Multiply the second equation by $\displaystyle-{1}\usepackage{cmbright}\Huge\$ , then add to the first: $\displaystyle{y}-{z}={1}\usepackage{cmbright}\Huge\$

Add this to the third equation to get: $\displaystyle{2}{y}={6}\usepackage{cmbright}\Huge\$ , so $\displaystyle{y}={3}\usepackage{cmbright}\Huge\$

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

Substitute $\displaystyle{z}={2}\usepackage{cmbright}\Huge\$ into the second equation: $\displaystyle-{3}{x}+{6}={3}\usepackage{cmbright}\Huge\$ , so $\displaystyle{x}={1}\usepackage{cmbright}\Huge\$

Check your solution in all three equations.