Solve {c}{x+y+z=4}{x-2y-z=1}{2x-y-2z=-1}

$$\left\{ \begin{array} { c } { x + y + z = 4 } \\ { x - 2 y - z = 1 } \\ { 2 x - y - 2 z = - 1 } \end{array} \right.$$

Solve for x, y, z

$$x=2$$

$$y=-1$$

$$z=3$$

Short Steps Using Substitution

Solve x+y+z=4 for x.

$$x=-y-z+4$$

Substitute -y-z+4 for x in the second and third equation.

$$-y-z+4-2y-z=1 2\left(-y-z+4\right)-y-2z=-1$$

Solve these equations for y and z respectively.

$$y=-\frac{2}{3}z+1 z=\frac{9}{4}-\frac{3}{4}y$$

Substitute -\frac{2}{3}z+1 for y in the equation z=\frac{9}{4}-\frac{3}{4}y.

$$z=\frac{9}{4}-\frac{3}{4}\left(-\frac{2}{3}z+1\right)$$

Solve z=\frac{9}{4}-\frac{3}{4}\left(-\frac{2}{3}z+1\right) for z.

$$z=3$$

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

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

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

$$y=-1$$

Substitute -1 for y and 3 for z in the equation x=-y-z+4.

$$x=-\left(-1\right)-3+4$$

Calculate x from x=-\left(-1\right)-3+4.

$$x=2$$

The system is now solved.

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