Solve {l}{x+y+z=0}{4x+2y+z=3}{9x+3y+z=6}

$$\left\{ \begin{array} { l } { x + y + z = 0 } \\ { 4 x + 2 y + z = 3 } \\ { 9 x + 3 y + z = 6 } \end{array} \right.$$

Solve for x, y, z

$$x=0$$

$$y=3$$

$$z=-3$$

Short Steps Using Substitution

Solve x+y+z=0 for x.

$$x=-y-z$$

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

$$4\left(-y-z\right)+2y+z=3 9\left(-y-z\right)+3y+z=6$$

Solve these equations for y and z respectively.

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

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

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

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

$$z=-3$$

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

$$y=-\frac{3}{2}-\frac{3}{2}\left(-3\right)$$

Calculate y from y=-\frac{3}{2}-\frac{3}{2}\left(-3\right).

$$y=3$$

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

$$x=-3-\left(-3\right)$$

Calculate x from x=-3-\left(-3\right).

$$x=0$$

The system is now solved.

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