Solve {l}{x-2y+3z=12}{2x-y-2z=5}{2x+2y-z=4}

$$\left. \begin{array} { l } { x - 2 y + 3 z = 12 } \\ { 2 x - y - 2 z = 5 } \\ { 2 x + 2 y - z = 4 } \end{array} \right.$$

Solve for x, y, z

$$x=4$$

$$y=-1$$

$$z=2$$

Short Steps Using Substitution

Solve x-2y+3z=12 for x.

$$x=2y-3z+12$$

Substitute 2y-3z+12 for x in the second and third equation.

$$2\left(2y-3z+12\right)-y-2z=5 2\left(2y-3z+12\right)+2y-z=4$$

Solve these equations for y and z respectively.

$$y=\frac{8}{3}z-\frac{19}{3} z=\frac{20}{7}+\frac{6}{7}y$$

Substitute \frac{8}{3}z-\frac{19}{3} for y in the equation z=\frac{20}{7}+\frac{6}{7}y.

$$z=\frac{20}{7}+\frac{6}{7}\left(\frac{8}{3}z-\frac{19}{3}\right)$$

Solve z=\frac{20}{7}+\frac{6}{7}\left(\frac{8}{3}z-\frac{19}{3}\right) for z.

$$z=2$$

Substitute 2 for z in the equation y=\frac{8}{3}z-\frac{19}{3}.

$$y=\frac{8}{3}\times 2-\frac{19}{3}$$

Calculate y from y=\frac{8}{3}\times 2-\frac{19}{3}.

$$y=-1$$

Substitute -1 for y and 2 for z in the equation x=2y-3z+12.

$$x=2\left(-1\right)-3\times 2+12$$

Calculate x from x=2\left(-1\right)-3\times 2+12.

$$x=4$$

The system is now solved.

$$x=4 y=-1 z=2$$