Solve {l}{5x+4y+z=0}{3x+y-4z=1}{x+y+z=-2}

$$\left\{ \begin{array} { l } { 5 x + 4 y + z = 0 } \\ { 3 x + y - 4 z = 1 } \\ { x + y + z = - 2 } \end{array} \right.$$

Solve for x, y, z

$$x=-31$$

$$y=42$$

$$z=-13$$

Short Steps Using Substitution

Solve 5x+4y+z=0 for z.

$$z=-5x-4y$$

Substitute -5x-4y for z in the second and third equation.

$$3x+y-4\left(-5x-4y\right)=1 x+y-5x-4y=-2$$

Solve these equations for y and x respectively.

$$y=-\frac{23}{17}x+\frac{1}{17} x=-\frac{3}{4}y+\frac{1}{2}$$

Substitute -\frac{23}{17}x+\frac{1}{17} for y in the equation x=-\frac{3}{4}y+\frac{1}{2}.

$$x=-\frac{3}{4}\left(-\frac{23}{17}x+\frac{1}{17}\right)+\frac{1}{2}$$

Solve x=-\frac{3}{4}\left(-\frac{23}{17}x+\frac{1}{17}\right)+\frac{1}{2} for x.

$$x=-31$$

Substitute -31 for x in the equation y=-\frac{23}{17}x+\frac{1}{17}.

$$y=-\frac{23}{17}\left(-31\right)+\frac{1}{17}$$

Calculate y from y=-\frac{23}{17}\left(-31\right)+\frac{1}{17}.

$$y=42$$

Substitute 42 for y and -31 for x in the equation z=-5x-4y.

$$z=-5\left(-31\right)-4\times 42$$

Calculate z from z=-5\left(-31\right)-4\times 42.

$$z=-13$$

The system is now solved.

$$x=-31 y=42 z=-13$$