Solve {c}{-1/3x+1/3y-1/6z=-3}{1/4x+frac{1}{3}y-frac{1}{2}z=2}{frac{1}{6}x-frac{1}{3}y+frac{1}{3}z=2}

$$\left. \begin{array} { c } { - \frac { 1 } { 3 } x + \frac { 1 } { 3 } y - \frac { 1 } { 6 } z = - 3 } \\ { \frac { 1 } { 4 } x + \frac { 1 } { 3 } y - \frac { 1 } { 2 } z = 2 } \\ { \frac { 1 } { 6 } x - \frac { 1 } { 3 } y + \frac { 1 } { 3 } z = 2 } \end{array} \right.$$

Solve for x, y, z

$$x=12$$

$$y=6$$

$$z=6$$

Short Steps Using Substitution

Solve -\frac{1}{3}x+\frac{1}{3}y-\frac{1}{6}z=-3 for x.

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

Substitute y-\frac{1}{2}z+9 for x in the second and third equation.

$$\frac{1}{4}\left(y-\frac{1}{2}z+9\right)+\frac{1}{3}y-\frac{1}{2}z=2 \frac{1}{6}\left(y-\frac{1}{2}z+9\right)-\frac{1}{3}y+\frac{1}{3}z=2$$

Solve these equations for y and z respectively.

$$y=-\frac{3}{7}+\frac{15}{14}z z=2+\frac{2}{3}y$$

Substitute -\frac{3}{7}+\frac{15}{14}z for y in the equation z=2+\frac{2}{3}y.

$$z=2+\frac{2}{3}\left(-\frac{3}{7}+\frac{15}{14}z\right)$$

Solve z=2+\frac{2}{3}\left(-\frac{3}{7}+\frac{15}{14}z\right) for z.

$$z=6$$

Substitute 6 for z in the equation y=-\frac{3}{7}+\frac{15}{14}z.

$$y=-\frac{3}{7}+\frac{15}{14}\times 6$$

Calculate y from y=-\frac{3}{7}+\frac{15}{14}\times 6.

$$y=6$$

Substitute 6 for y and 6 for z in the equation x=y-\frac{1}{2}z+9.

$$x=6-\frac{1}{2}\times 6+9$$

Calculate x from x=6-\frac{1}{2}\times 6+9.

$$x=12$$

The system is now solved.

$$x=12 y=6 z=6$$