Question

Cho \(\frac{12x-15y}{7}\)= \(\frac{20z-12x}{9}\)= \(\frac{15y-20z}{11}\)

Chứng minh \(\frac{x}{5}\)= \(\frac{y}{4}\)=\(\frac{z}{3}\)

Answer 1

Ta có: \(\frac{12x-15y}{7}=\frac{20z-12x}{9}=\frac{15y-20z}{11}.\)

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

\(\frac{12x-15y}{7}=\frac{20z-12x}{9}=\frac{15y-20z}{11}=\frac{12x-15y+20z-12x+15y-20z}{7+9+11}=\frac{\left(12x-12x\right)-\left(15y-15y\right)+\left(20z-20z\right)}{27}=0.\)

\(\Rightarrow\left\{{}\begin{matrix}\frac{12x-15y}{7}=0\Rightarrow12x-15y=0\Rightarrow12x=15y\\\frac{20z-12x}{9}=0\Rightarrow20z-12x=0\Rightarrow20z=12x\\\frac{15y-20z}{11}=0\Rightarrow15y-20z=0\Rightarrow15y=20z\end{matrix}\right.\)

\(\Rightarrow12x=15y=20z\)

\(\Rightarrow\frac{12x}{60}=\frac{15y}{60}=\frac{20z}{60}.\)

\(\Rightarrow\frac{x}{5}=\frac{y}{4}=\frac{z}{3}\left(đpcm\right).\)

Chúc bạn học tốt!