Solve {l}{3x+2y+4z=0}{2x-8y+7z=0}{8y+7z=-2}

$$\left. \begin{array} { l } { 3 x + 2 y + 4 z = 0 } \\ { 2 x - 8 y + 7 z = 0 } \\ { 8 y + 7 z = - 2 } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{23}{75}\approx 0.306666667$$

$$y=-\frac{13}{150}\approx -0.086666667$$

$$z=-\frac{14}{75}\approx -0.186666667$$

Short Steps Using Substitution

Solve 3x+2y+4z=0 for x.

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

Substitute -\frac{2}{3}y-\frac{4}{3}z for x in the equation 2x-8y+7z=0.

$$2\left(-\frac{2}{3}y-\frac{4}{3}z\right)-8y+7z=0$$

Solve the second equation for y and the third equation for z.

$$y=\frac{13}{28}z z=-\frac{2}{7}-\frac{8}{7}y$$

Substitute \frac{13}{28}z for y in the equation z=-\frac{2}{7}-\frac{8}{7}y.

$$z=-\frac{2}{7}-\frac{8}{7}\times \frac{13}{28}z$$

Solve z=-\frac{2}{7}-\frac{8}{7}\times \frac{13}{28}z for z.

$$z=-\frac{14}{75}$$

Substitute -\frac{14}{75} for z in the equation y=\frac{13}{28}z.

$$y=\frac{13}{28}\left(-\frac{14}{75}\right)$$

Calculate y from y=\frac{13}{28}\left(-\frac{14}{75}\right).

$$y=-\frac{13}{150}$$

Substitute -\frac{13}{150} for y and -\frac{14}{75} for z in the equation x=-\frac{2}{3}y-\frac{4}{3}z.

$$x=-\frac{2}{3}\left(-\frac{13}{150}\right)-\frac{4}{3}\left(-\frac{14}{75}\right)$$

Calculate x from x=-\frac{2}{3}\left(-\frac{13}{150}\right)-\frac{4}{3}\left(-\frac{14}{75}\right).

$$x=\frac{23}{75}$$

The system is now solved.

$$x=\frac{23}{75} y=-\frac{13}{150} z=-\frac{14}{75}$$