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

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

Solve for x, y, z

$$x=1$$

$$y=-3$$

$$z=3$$

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.

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

Solve these equations for y and z respectively.

$$y=-\frac{9}{2}+\frac{1}{2}z z=\frac{4}{5}y+\frac{27}{5}$$

Substitute -\frac{9}{2}+\frac{1}{2}z for y in the equation z=\frac{4}{5}y+\frac{27}{5}.

$$z=\frac{4}{5}\left(-\frac{9}{2}+\frac{1}{2}z\right)+\frac{27}{5}$$

Solve z=\frac{4}{5}\left(-\frac{9}{2}+\frac{1}{2}z\right)+\frac{27}{5} for z.

$$z=3$$

Substitute 3 for z in the equation y=-\frac{9}{2}+\frac{1}{2}z.

$$y=-\frac{9}{2}+\frac{1}{2}\times 3$$

Calculate y from y=-\frac{9}{2}+\frac{1}{2}\times 3.

$$y=-3$$

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

$$x=-3\left(-3\right)+3-11$$

Calculate x from x=-3\left(-3\right)+3-11.

$$x=1$$

The system is now solved.

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