Solve {r}{x+y+z=0}{2x-y+z=1}{x+y-2z=2}

$$\left. \begin{array} { r } { x + y + z = 0 } \\ { 2 x - y + z = 1 } \\ { x + y - 2 z = 2 } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{7}{9}\approx 0.777777778$$

$$y=-\frac{1}{9}\approx -0.111111111$$

$$z=-\frac{2}{3}\approx -0.666666667$$

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.

$$2\left(-y-z\right)-y+z=1 -y-z+y-2z=2$$

Solve these equations for y and z respectively.

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

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

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

Calculate y from y=-\frac{1}{3}-\frac{1}{3}\left(-\frac{2}{3}\right).

$$y=-\frac{1}{9}$$

Substitute -\frac{1}{9} for y and -\frac{2}{3} for z in the equation x=-y-z.

$$x=-\left(-\frac{1}{9}\right)-\left(-\frac{2}{3}\right)$$

Calculate x from x=-\left(-\frac{1}{9}\right)-\left(-\frac{2}{3}\right).

$$x=\frac{7}{9}$$

The system is now solved.

$$x=\frac{7}{9} y=-\frac{1}{9} z=-\frac{2}{3}$$