Solve {l}{-5x-3y-2z=-8}{-4x-y+6z=-7}{-6x-y+z=0}

$$\left. \begin{array} { l } { - 5 x - 3 y - 2 z = - 8 } \\ { - 4 x - y + 6 z = - 7 } \\ { - 6 x - y + z = 0 } \end{array} \right.$$

Solve for x, y, z

$$x=-1$$

$$y=5$$

$$z=-1$$

Short Steps Using Substitution

Reorder the equations.

$$-4x-y+6z=-7 -5x-3y-2z=-8 -6x-y+z=0$$

Solve -4x-y+6z=-7 for y.

$$y=-4x+6z+7$$

Substitute -4x+6z+7 for y in the second and third equation.

$$-5x-3\left(-4x+6z+7\right)-2z=-8 -6x-\left(-4x+6z+7\right)+z=0$$

Solve these equations for x and z respectively.

$$x=\frac{20}{7}z+\frac{13}{7} z=-\frac{2}{5}x-\frac{7}{5}$$

Substitute \frac{20}{7}z+\frac{13}{7} for x in the equation z=-\frac{2}{5}x-\frac{7}{5}.

$$z=-\frac{2}{5}\left(\frac{20}{7}z+\frac{13}{7}\right)-\frac{7}{5}$$

Solve z=-\frac{2}{5}\left(\frac{20}{7}z+\frac{13}{7}\right)-\frac{7}{5} for z.

$$z=-1$$

Substitute -1 for z in the equation x=\frac{20}{7}z+\frac{13}{7}.

$$x=\frac{20}{7}\left(-1\right)+\frac{13}{7}$$

Calculate x from x=\frac{20}{7}\left(-1\right)+\frac{13}{7}.

$$x=-1$$

Substitute -1 for x and -1 for z in the equation y=-4x+6z+7.

$$y=-4\left(-1\right)+6\left(-1\right)+7$$

Calculate y from y=-4\left(-1\right)+6\left(-1\right)+7.

$$y=5$$

The system is now solved.

$$x=-1 y=5 z=-1$$