Solve {l}{3x+9y-4z=108}{9x-9y+7z=0}{4x-11y+2=0}

$$\left\{ \begin{array} { l } { 3 x + 9 y - 4 z = 108 } \\ { 9 x - 9 y + 7 z = 0 } \\ { 4 x - 11 y + 2 = 0 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{2754}{245} = 11\frac{59}{245} \approx 11.240816327$$

$$y = \frac{1046}{245} = 4\frac{66}{245} \approx 4.269387755$$

$$z = -\frac{2196}{245} = -8\frac{236}{245} \approx -8.963265306$$

Short Steps Using Substitution

Solve 3x+9y-4z=108 for x.

$$x=-3y+\frac{4}{3}z+36$$

Substitute -3y+\frac{4}{3}z+36 for x in the second and third equation.

$$9\left(-3y+\frac{4}{3}z+36\right)-9y+7z=0 4\left(-3y+\frac{4}{3}z+36\right)-11y+2=0$$

Solve these equations for y and z respectively.

$$y=\frac{19}{36}z+9 z=-\frac{219}{8}+\frac{69}{16}y$$

Substitute \frac{19}{36}z+9 for y in the equation z=-\frac{219}{8}+\frac{69}{16}y.

$$z=-\frac{219}{8}+\frac{69}{16}\left(\frac{19}{36}z+9\right)$$

Solve z=-\frac{219}{8}+\frac{69}{16}\left(\frac{19}{36}z+9\right) for z.

$$z=-\frac{2196}{245}$$

Substitute -\frac{2196}{245} for z in the equation y=\frac{19}{36}z+9.

$$y=\frac{19}{36}\left(-\frac{2196}{245}\right)+9$$

Calculate y from y=\frac{19}{36}\left(-\frac{2196}{245}\right)+9.

$$y=\frac{1046}{245}$$

Substitute \frac{1046}{245} for y and -\frac{2196}{245} for z in the equation x=-3y+\frac{4}{3}z+36.

$$x=-3\times \frac{1046}{245}+\frac{4}{3}\left(-\frac{2196}{245}\right)+36$$

Calculate x from x=-3\times \frac{1046}{245}+\frac{4}{3}\left(-\frac{2196}{245}\right)+36.

$$x=\frac{2754}{245}$$

The system is now solved.

$$x=\frac{2754}{245} y=\frac{1046}{245} z=-\frac{2196}{245}$$