Solve {c}{3x+2y-2z=-8}{4x+2y-3z=-10}{-3x+8y-z=2}

$$\left. \begin{array} { c } { 3 x + 2 y - 2 z = - 8 } \\ { 4 x + 2 y - 3 z = - 10 } \\ { - 3 x + 8 y - z = 2 } \end{array} \right.$$

Solve for x, y, z

$$x = -\frac{5}{2} = -2\frac{1}{2} = -2.5$$

$$y=-\frac{3}{4}=-0.75$$

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

Short Steps Using Substitution

Reorder the equations.

$$-3x+8y-z=2 4x+2y-3z=-10 3x+2y-2z=-8$$

Solve -3x+8y-z=2 for z.

$$z=-3x+8y-2$$

Substitute -3x+8y-2 for z in the second and third equation.

$$4x+2y-3\left(-3x+8y-2\right)=-10 3x+2y-2\left(-3x+8y-2\right)=-8$$

Solve these equations for y and x respectively.

$$y=\frac{13}{22}x+\frac{8}{11} x=-\frac{4}{3}+\frac{14}{9}y$$

Substitute \frac{13}{22}x+\frac{8}{11} for y in the equation x=-\frac{4}{3}+\frac{14}{9}y.

$$x=-\frac{4}{3}+\frac{14}{9}\left(\frac{13}{22}x+\frac{8}{11}\right)$$

Solve x=-\frac{4}{3}+\frac{14}{9}\left(\frac{13}{22}x+\frac{8}{11}\right) for x.

$$x=-\frac{5}{2}$$

Substitute -\frac{5}{2} for x in the equation y=\frac{13}{22}x+\frac{8}{11}.

$$y=\frac{13}{22}\left(-\frac{5}{2}\right)+\frac{8}{11}$$

Calculate y from y=\frac{13}{22}\left(-\frac{5}{2}\right)+\frac{8}{11}.

$$y=-\frac{3}{4}$$

Substitute -\frac{3}{4} for y and -\frac{5}{2} for x in the equation z=-3x+8y-2.

$$z=-3\left(-\frac{5}{2}\right)+8\left(-\frac{3}{4}\right)-2$$

Calculate z from z=-3\left(-\frac{5}{2}\right)+8\left(-\frac{3}{4}\right)-2.

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

The system is now solved.

$$x=-\frac{5}{2} y=-\frac{3}{4} z=-\frac{1}{2}$$