Question

Given that 5x=2y, 2x=3z and xy=90, where x,y,z are positive. Calculate: x+y+z=....

Answer 1

\(5x=2y\Rightarrow\frac{x}{2}=\frac{y}{5}\Rightarrow\frac{x}{6}=\frac{y}{15}\)

\(2x=3z\Rightarrow\frac{x}{3}=\frac{z}{2}\Rightarrow\frac{x}{6}=\frac{z}{4}\)

\(\Rightarrow\frac{y}{15}=\frac{x}{6}=\frac{z}{4}=k\)

\(\Rightarrow\left\{\begin{matrix}y=15k\\x=6k\end{matrix}\right.\Rightarrow xy=15k\cdot6k\Rightarrow90k^2=90\Rightarrow k^2=1\)

Because x,y,z are positive

\(\Rightarrow k=\sqrt{1}=1\)

\(\Rightarrow\left\{\begin{matrix}\frac{x}{6}=1\rightarrow x=6\\\frac{y}{15}=1\rightarrow y=15\\\frac{z}{4}=1\rightarrow z=4\end{matrix}\right.\)

\(\Rightarrow x+y+z=6+15+4=25\)