a-2003b}{2002c-2003d}\) =>\(\dfrac{2002a+2003b}{2002a-2003b}=\dfrac{2002c+2003d}{2002c-2003d}\)

Ta có : \(\frac{a}{b}=\frac{c}{d}\)

\(\Rightarrow\frac{a}{c}=\frac{b}{d}=\frac{2002a}{2002c}=\frac{2003b}{2003d}=\frac{2002a+2003b}{2002c+2003d}=\frac{2002a-2003b}{2002c-2003d}\)

Suy ra : \(\frac{2002a+2003b}{2002a-2003b}=\frac{2002c+2003d}{2002c-2003d}\) (đpcm)

a, Áp dụng t/c dtsbn:

\(\dfrac{a}{b}=\dfrac{c}{d}\Rightarrow\dfrac{a}{c}=\dfrac{b}{d}=\dfrac{a+b}{c+d}=\dfrac{a-b}{c-d}\Rightarrow\dfrac{a+b}{a-b}=\dfrac{c+d}{c-d}\)

b, Áp dụng t/c dtsbn:

\(\dfrac{a}{b}=\dfrac{c}{d}\Rightarrow\dfrac{a}{c}=\dfrac{b}{d}=\dfrac{2a}{2c}=\dfrac{5b}{5d}=\dfrac{3a}{4c}=\dfrac{4b}{4d}=\dfrac{2a+5b}{2c+5d}=\dfrac{3a-4b}{3c-4d}\Rightarrow\dfrac{2a+5b}{3a-4b}=\dfrac{2c+5d}{3c-4d}\)

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

Ta có \(\dfrac{ab}{cd}=\dfrac{bk\cdot b}{dk\cdot d}=\dfrac{b^2k}{d^2k}=\dfrac{b^2}{d^2}\)

\(\dfrac{\left(a-b\right)^2}{\left(c-d\right)^2}=\dfrac{\left(bk-b\right)^2}{\left(dk-d\right)^2}=\dfrac{b^2\left(k-1\right)^2}{d^2\left(k-1\right)^2}=\dfrac{b^2}{d^2}\)

Do đó \(\dfrac{ab}{cd}=\dfrac{\left(a-b\right)^2}{\left(c-d\right)^2}\)

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

Ta có \(\dfrac{ac}{bd}=\dfrac{bk\cdot dk}{bd}=k^2\)

\(\dfrac{a^2+c^2}{b^2+d^2}=\dfrac{b^2k^2+d^2k^2}{b^2+d^2}=\dfrac{k^2\left(b^2+d^2\right)}{b^2+d^2}=k^2\)

Do đó \(\dfrac{ac}{bd}=\dfrac{a^2+c^2}{b^2+d^2}\)

\(\Leftrightarrow\dfrac{a+b}{c+d}=\dfrac{a-b}{c-d}=\dfrac{a}{c}=\dfrac{b}{d}=>\dfrac{a}{b}=\dfrac{c}{d}\)

Lời giải:

$\frac{a+b}{a-b}=\frac{c+d}{c-d}$

$\Rightarrow (a+b)(c-d)=(a-b)(c+d)$

$\Rightarrow ac-ad+bc-bd=ac+ad-bc-bd$

$\Rightarrow 2ad=2bc$

$\Rightarrow ad=bc$

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

\(\dfrac{a+b}{a-b}=\dfrac{c+d}{c-d}\Rightarrow\dfrac{a+b}{c+d}=\dfrac{a-b}{c-d}\)

Áp dụng t/c dãy tỉ số bằng nhau, ta có:

\(\dfrac{a+b}{c+d}=\dfrac{a-b}{c-d}=\dfrac{a+b+a-b}{c+d+c-d}=\dfrac{2a}{2c}=\dfrac{a}{c}\left(1\right)\)

\(\dfrac{a+b}{c+d}=\dfrac{a-b}{c-d}=\dfrac{a+b-a+b}{c+d-c+d}=\dfrac{2b}{2d}=\dfrac{b}{d}\left(2\right)\)

