Question

Cho \(\frac{2a+b}{a-2b}\)= \(\frac{2c+d}{c-2d}\). Chứng minh \(\frac{a}{b}\)= \(\frac{c}{d}\)

Answer 1

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

=> (2a + b)(c - 2d) = (a - 2b)(2c + d)

=> 2ac - 4ad + bc - 2bd = 2ac + ad - 4bc  - 2bd

=> -4ad + bc = ad - 4bc

=> -4ad - ad = -4bc - bc

=> -5ad = - 5bc

=> ad = bc

=> \(\frac{a}{b}=\frac{c}{d}\)(đpcm)

Answer 2

Theo bài ra ta có : 

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

\(\Leftrightarrow2ac-4ad+bc-2db=2ca-4bc+da-2bd\)

\(\Leftrightarrow-5ad+5bc=0\Leftrightarrow-5ab=-5bc\)

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

Answer 3

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

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

<=> 2ac - 4ad + bc - 2bd = 2ac + ad - 4bc - 2bd

<=> 2ac - 4ad + bc - 2bd - 2ac - ad + 4bc + 2bd = 0

<=> -5ad + 5bc = 0

<=> -5ad = -5bc

<=> ad = bc ( chia cả hai vế cho -5 )

<=> a/b = c/d ( đpcm )

Answer 4

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

\(\Leftrightarrow\left(2a+b\right)\left(c-2d\right)=\left(a-2b\right)\left(2c+d\right)\)

\(\Leftrightarrow2ac-4ad+bc-2bd=2ac+ad-4bc-2bd\)

\(\Leftrightarrow2ac-4ad+bc-2bd-2ac-ad+4bc+2bd=0\)

\(\Leftrightarrow-5ad+5bc=0\)

\(\Leftrightarrow-5ad=-5bc\)

\(\Leftrightarrow ad=bc\)

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