Solve {l}{x+2y-z=4}{2x+y+z=-2}{x+2y+z=2}

$$\left. \begin{array} { l } { x + 2 y - z = 4 } \\ { 2 x + y + z = - 2 } \\ { x + 2 y + z = 2 } \end{array} \right.$$

Solve for x, y, z

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

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

$$z=-1$$

Short Steps Using Substitution

Solve x+2y-z=4 for x.

$$x=-2y+z+4$$

Substitute -2y+z+4 for x in the second and third equation.

$$2\left(-2y+z+4\right)+y+z=-2 -2y+z+4+2y+z=2$$

Solve these equations for y and z respectively.

$$y=z+\frac{10}{3} z=-1$$

Substitute -1 for z in the equation y=z+\frac{10}{3}.

$$y=-1+\frac{10}{3}$$

Calculate y from y=-1+\frac{10}{3}.

$$y=\frac{7}{3}$$

Substitute \frac{7}{3} for y and -1 for z in the equation x=-2y+z+4.

$$x=-2\times \frac{7}{3}-1+4$$

Calculate x from x=-2\times \frac{7}{3}-1+4.

$$x=-\frac{5}{3}$$

The system is now solved.

$$x=-\frac{5}{3} y=\frac{7}{3} z=-1$$