Add the first two equations to eliminate y: $\displaystyle{2}{x}+{z}={8}\usepackage{cmbright}\Huge\$

Add the third equation, $\displaystyle-{2}{x}+{z}=-{4}\usepackage{cmbright}\Huge\$ , to this: $\displaystyle{2}{z}={4}\usepackage{cmbright}\Huge\$

Solve for z: $\displaystyle{z}={2}\usepackage{cmbright}\Huge\$

Substitute into the third equation: $\displaystyle-{2}{x}+{2}=-{4}\usepackage{cmbright}\Huge\$ , or $\displaystyle-{2}{x}=-{6}\usepackage{cmbright}\Huge\$ , so $\displaystyle{x}={3}\usepackage{cmbright}\Huge\$

Substitute into the second equation: $\displaystyle{3}+{y}={5}\usepackage{cmbright}\Huge\$ , so $\displaystyle{y}={2}\usepackage{cmbright}\Huge\$