\(\left(1\right),\left(2\right)\Rightarrow\dfrac{a}{c}=\dfrac{b}{d}\Rightarrow\dfrac{a}{b}=\dfrac{c}{d}\)

Bài 2: 

Đặt a/b=c/d=k

=>a=bk; c=dk

a: \(\dfrac{a}{a+b}=\dfrac{bk}{bk+b}=\dfrac{k}{k+1}\)

\(\dfrac{c}{c+d}=\dfrac{dk}{dk+d}=\dfrac{k}{k+1}\)

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

b: \(\dfrac{7a^2+5ac}{7a^2-5ac}=\dfrac{7\cdot b^2k^2+5\cdot bk\cdot dk}{7\cdot b^2k^2-5\cdot bk\cdot dk}\)

\(=\dfrac{7b^2k^2+5bdk^2}{7b^2k^2-5bdk^2}=\dfrac{7b^2+5bd}{7b^2-5bd}\)(đpcm)

\(\dfrac{a}{b}=\dfrac{c}{d}=>\dfrac{a}{b}+1=\dfrac{c}{d}+1=>\dfrac{a+b}{b}=\dfrac{c+d}{d}\)

\(\dfrac{a}{b}=\dfrac{c}{d}=>\dfrac{a}{b}-1=\dfrac{c}{d}-1=>\dfrac{a-b}{b}=\dfrac{c-d}{d}\)

\(\dfrac{a}{b}=\dfrac{c}{d}=>ad=cb=>ad+ac=cb+ac\)

\(=>a\left(c+d\right)=c\left(a+b\right)=>\dfrac{a}{c}=\dfrac{a+b}{c+d}=>\dfrac{a}{a+b}=\dfrac{c}{c+d}\)

a) \(\dfrac{a}{b}=\dfrac{c}{d}\)

nên \(\dfrac{a}{b}+1=\dfrac{c}{d}+1\)

hay \(\dfrac{a+b}{b}=\dfrac{c+d}{d}\)

b) \(\dfrac{a}{b}=\dfrac{c}{d}\)

nên \(\dfrac{a}{b}-1=\dfrac{c}{d}-1\)

hay \(\dfrac{a-b}{b}=\dfrac{c-d}{d}\)

Giải:

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

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

\(\dfrac{a}{c}=\dfrac{b}{d}=\dfrac{a+b}{c+d}=\dfrac{a-b}{c-d}\)

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

Vậy...

Đặt \(\dfrac{a}{b}=\dfrac{c}{d}=k\)

=>\(\left\{{}\begin{matrix}a=b.k\\c=d.k\end{matrix}\right.\) (1)

Thay (1) vào:

\(\dfrac{a+b}{a-b}=\dfrac{b.k+b}{b.k-b}=\dfrac{b.\left(k+1\right)}{b.\left(k-1\right)}=\dfrac{k+1}{k-1}\) (2)

\(\dfrac{c+d}{c-d}=\dfrac{d.k+d}{d.k-d}=\dfrac{d.\left(k+1\right)}{d.\left(k-1\right)}=\dfrac{k+1}{k-1}\) (3)

Từ (2) và (3) =>\(\dfrac{a+b}{a-b}=\dfrac{c+d}{c-d}=\dfrac{k+1}{k-1}\)

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

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

\(\dfrac{a}{c}=\dfrac{b}{d}=\dfrac{a+b}{c+d}=\dfrac{a-b}{c-d}\)

Áp dụng tính chất hoán vị ta được:

\(\dfrac{a+b}{c+d}=\dfrac{a-b}{c-d}\Rightarrow\dfrac{a+b}{a-b}=\dfrac{c+d}{c-d}\)

Vậy từ tỉ lệ thức \(\dfrac{a}{b}=\dfrac{c}{d}\Rightarrow\dfrac{a+b}{a-b}=\dfrac{c+d}{c-d}\)