Solve {l}{x+3y+z=1}{2x-y-z=0}{-x+2y+2z=0}

$$\left. \begin{array} { l } { x + 3 y + z = 1 } \\ { 2 x - y - z = 0 } \\ { - x + 2 y + 2 z = 0 } \end{array} \right.$$

Solve for x, y, z

$$x=0$$

$$y=\frac{1}{2}=0.5$$

$$z=-\frac{1}{2}=-0.5$$

Short Steps Using Substitution

Solve x+3y+z=1 for x.

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

Substitute -3y-z+1 for x in the second and third equation.

$$2\left(-3y-z+1\right)-y-z=0 -\left(-3y-z+1\right)+2y+2z=0$$

Solve these equations for y and z respectively.

$$y=-\frac{3}{7}z+\frac{2}{7} z=-\frac{5}{3}y+\frac{1}{3}$$

Substitute -\frac{3}{7}z+\frac{2}{7} for y in the equation z=-\frac{5}{3}y+\frac{1}{3}.

$$z=-\frac{5}{3}\left(-\frac{3}{7}z+\frac{2}{7}\right)+\frac{1}{3}$$

Solve z=-\frac{5}{3}\left(-\frac{3}{7}z+\frac{2}{7}\right)+\frac{1}{3} for z.

$$z=-\frac{1}{2}$$

Substitute -\frac{1}{2} for z in the equation y=-\frac{3}{7}z+\frac{2}{7}.

$$y=-\frac{3}{7}\left(-\frac{1}{2}\right)+\frac{2}{7}$$

Calculate y from y=-\frac{3}{7}\left(-\frac{1}{2}\right)+\frac{2}{7}.

$$y=\frac{1}{2}$$

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

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

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

$$x=0$$

The system is now solved.

$$x=0 y=\frac{1}{2} z=-\frac{1}{2}$$