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

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

Solve for y, z, x

$$x = \frac{55}{4} = 13\frac{3}{4} = 13.75$$

$$y = \frac{7}{4} = 1\frac{3}{4} = 1.75$$

$$z=-3$$

Short Steps Using Substitution

Reorder the equations.

$$x-5y+3z=-4 -4y-4z=5 -x+y-4z=0$$

Solve x-5y+3z=-4 for x.

$$x=5y-3z-4$$

Substitute 5y-3z-4 for x in the equation -x+y-4z=0.

$$-\left(5y-3z-4\right)+y-4z=0$$

Solve the second equation for z and the third equation for y.

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

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

$$y=-\frac{1}{4}\left(-y-\frac{5}{4}\right)+1$$

Solve y=-\frac{1}{4}\left(-y-\frac{5}{4}\right)+1 for y.

$$y=\frac{7}{4}$$

Substitute \frac{7}{4} for y in the equation z=-y-\frac{5}{4}.

$$z=-\frac{7}{4}-\frac{5}{4}$$

Calculate z from z=-\frac{7}{4}-\frac{5}{4}.

$$z=-3$$

Substitute -3 for z and \frac{7}{4} for y in the equation x=5y-3z-4.

$$x=5\times \frac{7}{4}-3\left(-3\right)-4$$

Calculate x from x=5\times \frac{7}{4}-3\left(-3\right)-4.

$$x=\frac{55}{4}$$

The system is now solved.

$$y=\frac{7}{4} z=-3 x=\frac{55}{4}$$