Solve {l}{x+2y+3z=0}{x-y+z=0}{x+y-2z=0}

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

Solve for x, y, z

$$x=0$$

$$y=0$$

$$z=0$$

Short Steps Using Substitution

Solve x+2y+3z=0 for x.

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

Substitute -2y-3z for x in the second and third equation.

$$-2y-3z-y+z=0 -2y-3z+y-2z=0$$

Solve these equations for y and z respectively.

$$y=-\frac{2}{3}z z=-\frac{1}{5}y$$

Substitute -\frac{2}{3}z for y in the equation z=-\frac{1}{5}y.

$$z=-\frac{1}{5}\left(-\frac{2}{3}\right)z$$

Solve z=-\frac{1}{5}\left(-\frac{2}{3}\right)z for z.

$$z=0$$

Substitute 0 for z in the equation y=-\frac{2}{3}z.

$$y=-\frac{2}{3}\times 0$$

Calculate y from y=-\frac{2}{3}\times 0.

$$y=0$$

Substitute 0 for y and 0 for z in the equation x=-2y-3z.

$$x=-2\times 0-3\times 0$$

Calculate x from x=-2\times 0-3\times 0.

$$x=0$$

The system is now solved.

$$x=0 y=0 z=0$$