Solve {c}{x+y+z=6}{-2x+y+z=-3}{2x+y-2z=6}

$$\left\{ \begin{array} { c } { x + y + z = 6 } \\ { - 2 x + y + z = - 3 } \\ { 2 x + y - 2 z = 6 } \end{array} \right.$$

Solve for x, y, z

$$x=3$$

$$y=2$$

$$z=1$$

Short Steps Using Substitution

Solve x+y+z=6 for x.

$$x=-y-z+6$$

Substitute -y-z+6 for x in the second and third equation.

$$-2\left(-y-z+6\right)+y+z=-3 2\left(-y-z+6\right)+y-2z=6$$

Solve these equations for y and z respectively.

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

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

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

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

$$z=1$$

Substitute 1 for z in the equation y=3-z.

$$y=3-1$$

Calculate y from y=3-1.

$$y=2$$

Substitute 2 for y and 1 for z in the equation x=-y-z+6.

$$x=-2-1+6$$

Calculate x from x=-2-1+6.

$$x=3$$

The system is now solved.

$$x=3 y=2 z=1$$