Solve {l}{2x+y+z=-2}{x+2y+z=1}{x-2y-3z=1}

$$\left. \begin{array} { l } { 2 x + y + z = - 2 } \\ { x + 2 y + z = 1 } \\ { x - 2 y - 3 z = 1 } \end{array} \right.$$

Solve for x, y, z

$$x=-1$$

$$y=2$$

$$z=-2$$

Short Steps Using Substitution

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

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

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

$$x+2\left(-2x-z-2\right)+z=1 x-2\left(-2x-z-2\right)-3z=1$$

Solve these equations for x and z respectively.

$$x=-\frac{1}{3}z-\frac{5}{3} z=5x+3$$

Substitute -\frac{1}{3}z-\frac{5}{3} for x in the equation z=5x+3.

$$z=5\left(-\frac{1}{3}z-\frac{5}{3}\right)+3$$

Solve z=5\left(-\frac{1}{3}z-\frac{5}{3}\right)+3 for z.

$$z=-2$$

Substitute -2 for z in the equation x=-\frac{1}{3}z-\frac{5}{3}.

$$x=-\frac{1}{3}\left(-2\right)-\frac{5}{3}$$

Calculate x from x=-\frac{1}{3}\left(-2\right)-\frac{5}{3}.

$$x=-1$$

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

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

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

$$y=2$$

The system is now solved.

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