Solve {l}{2x-y+z=-2}{3x-2y-z=4}{x-y+z=14}

$$\left. \begin{array} { l } { 2 x - y + z = - 2 } \\ { 3 x - 2 y - z = 4 } \\ { x - y + z = 14 } \end{array} \right.$$

Solve for x, y, z

$$x=-16$$

$$y = -\frac{82}{3} = -27\frac{1}{3} \approx -27.333333333$$

$$z = \frac{8}{3} = 2\frac{2}{3} \approx 2.666666667$$

Short Steps Using Substitution

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

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

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

$$3x-2\left(2x+z+2\right)-z=4 x-\left(2x+z+2\right)+z=14$$

Solve these equations for z and x respectively.

$$z=-\frac{1}{3}x-\frac{8}{3} x=-16$$

Substitute -16 for x in the equation z=-\frac{1}{3}x-\frac{8}{3}.

$$z=-\frac{1}{3}\left(-16\right)-\frac{8}{3}$$

Calculate z from z=-\frac{1}{3}\left(-16\right)-\frac{8}{3}.

$$z=\frac{8}{3}$$

Substitute \frac{8}{3} for z and -16 for x in the equation y=2x+z+2.

$$y=2\left(-16\right)+\frac{8}{3}+2$$

Calculate y from y=2\left(-16\right)+\frac{8}{3}+2.

$$y=-\frac{82}{3}$$

The system is now solved.

$$x=-16 y=-\frac{82}{3} z=\frac{8}{3}$$