Question

CMR nếu \(\frac{u+2}{u-2}=\frac{v+3}{v-3}\Rightarrow\frac{u}{2}=\frac{v}{3}\)

Answer 1

Có : \(\frac{u+2}{u-2}=\frac{v+3}{v-3}\)

\(\Leftrightarrow\frac{u+2}{v+3}=\frac{u-2}{v-3}\)

Theo tính chất dãy tỉ số , có :

\(\frac{u+2}{v+3}=\frac{u-2}{v-3}=\frac{u+2+u-2}{v+3+v-3}=\frac{u+2-u+2}{v+3-v+3}\)

\(\Rightarrow\frac{2u}{2v}=\frac{4}{6}\)

\(\Leftrightarrow\frac{u}{v}=\frac{2}{3}\Leftrightarrow\frac{u}{2}=\frac{v}{3}\)

Answer 2

Ta có:

  \(\frac{u+2}{u-2}=\frac{v+3}{v-3}\)

<=> (u+2).(v-3)=(u-2).(v+3)

<=>uv+2v-3u-6=uv-2v+3u-6

<=>2v-3u=3u-2v

<=>2v+2v=3u+3u

<=>4v=6u

<=>2v=3u

<=>\(\frac{u}{2}=\frac{v}{3}\)

Answer 3

\(\frac{u+2}{u-2}=\frac{v+3}{v-3}\)
nhân chéo,ta có
         (u+2)(v-3)=(u-2)(v+3)
<=> uv-3u+2v-6=uv+3u-2v-6          
=>   uv-3u+2v-6-uv-3u+2v+6=0
<=> -6u+4v=0
<=> 4v      =6u
<=> 2v      =3u
<=> \(\frac{u}{2}=\frac{v}{3}\)(đpcm)

Answer 4

\(\frac{u+2}{u-2}=\frac{v+3}{v-3}\)

\(\Rightarrow\left(u+2\right)\left(v-3\right)=\left(u-2\right)\left(v+3\right)\)

\(\Rightarrow uv-3u+2v-6=uv+3u-2v-6\)

\(\Rightarrow uv-3u+2v-6-\left(uv+3u-2v-6\right)=0\)

\(\Rightarrow uv-3u+2v-6-uv-3u+2v+6=0\)

\(\Rightarrow\left(uv-uv\right)+\left(2v+2v\right)+\left(-3u-3u\right)+\left(-6+6\right)=0\)

\(\Rightarrow4v-6u=0\)

\(\Rightarrow2v-3u=0\)

\(\Rightarrow2v=3u\)

\(\Rightarrow\frac{u}{2}=\frac{v}{3}\) (đpcm)