Solve {l}{x-2y-3z=-1}{2x+y-2z=6}{x+3y-2z=13}

$$\left\{ \begin{array} { l } { x - 2 y - 3 z = - 1 } \\ { 2 x + y - 2 z = 6 } \\ { x + 3 y - 2 z = 13 } \end{array} \right.$$

Solve for x, y, z

$$x=-\frac{3}{5}=-0.6$$

$$y = \frac{16}{5} = 3\frac{1}{5} = 3.2$$

$$z=-2$$

Short Steps Using Substitution

Solve x-2y-3z=-1 for x.

$$x=2y+3z-1$$

Substitute 2y+3z-1 for x in the second and third equation.

$$2\left(2y+3z-1\right)+y-2z=6 2y+3z-1+3y-2z=13$$

Solve these equations for y and z respectively.

$$y=\frac{8}{5}-\frac{4}{5}z z=-5y+14$$

Substitute \frac{8}{5}-\frac{4}{5}z for y in the equation z=-5y+14.

$$z=-5\left(\frac{8}{5}-\frac{4}{5}z\right)+14$$

Solve z=-5\left(\frac{8}{5}-\frac{4}{5}z\right)+14 for z.

$$z=-2$$

Substitute -2 for z in the equation y=\frac{8}{5}-\frac{4}{5}z.

$$y=\frac{8}{5}-\frac{4}{5}\left(-2\right)$$

Calculate y from y=\frac{8}{5}-\frac{4}{5}\left(-2\right).

$$y=\frac{16}{5}$$

Substitute \frac{16}{5} for y and -2 for z in the equation x=2y+3z-1.

$$x=2\times \frac{16}{5}+3\left(-2\right)-1$$

Calculate x from x=2\times \frac{16}{5}+3\left(-2\right)-1.

$$x=-\frac{3}{5}$$

The system is now solved.

$$x=-\frac{3}{5} y=\frac{16}{5} z=-2$$