1,
\(\frac{a}{b}=\frac{c}{d}\\ \Rightarrow\frac{a}{b}-1=\frac{c}{d}-1\\ \Rightarrow\frac{a}{b}-\frac{b}{b}=\frac{c}{d}-\frac{d}{d}\\ \Rightarrow\frac{a-b}{b}=\frac{c-d}{d}\\ \Rightarrow\frac{b}{a-b}=\frac{d}{c-d}\)
2,
Có: \(\frac{x}{y}=1,5=\frac{3}{2}\Rightarrow\frac{x}{3}=\frac{y}{2}\)
Đặt \(\frac{x}{3}=\frac{y}{2}=k\Rightarrow\left\{{}\begin{matrix}x=3k\\y=2k\end{matrix}\right.\)
Mà \(x\cdot y=24\)
\(\Rightarrow3k\cdot2k=24\\ \Rightarrow6k^2=24\\ \Rightarrow k^2=4\\ \Rightarrow\left[{}\begin{matrix}k=2\\k=-2\end{matrix}\right.\)
+ Với k = 2
\(\Rightarrow\left\{{}\begin{matrix}x=3\cdot2=6\\y=2\cdot2=4\end{matrix}\right.\)
+ Với k = -2
\(\Rightarrow\left\{{}\begin{matrix}x=3\cdot\left(-2\right)=-6\\y=2\cdot\left(-2\right)=-4\end{matrix}\right.\)
Vậy \(\left(x;y\right)\in\left\{\left(6;4\right);\left(-6;-4\right)\right\}\)
a) Ta có: \(\frac{a}{b}=\frac{c}{d}.\)
\(\Rightarrow\frac{a}{b}-1=\frac{c}{d}-1\)
\(\Rightarrow\frac{a}{b}-\frac{b}{b}=\frac{c}{d}-\frac{d}{d}.\)
\(\Rightarrow\frac{a-b}{b}=\frac{c-d}{d}\)
\(\Rightarrow\frac{b}{a-b}=\frac{d}{c-d}\left(đpcm\right).\)
b) Ta có: \(\frac{x}{y}=1,5.\)
Đổi \(1,5=\frac{3}{2}\)
\(\Rightarrow\frac{x}{y}=\frac{3}{2}.\)
\(\Rightarrow\frac{x}{3}=\frac{y}{2}\) và \(x.y=24.\)
Đặt \(\frac{x}{3}=\frac{y}{2}=k\Rightarrow\left\{{}\begin{matrix}x=3k\\y=2k\end{matrix}\right.\)
Có: \(x.y=24\)
=> \(3k.2k=24\)
=> \(6.k^2=24\)
=> \(k^2=24:6\)
=> \(k^2=4\)
=> \(k=\pm2.\)
TH1: \(k=2.\)
\(\Rightarrow\left\{{}\begin{matrix}x=3.2=6\\y=2.2=4\end{matrix}\right.\)
TH2: \(k=-2.\)
\(\Rightarrow\left\{{}\begin{matrix}x=3.\left(-2\right)=-6\\y=2.\left(-2\right)=-4\end{matrix}\right.\)
Vậy \(\left(x;y\right)=\left(6;4\right),\left(-6;-4\right).\)
Chúc bạn học tốt!
