Solve {l}{x+y+z=0}{2x-y+z=1}{2x+y-z=8}

$$\left. \begin{array} { l } { x + y + z = 0 } \\ { 2 x - y + z = 1 } \\ { 2 x + y - z = 8 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{9}{4} = 2\frac{1}{4} = 2.25$$

$$y=\frac{5}{8}=0.625$$

$$z = -\frac{23}{8} = -2\frac{7}{8} = -2.875$$

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.

$$2\left(-y-z\right)-y+z=1 2\left(-y-z\right)+y-z=8$$

Solve these equations for y and z respectively.

$$y=-\frac{1}{3}-\frac{1}{3}z z=-\frac{8}{3}-\frac{1}{3}y$$

Substitute -\frac{1}{3}-\frac{1}{3}z for y in the equation z=-\frac{8}{3}-\frac{1}{3}y.

$$z=-\frac{8}{3}-\frac{1}{3}\left(-\frac{1}{3}-\frac{1}{3}z\right)$$

Solve z=-\frac{8}{3}-\frac{1}{3}\left(-\frac{1}{3}-\frac{1}{3}z\right) for z.

$$z=-\frac{23}{8}$$

Substitute -\frac{23}{8} for z in the equation y=-\frac{1}{3}-\frac{1}{3}z.

$$y=-\frac{1}{3}-\frac{1}{3}\left(-\frac{23}{8}\right)$$

Calculate y from y=-\frac{1}{3}-\frac{1}{3}\left(-\frac{23}{8}\right).

$$y=\frac{5}{8}$$

Substitute \frac{5}{8} for y and -\frac{23}{8} for z in the equation x=-y-z.

$$x=-\frac{5}{8}-\left(-\frac{23}{8}\right)$$

Calculate x from x=-\frac{5}{8}-\left(-\frac{23}{8}\right).

$$x=\frac{9}{4}$$

The system is now solved.

$$x=\frac{9}{4} y=\frac{5}{8} z=-\frac{23}{8}$$