Solve {r}{3x+0y+15z=0}{-x+3y-z=0}{x-y+3z=0}

$$\left. \begin{array} { r } { 3 x + 0 y + 15 z = 0 } \\ { - x + 3 y - z = 0 } \\ { x - y + 3 z = 0 } \end{array} \right.$$

Solve for x, y, z

$$x=0$$

$$y=0$$

$$z=0$$

Short Steps Using Substitution

Reorder the equations.

$$-x+3y-z=0 3x+0y+15z=0 x-y+3z=0$$

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

$$x=3y-z$$

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

$$3\left(3y-z\right)+0y+15z=0 3y-z-y+3z=0$$

Solve these equations for y and z respectively.

$$y=-\frac{4}{3}z z=-y$$

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

$$z=-\left(-\frac{4}{3}\right)z$$

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

$$z=0$$

Substitute 0 for z in the equation y=-\frac{4}{3}z.

$$y=-\frac{4}{3}\times 0$$

Calculate y from y=-\frac{4}{3}\times 0.

$$y=0$$

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

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

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

$$x=0$$

The system is now solved.

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