Solve {c}{3x-2y=10}{-2x+6y-4z=0}{-4y+9z=-30}

$$\left. \begin{array} { c } { 3 x - 2 y = 10 } \\ { - 2 x + 6 y - 4 z = 0 } \\ { - 4 y + 9 z = - 30 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{70}{39} = 1\frac{31}{39} \approx 1.794871795$$

$$y = -\frac{30}{13} = -2\frac{4}{13} \approx -2.307692308$$

$$z = -\frac{170}{39} = -4\frac{14}{39} \approx -4.358974359$$

Short Steps Using Substitution

Solve 3x-2y=10 for x.

$$x=\frac{10}{3}+\frac{2}{3}y$$

Substitute \frac{10}{3}+\frac{2}{3}y for x in the equation -2x+6y-4z=0.

$$-2\left(\frac{10}{3}+\frac{2}{3}y\right)+6y-4z=0$$

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

$$y=\frac{10}{7}+\frac{6}{7}z z=-\frac{10}{3}+\frac{4}{9}y$$

Substitute \frac{10}{7}+\frac{6}{7}z for y in the equation z=-\frac{10}{3}+\frac{4}{9}y.

$$z=-\frac{10}{3}+\frac{4}{9}\left(\frac{10}{7}+\frac{6}{7}z\right)$$

Solve z=-\frac{10}{3}+\frac{4}{9}\left(\frac{10}{7}+\frac{6}{7}z\right) for z.

$$z=-\frac{170}{39}$$

Substitute -\frac{170}{39} for z in the equation y=\frac{10}{7}+\frac{6}{7}z.

$$y=\frac{10}{7}+\frac{6}{7}\left(-\frac{170}{39}\right)$$

Calculate y from y=\frac{10}{7}+\frac{6}{7}\left(-\frac{170}{39}\right).

$$y=-\frac{30}{13}$$

Substitute -\frac{30}{13} for y in the equation x=\frac{10}{3}+\frac{2}{3}y.

$$x=\frac{10}{3}+\frac{2}{3}\left(-\frac{30}{13}\right)$$

Calculate x from x=\frac{10}{3}+\frac{2}{3}\left(-\frac{30}{13}\right).

$$x=\frac{70}{39}$$

The system is now solved.

$$x=\frac{70}{39} y=-\frac{30}{13} z=-\frac{170}{39}$$