Solve {l}{-x+2y-z=1}{3x-y+2z=-2}{x+y+1=z}

$$\left. \begin{array} { l } { - x + 2 y - z = 1 } \\ { 3 x - y + 2 z = - 2 } \\ { x + y + 1 = z } \end{array} \right.$$

Solve for x, y, z

$$x=-\frac{6}{7}\approx -0.857142857$$

$$y=\frac{2}{7}\approx 0.285714286$$

$$z=\frac{3}{7}\approx 0.428571429$$

Short Steps Using Substitution

Reorder the equations.

$$x+y+1=z 3x-y+2z=-2 -x+2y-z=1$$

Solve x+y+1=z for z.

$$z=x+y+1$$

Substitute x+y+1 for z in the second and third equation.

$$3x-y+2\left(x+y+1\right)=-2 -x+2y-\left(x+y+1\right)=1$$

Solve these equations for y and x respectively.

$$y=-5x-4 x=\frac{1}{2}y-1$$

Substitute -5x-4 for y in the equation x=\frac{1}{2}y-1.

$$x=\frac{1}{2}\left(-5x-4\right)-1$$

Solve x=\frac{1}{2}\left(-5x-4\right)-1 for x.

$$x=-\frac{6}{7}$$

Substitute -\frac{6}{7} for x in the equation y=-5x-4.

$$y=-5\left(-\frac{6}{7}\right)-4$$

Calculate y from y=-5\left(-\frac{6}{7}\right)-4.

$$y=\frac{2}{7}$$

Substitute \frac{2}{7} for y and -\frac{6}{7} for x in the equation z=x+y+1.

$$z=-\frac{6}{7}+\frac{2}{7}+1$$

Calculate z from z=-\frac{6}{7}+\frac{2}{7}+1.

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

The system is now solved.

$$x=-\frac{6}{7} y=\frac{2}{7} z=\frac{3}{7}$$