Solve {l}{2x+3y+4z=0}{3x+2y+8z=4}{9x+y+6z=4}

$$\left. \begin{array} { l } { 2 x + 3 y + 4 z = 0 } \\ { 3 x + 2 y + 8 z = 4 } \\ { 9 x + y + 6 z = 4 } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{4}{55}\approx 0.072727273$$

$$y = -\frac{56}{55} = -1\frac{1}{55} \approx -1.018181818$$

$$z=\frac{8}{11}\approx 0.727272727$$

Short Steps Using Substitution

Reorder the equations.

$$9x+y+6z=4 3x+2y+8z=4 2x+3y+4z=0$$

Solve 9x+y+6z=4 for y.

$$y=-9x-6z+4$$

Substitute -9x-6z+4 for y in the second and third equation.

$$3x+2\left(-9x-6z+4\right)+8z=4 2x+3\left(-9x-6z+4\right)+4z=0$$

Solve these equations for x and z respectively.

$$x=\frac{4}{15}-\frac{4}{15}z z=-\frac{25}{14}x+\frac{6}{7}$$

Substitute \frac{4}{15}-\frac{4}{15}z for x in the equation z=-\frac{25}{14}x+\frac{6}{7}.

$$z=-\frac{25}{14}\left(\frac{4}{15}-\frac{4}{15}z\right)+\frac{6}{7}$$

Solve z=-\frac{25}{14}\left(\frac{4}{15}-\frac{4}{15}z\right)+\frac{6}{7} for z.

$$z=\frac{8}{11}$$

Substitute \frac{8}{11} for z in the equation x=\frac{4}{15}-\frac{4}{15}z.

$$x=\frac{4}{15}-\frac{4}{15}\times \frac{8}{11}$$

Calculate x from x=\frac{4}{15}-\frac{4}{15}\times \frac{8}{11}.

$$x=\frac{4}{55}$$

Substitute \frac{4}{55} for x and \frac{8}{11} for z in the equation y=-9x-6z+4.

$$y=-9\times \frac{4}{55}-6\times \frac{8}{11}+4$$

Calculate y from y=-9\times \frac{4}{55}-6\times \frac{8}{11}+4.

$$y=-\frac{56}{55}$$

The system is now solved.

$$x=\frac{4}{55} y=-\frac{56}{55} z=\frac{8}{11}$$