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

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

Solve for x, y, z

$$x=-1$$

$$y=1$$

$$z=-2$$

Short Steps Using Substitution

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

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

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

$$-5\left(-2x-2z-5\right)-4z=3 3x-3\left(-2x-2z-5\right)-z=-4$$

Solve these equations for x and z respectively.

$$x=-\frac{11}{5}-\frac{3}{5}z z=-\frac{9}{5}x-\frac{19}{5}$$

Substitute -\frac{11}{5}-\frac{3}{5}z for x in the equation z=-\frac{9}{5}x-\frac{19}{5}.

$$z=-\frac{9}{5}\left(-\frac{11}{5}-\frac{3}{5}z\right)-\frac{19}{5}$$

Solve z=-\frac{9}{5}\left(-\frac{11}{5}-\frac{3}{5}z\right)-\frac{19}{5} for z.

$$z=-2$$

Substitute -2 for z in the equation x=-\frac{11}{5}-\frac{3}{5}z.

$$x=-\frac{11}{5}-\frac{3}{5}\left(-2\right)$$

Calculate x from x=-\frac{11}{5}-\frac{3}{5}\left(-2\right).

$$x=-1$$

Substitute -1 for x and -2 for z in the equation y=-2x-2z-5.

$$y=-2\left(-1\right)-2\left(-2\right)-5$$

Calculate y from y=-2\left(-1\right)-2\left(-2\right)-5.

$$y=1$$

The system is now solved.

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