Solve {l}{2x+3y-z=-11}{-3x+2y+4z=11}{-x+4y+2z=-1}

$$\left\{ \begin{array} { l } { 2 x + 3 y - z = - 11 } \\ { - 3 x + 2 y + 4 z = 11 } \\ { - x + 4 y + 2 z = - 1 } \end{array} \right.$$

Solve for x, y, z

$$x=-1$$

$$y=-2$$

$$z=3$$

Short Steps Using Substitution

Solve 2x+3y-z=-11 for z.

$$z=2x+3y+11$$

Substitute 2x+3y+11 for z in the second and third equation.

$$-3x+2y+4\left(2x+3y+11\right)=11 -x+4y+2\left(2x+3y+11\right)=-1$$

Solve these equations for y and x respectively.

$$y=-\frac{5}{14}x-\frac{33}{14} x=-\frac{10}{3}y-\frac{23}{3}$$

Substitute -\frac{5}{14}x-\frac{33}{14} for y in the equation x=-\frac{10}{3}y-\frac{23}{3}.

$$x=-\frac{10}{3}\left(-\frac{5}{14}x-\frac{33}{14}\right)-\frac{23}{3}$$

Solve x=-\frac{10}{3}\left(-\frac{5}{14}x-\frac{33}{14}\right)-\frac{23}{3} for x.

$$x=-1$$

Substitute -1 for x in the equation y=-\frac{5}{14}x-\frac{33}{14}.

$$y=-\frac{5}{14}\left(-1\right)-\frac{33}{14}$$

Calculate y from y=-\frac{5}{14}\left(-1\right)-\frac{33}{14}.

$$y=-2$$

Substitute -2 for y and -1 for x in the equation z=2x+3y+11.

$$z=2\left(-1\right)+3\left(-2\right)+11$$

Calculate z from z=2\left(-1\right)+3\left(-2\right)+11.

$$z=3$$

The system is now solved.

$$x=-1 y=-2 z=3$$