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

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

Solve for x, y, z

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

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

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

Short Steps Using Substitution

Multiply each equation by the least common multiple of denominators in it. Simplify.

$$x+2y-z+1=0 2x+2=y-2 y-2=-2z$$

Solve x+2y-z+1=0 for x.

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

Substitute -2y+z-1 for x in the equation 2x+2=y-2.

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

Solve the second equation for y and the third equation for z.

$$y=\frac{2}{5}+\frac{2}{5}z z=-\frac{1}{2}y+1$$

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

$$z=-\frac{1}{2}\left(\frac{2}{5}+\frac{2}{5}z\right)+1$$

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

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

Substitute \frac{2}{3} for z in the equation y=\frac{2}{5}+\frac{2}{5}z.

$$y=\frac{2}{5}+\frac{2}{5}\times \frac{2}{3}$$

Calculate y from y=\frac{2}{5}+\frac{2}{5}\times \frac{2}{3}.

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

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

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

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

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

The system is now solved.

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