Solve {l}{-6x-2y-z=-17}{5x+y-6z=19}{-4x-6y-6z=-20}

$$\left. \begin{array} { l } { - 6 x - 2 y - z = - 17 } \\ { 5 x + y - 6 z = 19 } \\ { - 4 x - 6 y - 6 z = - 20 } \end{array} \right.$$

Solve for x, y, z

$$x=2$$

$$y=3$$

$$z=-1$$

Short Steps Using Substitution

Solve -6x-2y-z=-17 for z.

$$z=-6x-2y+17$$

Substitute -6x-2y+17 for z in the second and third equation.

$$5x+y-6\left(-6x-2y+17\right)=19 -4x-6y-6\left(-6x-2y+17\right)=-20$$

Solve these equations for y and x respectively.

$$y=-\frac{41}{13}x+\frac{121}{13} x=\frac{41}{16}-\frac{3}{16}y$$

Substitute -\frac{41}{13}x+\frac{121}{13} for y in the equation x=\frac{41}{16}-\frac{3}{16}y.

$$x=\frac{41}{16}-\frac{3}{16}\left(-\frac{41}{13}x+\frac{121}{13}\right)$$

Solve x=\frac{41}{16}-\frac{3}{16}\left(-\frac{41}{13}x+\frac{121}{13}\right) for x.

$$x=2$$

Substitute 2 for x in the equation y=-\frac{41}{13}x+\frac{121}{13}.

$$y=-\frac{41}{13}\times 2+\frac{121}{13}$$

Calculate y from y=-\frac{41}{13}\times 2+\frac{121}{13}.

$$y=3$$

Substitute 3 for y and 2 for x in the equation z=-6x-2y+17.

$$z=-6\times 2-2\times 3+17$$

Calculate z from z=-6\times 2-2\times 3+17.

$$z=-1$$

The system is now solved.

$$x=2 y=3 z=-1$$