Solve {l}{6x+5y-2z=8}{-6x-6y+5z=-24}{-2x+4y+3z=-28}

$$\left. \begin{array} { l } { 6 x + 5 y - 2 z = 8 } \\ { - 6 x - 6 y + 5 z = - 24 } \\ { - 2 x + 4 y + 3 z = - 28 } \end{array} \right.$$

Solve for x, y, z

$$x=1$$

$$y=-2$$

$$z=-6$$

Short Steps Using Substitution

Solve 6x+5y-2z=8 for x.

$$x=-\frac{5}{6}y+\frac{1}{3}z+\frac{4}{3}$$

Substitute -\frac{5}{6}y+\frac{1}{3}z+\frac{4}{3} for x in the second and third equation.

$$-6\left(-\frac{5}{6}y+\frac{1}{3}z+\frac{4}{3}\right)-6y+5z=-24 -2\left(-\frac{5}{6}y+\frac{1}{3}z+\frac{4}{3}\right)+4y+3z=-28$$

Solve these equations for y and z respectively.

$$y=16+3z z=-\frac{76}{7}-\frac{17}{7}y$$

Substitute 16+3z for y in the equation z=-\frac{76}{7}-\frac{17}{7}y.

$$z=-\frac{76}{7}-\frac{17}{7}\left(16+3z\right)$$

Solve z=-\frac{76}{7}-\frac{17}{7}\left(16+3z\right) for z.

$$z=-6$$

Substitute -6 for z in the equation y=16+3z.

$$y=16+3\left(-6\right)$$

Calculate y from y=16+3\left(-6\right).

$$y=-2$$

Substitute -2 for y and -6 for z in the equation x=-\frac{5}{6}y+\frac{1}{3}z+\frac{4}{3}.

$$x=-\frac{5}{6}\left(-2\right)+\frac{1}{3}\left(-6\right)+\frac{4}{3}$$

Calculate x from x=-\frac{5}{6}\left(-2\right)+\frac{1}{3}\left(-6\right)+\frac{4}{3}.

$$x=1$$

The system is now solved.

$$x=1 y=-2 z=-6$$