Solve {l}{-8=1x+2y+27}{13=-1x+y-2z}{12=1x-y-7}

$$\left. \begin{array} { l } { - 8 = 1 x + 2 y + 27 } \\ { 13 = - 1 x + y - 2 z } \\ { 12 = 1 x - y - 7 } \end{array} \right.$$

Solve for x, y, z

$$x=1$$

$$y=-18$$

$$z=-16$$

Short Steps Using Substitution

Reorder the equations.

$$13=-x+y-2z -8=1x+2y+27 12=1x-y-7$$

Solve 13=-x+y-2z for x.

$$x=y-13-2z$$

Substitute y-13-2z for x in the second and third equation.

$$-8=1\left(y-13-2z\right)+2y+27 12=1\left(y-13-2z\right)-y-7$$

Solve these equations for y and z respectively.

$$y=-\frac{22}{3}+\frac{2}{3}z z=-16$$

Substitute -16 for z in the equation y=-\frac{22}{3}+\frac{2}{3}z.

$$y=-\frac{22}{3}+\frac{2}{3}\left(-16\right)$$

Calculate y from y=-\frac{22}{3}+\frac{2}{3}\left(-16\right).

$$y=-18$$

Substitute -18 for y and -16 for z in the equation x=y-13-2z.

$$x=-18-13-2\left(-16\right)$$

Calculate x from x=-18-13-2\left(-16\right).

$$x=1$$

The system is now solved.

$$x=1 y=-18 z=-16$$