Solve {l}{x+y+z=2}{x-y=1}{2x+y-z=20}

$$\left\{ \begin{array} { l } { x + y + z = 2 } \\ { x - y = 1 } \\ { 2 x + y - z = 20 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{24}{5} = 4\frac{4}{5} = 4.8$$

$$y = \frac{19}{5} = 3\frac{4}{5} = 3.8$$

$$z = -\frac{33}{5} = -6\frac{3}{5} = -6.6$$

Short Steps Using Substitution

Solve x+y+z=2 for x.

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

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

$$-y-z+2-y=1 2\left(-y-z+2\right)+y-z=20$$

Solve these equations for y and z respectively.

$$y=\frac{1}{2}-\frac{1}{2}z z=-\frac{16}{3}-\frac{1}{3}y$$

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

$$z=-\frac{16}{3}-\frac{1}{3}\left(\frac{1}{2}-\frac{1}{2}z\right)$$

Solve z=-\frac{16}{3}-\frac{1}{3}\left(\frac{1}{2}-\frac{1}{2}z\right) for z.

$$z=-\frac{33}{5}$$

Substitute -\frac{33}{5} for z in the equation y=\frac{1}{2}-\frac{1}{2}z.

$$y=\frac{1}{2}-\frac{1}{2}\left(-\frac{33}{5}\right)$$

Calculate y from y=\frac{1}{2}-\frac{1}{2}\left(-\frac{33}{5}\right).

$$y=\frac{19}{5}$$

Substitute \frac{19}{5} for y and -\frac{33}{5} for z in the equation x=-y-z+2.

$$x=-\frac{19}{5}-\left(-\frac{33}{5}\right)+2$$

Calculate x from x=-\frac{19}{5}-\left(-\frac{33}{5}\right)+2.

$$x=\frac{24}{5}$$

The system is now solved.

$$x=\frac{24}{5} y=\frac{19}{5} z=-\frac{33}{5}$$