Solve {l}{-4x-5y+6z=-3}{-5x-6y-2z=20}{-2x+5y+z=12}

$$\left\{ \begin{array} { l } { - 4 x - 5 y + 6 z = - 3 } \\ { - 5 x - 6 y - 2 z = 20 } \\ { - 2 x + 5 y + z = 12 } \end{array} \right.$$

Solve for x, y, z

$$x = -\frac{1240}{283} = -4\frac{108}{283} \approx -4.381625442$$

$$y = \frac{323}{283} = 1\frac{40}{283} \approx 1.141342756$$

$$z = -\frac{699}{283} = -2\frac{133}{283} \approx -2.469964664$$

Short Steps Using Substitution

Reorder the equations.

$$-2x+5y+z=12 -5x-6y-2z=20 -4x-5y+6z=-3$$

Solve -2x+5y+z=12 for z.

$$z=2x-5y+12$$

Substitute 2x-5y+12 for z in the second and third equation.

$$-5x-6y-2\left(2x-5y+12\right)=20 -4x-5y+6\left(2x-5y+12\right)=-3$$

Solve these equations for y and x respectively.

$$y=\frac{9}{4}x+11 x=-\frac{75}{8}+\frac{35}{8}y$$

Substitute \frac{9}{4}x+11 for y in the equation x=-\frac{75}{8}+\frac{35}{8}y.

$$x=-\frac{75}{8}+\frac{35}{8}\left(\frac{9}{4}x+11\right)$$

Solve x=-\frac{75}{8}+\frac{35}{8}\left(\frac{9}{4}x+11\right) for x.

$$x=-\frac{1240}{283}$$

Substitute -\frac{1240}{283} for x in the equation y=\frac{9}{4}x+11.

$$y=\frac{9}{4}\left(-\frac{1240}{283}\right)+11$$

Calculate y from y=\frac{9}{4}\left(-\frac{1240}{283}\right)+11.

$$y=\frac{323}{283}$$

Substitute \frac{323}{283} for y and -\frac{1240}{283} for x in the equation z=2x-5y+12.

$$z=2\left(-\frac{1240}{283}\right)-5\times \frac{323}{283}+12$$

Calculate z from z=2\left(-\frac{1240}{283}\right)-5\times \frac{323}{283}+12.

$$z=-\frac{699}{283}$$

The system is now solved.

$$x=-\frac{1240}{283} y=\frac{323}{283} z=-\frac{699}{283}$$