Solve {l}{x-6y-2z=7}{-2x+5y-2z=-1}{-5x+6y+5z=-6}

$$\left\{ \begin{array} { l } { x - 6 y - 2 z = 7 } \\ { - 2 x + 5 y - 2 z = - 1 } \\ { - 5 x + 6 y + 5 z = - 6 } \end{array} \right.$$

Solve for x, y, z

$$x=-1$$

$$y=-1$$

$$z=-1$$

Short Steps Using Substitution

Solve x-6y-2z=7 for x.

$$x=6y+2z+7$$

Substitute 6y+2z+7 for x in the second and third equation.

$$-2\left(6y+2z+7\right)+5y-2z=-1 -5\left(6y+2z+7\right)+6y+5z=-6$$

Solve these equations for y and z respectively.

$$y=-\frac{6}{7}z-\frac{13}{7} z=-\frac{24}{5}y-\frac{29}{5}$$

Substitute -\frac{6}{7}z-\frac{13}{7} for y in the equation z=-\frac{24}{5}y-\frac{29}{5}.

$$z=-\frac{24}{5}\left(-\frac{6}{7}z-\frac{13}{7}\right)-\frac{29}{5}$$

Solve z=-\frac{24}{5}\left(-\frac{6}{7}z-\frac{13}{7}\right)-\frac{29}{5} for z.

$$z=-1$$

Substitute -1 for z in the equation y=-\frac{6}{7}z-\frac{13}{7}.

$$y=-\frac{6}{7}\left(-1\right)-\frac{13}{7}$$

Calculate y from y=-\frac{6}{7}\left(-1\right)-\frac{13}{7}.

$$y=-1$$

Substitute -1 for y and -1 for z in the equation x=6y+2z+7.

$$x=6\left(-1\right)+2\left(-1\right)+7$$

Calculate x from x=6\left(-1\right)+2\left(-1\right)+7.

$$x=-1$$

The system is now solved.

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