Solve {c}{x+2y-z+1=0}{2x+y-z=0}{x+y+3z-6=0}

$$\left. \begin{array} { c } { x + 2 y - z + 1 = 0 } \\ { 2 x + y - z = 0 } \\ { x + y + 3 z - 6 = 0 } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{10}{11}\approx 0.909090909$$

$$y=-\frac{1}{11}\approx -0.090909091$$

$$z = \frac{19}{11} = 1\frac{8}{11} \approx 1.727272727$$

Short Steps Using Substitution

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

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

Substitute -2y+z-1 for x in the second and third equation.

$$2\left(-2y+z-1\right)+y-z=0 -2y+z-1+y+3z-6=0$$

Solve these equations for y and z respectively.

$$y=-\frac{2}{3}+\frac{1}{3}z z=\frac{7}{4}+\frac{1}{4}y$$

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

$$z=\frac{7}{4}+\frac{1}{4}\left(-\frac{2}{3}+\frac{1}{3}z\right)$$

Solve z=\frac{7}{4}+\frac{1}{4}\left(-\frac{2}{3}+\frac{1}{3}z\right) for z.

$$z=\frac{19}{11}$$

Substitute \frac{19}{11} for z in the equation y=-\frac{2}{3}+\frac{1}{3}z.

$$y=-\frac{2}{3}+\frac{1}{3}\times \frac{19}{11}$$

Calculate y from y=-\frac{2}{3}+\frac{1}{3}\times \frac{19}{11}.

$$y=-\frac{1}{11}$$

Substitute -\frac{1}{11} for y and \frac{19}{11} for z in the equation x=-2y+z-1.

$$x=-2\left(-\frac{1}{11}\right)+\frac{19}{11}-1$$

Calculate x from x=-2\left(-\frac{1}{11}\right)+\frac{19}{11}-1.

$$x=\frac{10}{11}$$

The system is now solved.

$$x=\frac{10}{11} y=-\frac{1}{11} z=\frac{19}{11}$$