Solve {l}{x+4y+z=0}{x+y=0}{3x-3y-z=0}

$$\left\{ \begin{array} { l } { x + 4 y + z = 0 } \\ { x + y = 0 } \\ { 3 x - 3 y - z = 0 } \end{array} \right.$$

Solve for x, y, z

$$x=0$$

$$y=0$$

$$z=0$$

Short Steps Using Substitution

Solve x+4y+z=0 for x.

$$x=-4y-z$$

Substitute -4y-z for x in the second and third equation.

$$-4y-z+y=0 3\left(-4y-z\right)-3y-z=0$$

Solve these equations for y and z respectively.

$$y=-\frac{1}{3}z z=-\frac{15}{4}y$$

Substitute -\frac{1}{3}z for y in the equation z=-\frac{15}{4}y.

$$z=-\frac{15}{4}\left(-\frac{1}{3}\right)z$$

Solve z=-\frac{15}{4}\left(-\frac{1}{3}\right)z for z.

$$z=0$$

Substitute 0 for z in the equation y=-\frac{1}{3}z.

$$y=-\frac{1}{3}\times 0$$

Calculate y from y=-\frac{1}{3}\times 0.

$$y=0$$

Substitute 0 for y and 0 for z in the equation x=-4y-z.

$$x=-4\times 0-0$$

Calculate x from x=-4\times 0-0.

$$x=0$$

The system is now solved.

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