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

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

Solve for x, y, z

$$x = \frac{5}{3} = 1\frac{2}{3} \approx 1.666666667$$

$$y=\frac{1}{3}\approx 0.333333333$$

$$z=1$$

Short Steps Using Substitution

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

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

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

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

Solve these equations for y and z respectively.

$$y=\frac{1}{2}z-\frac{1}{6} z=\frac{3}{4}+\frac{3}{4}y$$

Substitute \frac{1}{2}z-\frac{1}{6} for y in the equation z=\frac{3}{4}+\frac{3}{4}y.

$$z=\frac{3}{4}+\frac{3}{4}\left(\frac{1}{2}z-\frac{1}{6}\right)$$

Solve z=\frac{3}{4}+\frac{3}{4}\left(\frac{1}{2}z-\frac{1}{6}\right) for z.

$$z=1$$

Substitute 1 for z in the equation y=\frac{1}{2}z-\frac{1}{6}.

$$y=\frac{1}{2}\times 1-\frac{1}{6}$$

Calculate y from y=\frac{1}{2}\times 1-\frac{1}{6}.

$$y=\frac{1}{3}$$

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

$$x=2\times \frac{1}{3}-1+2$$

Calculate x from x=2\times \frac{1}{3}-1+2.

$$x=\frac{5}{3}$$

The system is now solved.

$$x=\frac{5}{3} y=\frac{1}{3} z=1$$