Solve {l}{2x-2y+0z=10}{-2x+9y-4z=0}{0x-4y+9z=-30}

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

Solve for x, y, z

$$x = \frac{205}{47} = 4\frac{17}{47} \approx 4.361702128$$

$$y=-\frac{30}{47}\approx -0.638297872$$

$$z = -\frac{170}{47} = -3\frac{29}{47} \approx -3.617021277$$

Short Steps Using Substitution

Solve 2x-2y+0z=10 for x.

$$x=5+y$$

Substitute 5+y for x in the second and third equation.

$$-2\left(5+y\right)+9y-4z=0 0\left(5+y\right)-4y+9z=-30$$

Solve these equations for y and z respectively.

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

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

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

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

$$z=-\frac{170}{47}$$

Substitute -\frac{170}{47} for z in the equation y=\frac{10}{7}+\frac{4}{7}z.

$$y=\frac{10}{7}+\frac{4}{7}\left(-\frac{170}{47}\right)$$

Calculate y from y=\frac{10}{7}+\frac{4}{7}\left(-\frac{170}{47}\right).

$$y=-\frac{30}{47}$$

Substitute -\frac{30}{47} for y in the equation x=5+y.

$$x=5-\frac{30}{47}$$

Calculate x from x=5-\frac{30}{47}.

$$x=\frac{205}{47}$$

The system is now solved.

$$x=\frac{205}{47} y=-\frac{30}{47} z=-\frac{170}{47}$$