2x = 3y = 5z

$\Rightarrow \frac{x}{\frac{1}{2}}=\frac{y}{\frac{1}{3}}=\frac{z}{\frac{1}{5}}=\frac{x+y-z}{\frac{1}{2}+\frac{1}{3}-\frac{1}{5}}=\frac{57}{\frac{19}{30}}=90$

$\Rightarrow \frac{{{x}^{2}}}{{{\left( \frac{1}{2} \right)}^{2}}}=\frac{{{y}^{2}}}{{{\left( \frac{1}{3} \right)}^{2}}}=\frac{{{z}^{2}}}{{{\left( \frac{1}{5} \right)}^{2}}}={{90}^{2}}$

$\Rightarrow \frac{{{x}^{2}}-{{y}^{2}}+{{z}^{2}}}{{{\left( \frac{1}{2} \right)}^{2}}-{{\left( \frac{1}{3} \right)}^{2}}+{{\left( \frac{1}{5} \right)}^{2}}}={{90}^{2}}$

$\Rightarrow \frac{{{x}^{2}}-{{y}^{2}}+{{z}^{2}}}{\frac{161}{900}}={{90}^{2}}$

$\Rightarrow {{x}^{2}}-{{y}^{2}}+{{z}^{2}}={{90}^{2}}.\frac{161}{900}=1449$