Solve {l}{x+y+2z=-1}{2x-y+2z=-4}{4x+y+4z=-2}

$$\left. \begin{array} { l } { x + y + 2 z = - 1 } \\ { 2 x - y + 2 z = - 4 } \\ { 4 x + y + 4 z = - 2 } \end{array} \right.$$

Solve for x, y, z

$$x=1$$

$$y=2$$

$$z=-2$$

Short Steps Using Substitution

Solve x+y+2z=-1 for x.

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

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

$$2\left(-y-2z-1\right)-y+2z=-4 4\left(-y-2z-1\right)+y+4z=-2$$

Solve these equations for y and z respectively.

$$y=\frac{2}{3}-\frac{2}{3}z z=-\frac{3}{4}y-\frac{1}{2}$$

Substitute \frac{2}{3}-\frac{2}{3}z for y in the equation z=-\frac{3}{4}y-\frac{1}{2}.

$$z=-\frac{3}{4}\left(\frac{2}{3}-\frac{2}{3}z\right)-\frac{1}{2}$$

Solve z=-\frac{3}{4}\left(\frac{2}{3}-\frac{2}{3}z\right)-\frac{1}{2} for z.

$$z=-2$$

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

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

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

$$y=2$$

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

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

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

$$x=1$$

The system is now solved.

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