Solve {l}{2x+y+z=3}{x+2y-z=4}{x+y-4z=-11}

$$\left\{ \begin{array} { l } { 2 x + y + z = 3 } \\ { x + 2 y - z = 4 } \\ { x + y - 4 z = - 11 } \end{array} \right.$$

Solve for x, y, z

$$x = -\frac{8}{3} = -2\frac{2}{3} \approx -2.666666667$$

$$y=5$$

$$z = \frac{10}{3} = 3\frac{1}{3} \approx 3.333333333$$

Short Steps Using Substitution

Solve 2x+y+z=3 for y.

$$y=-2x-z+3$$

Substitute -2x-z+3 for y in the second and third equation.

$$x+2\left(-2x-z+3\right)-z=4 x-2x-z+3-4z=-11$$

Solve these equations for x and z respectively.

$$x=-z+\frac{2}{3} z=-\frac{1}{5}x+\frac{14}{5}$$

Substitute -z+\frac{2}{3} for x in the equation z=-\frac{1}{5}x+\frac{14}{5}.

$$z=-\frac{1}{5}\left(-z+\frac{2}{3}\right)+\frac{14}{5}$$

Solve z=-\frac{1}{5}\left(-z+\frac{2}{3}\right)+\frac{14}{5} for z.

$$z=\frac{10}{3}$$

Substitute \frac{10}{3} for z in the equation x=-z+\frac{2}{3}.

$$x=-\frac{10}{3}+\frac{2}{3}$$

Calculate x from x=-\frac{10}{3}+\frac{2}{3}.

$$x=-\frac{8}{3}$$

Substitute -\frac{8}{3} for x and \frac{10}{3} for z in the equation y=-2x-z+3.

$$y=-2\left(-\frac{8}{3}\right)-\frac{10}{3}+3$$

Calculate y from y=-2\left(-\frac{8}{3}\right)-\frac{10}{3}+3.

$$y=5$$

The system is now solved.

$$x=-\frac{8}{3} y=5 z=\frac{10}{3}$$