Solve {l}{-2x-2y-4z=28}{-2x-y-5z=31}{-5x-5y+5z=-5}

$$\left. \begin{array}{l}{ - 2 x - 2 y - 4 z = 28 }\\{ - 2 x - y - 5 z = 31 }\\{ - 5 x - 5 y + 5 z = - 5 }\end{array} \right.$$

Solve for x, y, z

$$x=-2$$

$$y=-2$$

$$z=-5$$

Short Steps Using Substitution

Reorder the equations.

$$-2x-y-5z=31 -2x-2y-4z=28 -5x-5y+5z=-5$$

Solve -2x-y-5z=31 for y.

$$y=-2x-5z-31$$

Substitute -2x-5z-31 for y in the second and third equation.

$$-2x-2\left(-2x-5z-31\right)-4z=28 -5x-5\left(-2x-5z-31\right)+5z=-5$$

Solve these equations for x and z respectively.

$$x=-17-3z z=-\frac{16}{3}-\frac{1}{6}x$$

Substitute -17-3z for x in the equation z=-\frac{16}{3}-\frac{1}{6}x.

$$z=-\frac{16}{3}-\frac{1}{6}\left(-17-3z\right)$$

Solve z=-\frac{16}{3}-\frac{1}{6}\left(-17-3z\right) for z.

$$z=-5$$

Substitute -5 for z in the equation x=-17-3z.

$$x=-17-3\left(-5\right)$$

Calculate x from x=-17-3\left(-5\right).

$$x=-2$$

Substitute -2 for x and -5 for z in the equation y=-2x-5z-31.

$$y=-2\left(-2\right)-5\left(-5\right)-31$$

Calculate y from y=-2\left(-2\right)-5\left(-5\right)-31.

$$y=-2$$

The system is now solved.

$$x=-2 y=-2 z=-5$$