Solve {l}{-4x-5y-z=18}{-2x-5y-2z=12}{-2x+5y+2z=4}

$$\left\{ \begin{array} { l } { - 4 x - 5 y - z = 18 } \\ { - 2 x - 5 y - 2 z = 12 } \\ { - 2 x + 5 y + 2 z = 4 } \end{array} \right.$$

Solve for x, y, z

$$x=-4$$

$$y=0$$

$$z=-2$$

Short Steps Using Substitution

Solve -4x-5y-z=18 for z.

$$z=-4x-5y-18$$

Substitute -4x-5y-18 for z in the second and third equation.

$$-2x-5y-2\left(-4x-5y-18\right)=12 -2x+5y+2\left(-4x-5y-18\right)=4$$

Solve these equations for y and x respectively.

$$y=-\frac{24}{5}-\frac{6}{5}x x=-4-\frac{1}{2}y$$

Substitute -\frac{24}{5}-\frac{6}{5}x for y in the equation x=-4-\frac{1}{2}y.

$$x=-4-\frac{1}{2}\left(-\frac{24}{5}-\frac{6}{5}x\right)$$

Solve x=-4-\frac{1}{2}\left(-\frac{24}{5}-\frac{6}{5}x\right) for x.

$$x=-4$$

Substitute -4 for x in the equation y=-\frac{24}{5}-\frac{6}{5}x.

$$y=-\frac{24}{5}-\frac{6}{5}\left(-4\right)$$

Calculate y from y=-\frac{24}{5}-\frac{6}{5}\left(-4\right).

$$y=0$$

Substitute 0 for y and -4 for x in the equation z=-4x-5y-18.

$$z=-4\left(-4\right)-5\times 0-18$$

Calculate z from z=-4\left(-4\right)-5\times 0-18.

$$z=-2$$

The system is now solved.

$$x=-4 y=0 z=-2$$