Solve {l}{x+y+z=0}{2x-y-z=2}{x-z=3}

$$\left. \begin{array} { l } { x + y + z = 0 } \\ { 2 x - y - z = 2 } \\ { x - z = 3 } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{2}{3}\approx 0.666666667$$

$$y = \frac{5}{3} = 1\frac{2}{3} \approx 1.666666667$$

$$z = -\frac{7}{3} = -2\frac{1}{3} \approx -2.333333333$$

Short Steps Using Substitution

Solve x+y+z=0 for x.

$$x=-y-z$$

Substitute -y-z for x in the second and third equation.

$$2\left(-y-z\right)-y-z=2 -y-z-z=3$$

Solve these equations for y and z respectively.

$$y=-z-\frac{2}{3} z=-\frac{3}{2}-\frac{1}{2}y$$

Substitute -z-\frac{2}{3} for y in the equation z=-\frac{3}{2}-\frac{1}{2}y.

$$z=-\frac{3}{2}-\frac{1}{2}\left(-z-\frac{2}{3}\right)$$

Solve z=-\frac{3}{2}-\frac{1}{2}\left(-z-\frac{2}{3}\right) for z.

$$z=-\frac{7}{3}$$

Substitute -\frac{7}{3} for z in the equation y=-z-\frac{2}{3}.

$$y=-\left(-\frac{7}{3}\right)-\frac{2}{3}$$

Calculate y from y=-\left(-\frac{7}{3}\right)-\frac{2}{3}.

$$y=\frac{5}{3}$$

Substitute \frac{5}{3} for y and -\frac{7}{3} for z in the equation x=-y-z.

$$x=-\frac{5}{3}-\left(-\frac{7}{3}\right)$$

Calculate x from x=-\frac{5}{3}-\left(-\frac{7}{3}\right).

$$x=\frac{2}{3}$$

The system is now solved.

$$x=\frac{2}{3} y=\frac{5}{3} z=-\frac{7}{3}$$