Add the first and second equations: $\displaystyle-{x}+{z}=-{1}\usepackage{cmbright}\Huge\$

Add the first and third equations: $\displaystyle-{2}{x}-{z}={4}\usepackage{cmbright}\Huge\$

Add the two new equations: $\displaystyle-{3}{x}={3}\usepackage{cmbright}\Huge\$ , so $\displaystyle{x}=-{1}\usepackage{cmbright}\Huge\$

Substitute this into the first new equation: $\displaystyle{1}+{z}=-{1}\usepackage{cmbright}\Huge\$ , so $\displaystyle{z}=-{2}\usepackage{cmbright}\Huge\$

Substitute x and z into the second equation: $\displaystyle{3}+{y}+{4}={4}\usepackage{cmbright}\Huge\$ , so $\displaystyle{y}=-{3}\usepackage{cmbright}\Huge\$

Check your solution in all three equations.