Solve {l}{x+3y-6z=7}{2x-y+z=1}{x+2y+2z=-1}

$$\left. \begin{array} { l } { x + 3 y - 6 z = 7 } \\ { 2 x - y + z = 1 } \\ { x + 2 y + 2 z = - 1 } \end{array} \right.$$

Solve for x, y, z

$$x=1$$

$$y=0$$

$$z=-1$$

Short Steps Using Substitution

Solve x+3y-6z=7 for x.

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

Substitute -3y+6z+7 for x in the second and third equation.

$$2\left(-3y+6z+7\right)-y+z=1 -3y+6z+7+2y+2z=-1$$

Solve these equations for y and z respectively.

$$y=\frac{13}{7}+\frac{13}{7}z z=\frac{1}{8}y-1$$

Substitute \frac{13}{7}+\frac{13}{7}z for y in the equation z=\frac{1}{8}y-1.

$$z=\frac{1}{8}\left(\frac{13}{7}+\frac{13}{7}z\right)-1$$

Solve z=\frac{1}{8}\left(\frac{13}{7}+\frac{13}{7}z\right)-1 for z.

$$z=-1$$

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

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

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

$$y=0$$

Substitute 0 for y and -1 for z in the equation x=-3y+6z+7.

$$x=-3\times 0+6\left(-1\right)+7$$

Calculate x from x=-3\times 0+6\left(-1\right)+7.

$$x=1$$

The system is now solved.

$$x=1 y=0 z=-1$$