Solve {l}{-8x+3y-6z=83}{-5x-6y-3z=71}{-6x+y-z=45}

$$\left\{ \begin{array} { l } { - 8 x + 3 y - 6 z = 83 } \\ { - 5 x - 6 y - 3 z = 71 } \\ { - 6 x + y - z = 45 } \end{array} \right.$$

Solve for x, y, z

$$x=-7$$

$$y=-3$$

$$z=-6$$

Short Steps Using Substitution

Reorder the equations.

$$-6x+y-z=45 -5x-6y-3z=71 -8x+3y-6z=83$$

Solve -6x+y-z=45 for y.

$$y=6x+z+45$$

Substitute 6x+z+45 for y in the second and third equation.

$$-5x-6\left(6x+z+45\right)-3z=71 -8x+3\left(6x+z+45\right)-6z=83$$

Solve these equations for x and z respectively.

$$x=-\frac{9}{41}z-\frac{341}{41} z=\frac{52}{3}+\frac{10}{3}x$$

Substitute -\frac{9}{41}z-\frac{341}{41} for x in the equation z=\frac{52}{3}+\frac{10}{3}x.

$$z=\frac{52}{3}+\frac{10}{3}\left(-\frac{9}{41}z-\frac{341}{41}\right)$$

Solve z=\frac{52}{3}+\frac{10}{3}\left(-\frac{9}{41}z-\frac{341}{41}\right) for z.

$$z=-6$$

Substitute -6 for z in the equation x=-\frac{9}{41}z-\frac{341}{41}.

$$x=-\frac{9}{41}\left(-6\right)-\frac{341}{41}$$

Calculate x from x=-\frac{9}{41}\left(-6\right)-\frac{341}{41}.

$$x=-7$$

Substitute -7 for x and -6 for z in the equation y=6x+z+45.

$$y=6\left(-7\right)-6+45$$

Calculate y from y=6\left(-7\right)-6+45.

$$y=-3$$

The system is now solved.

$$x=-7 y=-3 z=-6$$