Solve {r}{0x+3y-4z=16}{2x-5y+7z=-27}{-x+0y-9z=9}

$$\left. \begin{array} { r } { 0 x + 3 y - 4 z = 16 } \\ { 2 x - 5 y + 7 z = - 27 } \\ { - x + 0 y - 9 z = 9 } \end{array} \right.$$

Solve for x, y, z

$$x=0$$

$$y=4$$

$$z=-1$$

Short Steps Using Substitution

Reorder the equations.

$$-x+0y-9z=9 2x-5y+7z=-27 0x+3y-4z=16$$

Solve -x+0y-9z=9 for x.

$$x=-9z-9$$

Substitute -9z-9 for x in the second and third equation.

$$2\left(-9z-9\right)-5y+7z=-27 0\left(-9z-9\right)+3y-4z=16$$

Solve these equations for y and z respectively.

$$y=\frac{9}{5}-\frac{11}{5}z z=\frac{3}{4}y-4$$

Substitute \frac{9}{5}-\frac{11}{5}z for y in the equation z=\frac{3}{4}y-4.

$$z=\frac{3}{4}\left(\frac{9}{5}-\frac{11}{5}z\right)-4$$

Solve z=\frac{3}{4}\left(\frac{9}{5}-\frac{11}{5}z\right)-4 for z.

$$z=-1$$

Substitute -1 for z in the equation y=\frac{9}{5}-\frac{11}{5}z.

$$y=\frac{9}{5}-\frac{11}{5}\left(-1\right)$$

Calculate y from y=\frac{9}{5}-\frac{11}{5}\left(-1\right).

$$y=4$$

Substitute -1 for z in the equation x=-9z-9.

$$x=-9\left(-1\right)-9$$

Calculate x from x=-9\left(-1\right)-9.

$$x=0$$

The system is now solved.

$$x=0 y=4 z=-1$$