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\(\dfrac{3a+4b}{5a-6b}=\dfrac{3c+4d}{5c-6d}\)
=> \(\dfrac{3a+4b}{3c+4d}=\dfrac{5a-6b}{5c-6d}\)
ta có
\(\dfrac{3a+4b}{3c+4d}=\dfrac{3a}{3c}=\dfrac{4b}{4d}=\dfrac{a}{c}=\dfrac{b}{d}=>\dfrac{a}{b}=\dfrac{c}{d}\)(đpcm)
Ta có:
\(\dfrac{3a+4b}{5a-6b}=\dfrac{3c+4d}{5c-6d}\)
\(\Leftrightarrow\left(3a+4b\right)\left(5c-6d\right)=\left(3c+4d\right)\left(5a-6b\right)\)
\(\Rightarrow15ac-18ad+20bc-24bd=15ac-18bc+20ad-24bd\)
\(\Rightarrow15ac-15ac-18ad-20ad=-24bd+24bd-18bc-20bc\)
\(\Rightarrow-38ad=-38bc\)
\(\Rightarrow ad=bc\)
\(\Rightarrow\dfrac{a}{b}=\dfrac{c}{d}\)
a/ Đặt :
\(\dfrac{a}{b}=\dfrac{c}{d}=k\)\(\Leftrightarrow\left\{{}\begin{matrix}a=bk\\c=dk\end{matrix}\right.\)
Ta có :
\(VT=\dfrac{a-b}{a+b}=\dfrac{bk-b}{bk+b}=\dfrac{b\left(k-1\right)}{b\left(k+1\right)}=\dfrac{k-1}{k+1}\)\(\left(1\right)\)
\(VP=\dfrac{c-d}{c+d}=\dfrac{dk-d}{dk+d}=\dfrac{d\left(k-1\right)}{d\left(k+1\right)}=\dfrac{k-1}{k+1}\left(2\right)\)
Từ \(\left(1\right)+\left(2\right)\Leftrightarrowđpcm\)
b/ Đặt :
\(\dfrac{a}{b}=\dfrac{c}{d}=k\)\(\Leftrightarrow\left\{{}\begin{matrix}a=bk\\c=dk\end{matrix}\right.\)
\(VT=\dfrac{2a+5b}{3a-4b}=\dfrac{2bk+5b}{3bk-4b}=\dfrac{b\left(2k+5\right)}{b\left(3k-4\right)}=\dfrac{2k+5}{3k-4}\left(1\right)\)
\(VP=\dfrac{2c+5d}{3c-4d}=\dfrac{2dk+5d}{3dk-4d}=\dfrac{d\left(2k+5\right)}{d\left(3k-4\right)}=\dfrac{2k+5}{3k-4}\left(2\right)\)
Từ \(\left(1\right)+\left(2\right)\Leftrightarrowđpcm\)
a) Từ \(\dfrac{a}{b}=\dfrac{c}{d}\) \(\Rightarrow\dfrac{a}{c}=\dfrac{b}{d}\)
Áp dụng tính chất của dãy tỉ số bằng nhau, ta có:
\(\dfrac{a}{c}=\dfrac{b}{d}=\dfrac{a+b}{c+d}=\dfrac{a-b}{c-d}\)
Từ \(\dfrac{a+b}{c+d}=\dfrac{a-b}{c-d}\) \(\Rightarrow\dfrac{c-d}{c+d}=\dfrac{a-b}{a+b}\)
b) Từ \(\dfrac{a}{b}=\dfrac{c}{d}\) \(\Rightarrow\dfrac{a}{c}=\dfrac{b}{d}=\dfrac{2a}{2c}=\dfrac{3a}{3c}=\dfrac{4b}{4d}=\dfrac{5b}{5d}\)
Áp dụng tính chất của dãy tỉ số bằng nhau, ta có:
\(\dfrac{2a}{2c}=\dfrac{3a}{3c}=\dfrac{4b}{4d}=\dfrac{5b}{5d}=\dfrac{2a+5b}{2c+5d}=\dfrac{3a-4b}{3c-4d}\)
Từ \(\dfrac{2a+5b}{2c+5d}=\dfrac{3a-4b}{3c-4d}\) \(\Rightarrow\dfrac{2a+5b}{3a-4b}=\dfrac{2c+5d}{3c-4d}\)
\(\dfrac{2a+13b}{3a-7b}=\dfrac{2c+13d}{3c-7d}\Rightarrow\dfrac{2a+13b}{2c+13d}=\dfrac{3a-7b}{3c-7d}\) (1)
Nhân tư và mẫu vế trái (1) với 3 và vế phải với 13 ta được:
\(\dfrac{2a+13b}{2c+13d}=\dfrac{14a+91b}{14c+91d}=\dfrac{39a-91b}{39c-91d}\)
=\(\dfrac{\left(14a+91b\right)+\left(39a-91b\right)}{\left(14c+91d\right)+\left(39c-91d\right)}=\dfrac{53a}{53c}=\dfrac{a}{c}\) (2)
Nhân tử và mẫu vế trái (1) với 3 và vế phải với 2 ta được:
\(\dfrac{2a+13b}{2c+13d}=\dfrac{6a+39b}{6c+39d}=\dfrac{6a-14b}{6c-14d}=\dfrac{53b}{53d}=\dfrac{b}{d}\) (3)
Từ (2) và (3) suy ra :
\(\dfrac{a}{c}=\dfrac{b}{d}\Rightarrow\dfrac{a}{b}=\dfrac{c}{d}\)
