Solve {l}{x+3y-z=11}{2x-y+4z=-5}{-x-3y+2z=-12}

$$\left\{ \begin{array} { l } { x + 3 y - z = 11 } \\ { 2 x - y + 4 z = - 5 } \\ { - x - 3 y + 2 z = - 12 } \end{array} \right.$$

Solve for x, y, z

$$x=1$$

$$y=3$$

$$z=-1$$

Short Steps Using Substitution

Solve x+3y-z=11 for x.

$$x=-3y+z+11$$

Substitute -3y+z+11 for x in the second and third equation.

$$2\left(-3y+z+11\right)-y+4z=-5 -\left(-3y+z+11\right)-3y+2z=-12$$

Solve these equations for y and z respectively.

$$y=\frac{27}{7}+\frac{6}{7}z z=-1$$

Substitute -1 for z in the equation y=\frac{27}{7}+\frac{6}{7}z.

$$y=\frac{27}{7}+\frac{6}{7}\left(-1\right)$$

Calculate y from y=\frac{27}{7}+\frac{6}{7}\left(-1\right).

$$y=3$$

Substitute 3 for y and -1 for z in the equation x=-3y+z+11.

$$x=-3\times 3-1+11$$

Calculate x from x=-3\times 3-1+11.

$$x=1$$

The system is now solved.

$$x=1 y=3 z=-1$$