Ta có:

$\frac{x}{y}=\frac{2}{3}\Rightarrow \frac{x}{2}=\frac{y}{3}\Rightarrow \frac{x}{2.3}=\frac{y}{3.3}$

$\frac{x}{3}=\frac{z}{5}\Rightarrow \frac{x}{3.2}=\frac{z}{5.2}$

Suy ra:

$\frac{x}{6}=\frac{y}{9}=\frac{z}{10}=k=\frac{x+2y-2z}{6+2.9-2.10}$ ( k>0 do x,y,z dương )

$\Rightarrow $ x = 6k; y = 9k ; z = 10k (với k nguyên dương) thay vào ${{x}^{2}}+{{y}^{2}}+{{z}^{2}}=\frac{217}{4}$

Ta được:

${{(6k)}^{2}}+{{(9k)}^{2}}+{{(10k)}^{2}}=\frac{217}{4}$

217${{k}^{2}}=\frac{217}{4}$

$\Rightarrow {{k}^{2}}=\frac{1}{4}$ 

$\Rightarrow k=\frac{1}{2}$ (vì k nguyên dương)

$\Rightarrow \frac{1}{2}=\frac{x+2y-2z}{4}\Rightarrow x+2y-2z=2$