Solve {l}{3x+2y+z=0}{x+y+5z=0}{x-y+2z=0}

$$\left. \begin{array} { l } { 3 x + 2 y + z = 0 } \\ { x + y + 5 z = 0 } \\ { x - y + 2 z = 0 } \end{array} \right.$$

Solve for x, y, z

$$x=0$$

$$y=0$$

$$z=0$$

Short Steps Using Substitution

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

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

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

$$x+y+5\left(-3x-2y\right)=0 x-y+2\left(-3x-2y\right)=0$$

Solve these equations for y and x respectively.

$$y=-\frac{14}{9}x x=-y$$

Substitute -\frac{14}{9}x for y in the equation x=-y.

$$x=-\left(-\frac{14}{9}\right)x$$

Solve x=-\left(-\frac{14}{9}\right)x for x.

$$x=0$$

Substitute 0 for x in the equation y=-\frac{14}{9}x.

$$y=-\frac{14}{9}\times 0$$

Calculate y from y=-\frac{14}{9}\times 0.

$$y=0$$

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

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

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

$$z=0$$

The system is now solved.

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