Solve {l}{2x+y+2z=-2}{-x+2y-2z=-5}{2x+y+4z=0}

$$\left. \begin{array}{l}{ 2 x + y + 2 z = - 2 }\\{ - x + 2 y - 2 z = - 5 }\\{ 2 x + y + 4 z = 0 }\end{array} \right.$$

Solve for x, y, z

$$x=-1$$

$$y=-2$$

$$z=1$$

Short Steps Using Substitution

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

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

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

$$-x+2\left(-2-2x-2z\right)-2z=-5 2x-2-2x-2z+4z=0$$

Solve these equations for x and z respectively.

$$x=\frac{1}{5}-\frac{6}{5}z z=1$$

Substitute 1 for z in the equation x=\frac{1}{5}-\frac{6}{5}z.

$$x=\frac{1}{5}-\frac{6}{5}$$

Calculate x from x=\frac{1}{5}-\frac{6}{5}.

$$x=-1$$

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

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

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

$$y=-2$$

The system is now solved.

$$x=-1 y=-2 z=1$$