Solve {l}{2x+y+z=0}{x+2y-z=-6}{x+5y+2z=0}

$$\left. \begin{array} { l } { 2 x + y + z = 0 } \\ { x + 2 y - z = - 6 } \\ { x + 5 y + 2 z = 0 } \end{array} \right.$$

Solve for x, y, z

$$x=-1$$

$$y=-1$$

$$z=3$$

Short Steps Using Substitution

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

$$y=-2x-z$$

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

$$x+2\left(-2x-z\right)-z=-6 x+5\left(-2x-z\right)+2z=0$$

Solve these equations for x and z respectively.

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

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

$$z=-3\left(2-z\right)$$

Solve z=-3\left(2-z\right) for z.

$$z=3$$

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

$$x=2-3$$

Calculate x from x=2-3.

$$x=-1$$

Substitute -1 for x and 3 for z in the equation y=-2x-z.

$$y=-2\left(-1\right)-3$$

Calculate y from y=-2\left(-1\right)-3.

$$y=-1$$

The system is now solved.

$$x=-1 y=-1 z=3$$