Solve {l}{x-y+z=6}{x+y-z=0}{2x-3y+8z=33}

$$\left\{ \begin{array} { l } { x - y + z = 6 } \\ { x + y - z = 0 } \\ { 2 x - 3 y + 8 z = 33 } \end{array} \right.$$

Solve for x, y, z

$$x=3$$

$$y=\frac{3}{5}=0.6$$

$$z = \frac{18}{5} = 3\frac{3}{5} = 3.6$$

Short Steps Using Substitution

Solve x-y+z=6 for x.

$$x=y-z+6$$

Substitute y-z+6 for x in the second and third equation.

$$y-z+6+y-z=0 2\left(y-z+6\right)-3y+8z=33$$

Solve these equations for y and z respectively.

$$y=-3+z z=\frac{1}{6}y+\frac{7}{2}$$

Substitute -3+z for y in the equation z=\frac{1}{6}y+\frac{7}{2}.

$$z=\frac{1}{6}\left(-3+z\right)+\frac{7}{2}$$

Solve z=\frac{1}{6}\left(-3+z\right)+\frac{7}{2} for z.

$$z=\frac{18}{5}$$

Substitute \frac{18}{5} for z in the equation y=-3+z.

$$y=-3+\frac{18}{5}$$

Calculate y from y=-3+\frac{18}{5}.

$$y=\frac{3}{5}$$

Substitute \frac{3}{5} for y and \frac{18}{5} for z in the equation x=y-z+6.

$$x=\frac{3}{5}-\frac{18}{5}+6$$

Calculate x from x=\frac{3}{5}-\frac{18}{5}+6.

$$x=3$$

The system is now solved.

$$x=3 y=\frac{3}{5} z=\frac{18}{5}$$