Ta có: \(\frac{3a+4b}{3a-4b}=\frac{3c+4d}{3c-4d}\)
\(\Rightarrow\frac{3a+4b}{3a-4b}-1=\frac{3c+4d}{3c-4d}-1\)
\(\Leftrightarrow\frac{8b}{3a-4b}=\frac{8d}{3c-4d}\)
\(\Rightarrow b\left(3c-4d\right)=d\left(3a-4b\right)\)
\(\Leftrightarrow3bc=3ad\)
\(\Rightarrow\frac{a}{b}=\frac{c}{d}\left(đpcm\right)\)
Ta có: \(\dfrac{2a+13b}{3a-7b}=\dfrac{2c+13d}{3c-7d}\)
\(\Rightarrow\left(2a+13b\right)\left(3c-7d\right)=\left(2c+13d\right)\left(3a-7b\right)\)
\(\Rightarrow6ac+39bc-14ad-91bd=6ac+39ad-14bc-91bd\)
\(\Rightarrow6ac-6ac+39bc+14bc-14ad-39ad-91bd+91bd=0\)
\(\Rightarrow53bc-53ad=0\)
\(\Rightarrow53bc=53ad\)
\(\Rightarrow bc=ad\)
\(\Rightarrow\dfrac{a}{b}=\dfrac{c}{d}\rightarrowđpcm.\)
\(\dfrac{2a+13b}{3a-7b}=\dfrac{2c+13d}{3c-7d}\)
\(\Leftrightarrow\)(2a+13b)(3c-7d)=(2c+13d)(3a-7b)
2a(3c-7d)+13b(3c-7d)=2c(3a-7b)+13d(3a-7b)
6ac-14ad+39bc-91bd=6ac-14bc+39ad+91bd
14ad+39bc+91bd=14bc+39ad+91bd
14ad+39bc=14bc+39ad
39bc=14bc+39ad-14ad
39bc=14bc+25ad
39bc-14bc=25ad
25bc=25ad
bc=ad
Ta có: Điều đề bài cho:
\(\dfrac{a}{b}=\dfrac{c}{d}\Leftrightarrow ad=bc\left(đpcm\right)\)
Ta có:
\(\dfrac{a}{b}=\dfrac{c}{d}\Rightarrow\dfrac{a}{c}=\dfrac{b}{d}\)
\(\Rightarrow\dfrac{a}{c}=\dfrac{b}{d}=\dfrac{2a+5b}{2c+5d}\)
\(\dfrac{a}{c}=\dfrac{b}{d}=\dfrac{3a-4b}{3c-4d}\)
\(\Rightarrow\dfrac{2a+5b}{2c+5d}=\dfrac{3a-4b}{3c-4d}=\dfrac{a}{c}=\dfrac{b}{d}\)
\(\Rightarrow\dfrac{2a+5b}{3a-4b}=\dfrac{2c+5d}{3c-4d}\left(dpcm\right)\)
Đặt \(\dfrac{a}{b}=\dfrac{c}{d}=k\)
\(\Rightarrow\left[{}\begin{matrix}a=bk\\c=dk\end{matrix}\right.\) \(\Rightarrow\dfrac{2bk+5b}{3bk-4b}=\dfrac{2dk+5d}{3dk-4d}\)
\(VT=\dfrac{2a+5b}{3a-4b}=\dfrac{2bk+5b}{3bk-4b}=\dfrac{b\left(2k+5\right)}{b\left(3k-4\right)}=\dfrac{2k+5}{3k-4}\left(1\right)\)
\(VP=\dfrac{2c+5d}{3c-4d}=\dfrac{2dk+5d}{3dk-4d}=\dfrac{d\left(2k+5\right)}{d\left(3k-4\right)}=\dfrac{2k+5}{3k-4}\left(2\right)\)
Từ \(\left(1\right)\) và \(\left(2\right)\) \(\Rightarrow\) Đpcm.
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)
\(\frac{a}{b}=\frac{c}{d}\)\(\Rightarrow\frac{a}{c}=\frac{b}{d}=\frac{a-b}{c-d}\)
\(\Rightarrow\frac{a^2}{c^2}=\frac{b^2}{d^2}=\frac{a^2-b^2}{c^2-d^2}=\frac{\left(a-b\right)^2}{\left(c-d\right)^2}\)
b) \(\frac{a}{b}=\frac{c}{d}\)\(\Rightarrow\frac{a}{c}=\frac{b}{d}\)\(\Rightarrow\frac{2a}{2c}=\frac{5b}{5d}=\frac{3a}{3c}=\frac{4b}{4d}=\frac{2a+5b}{2c+5d}=\frac{3a-4b}{3c-4d}\)
\(\Rightarrow\frac{2a+5b}{3a-4b}=\frac{2c+5d}{3c-4d}\)
Đặt \(\frac{a}{b}=\frac{c}{d}=k\)
=>a=bk; c=dk
\(\frac{5a-7b}{3a+4b}=\frac{5\cdot bk-7b}{3\cdot bk+4b}=\frac{b\left(5k-7\right)}{b\left(3k+4\right)}=\frac{5k-7}{3k+4}\)
\(\frac{5c-7d}{3c+4d}=\frac{5\cdot dk-7d}{3\cdot dk+4d}=\frac{d\left(5k-7\right)}{d\left(3k+4\right)}=\frac{5k-7}{3k+4}\)
Do đó: \(\frac{5a-7b}{3a+4b}=\frac{5c-7d}{3c+4d}\)