Solve {l}{x+y+z=22}{3x+2y+z=44}{-x+y-7z=99}

$$\left. \begin{array} { l } { x + y + z = 22 } \\ { 3 x + 2 y + z = 44 } \\ { - x + y - 7 z = 99 } \end{array} \right.$$

Solve for x, y, z

$$x = -\frac{77}{10} = -7\frac{7}{10} = -7.7$$

$$y = \frac{187}{5} = 37\frac{2}{5} = 37.4$$

$$z = -\frac{77}{10} = -7\frac{7}{10} = -7.7$$

Short Steps Using Substitution

Solve x+y+z=22 for x.

$$x=-y-z+22$$

Substitute -y-z+22 for x in the second and third equation.

$$3\left(-y-z+22\right)+2y+z=44 -\left(-y-z+22\right)+y-7z=99$$

Solve these equations for y and z respectively.

$$y=-2z+22 z=\frac{1}{3}y-\frac{121}{6}$$

Substitute -2z+22 for y in the equation z=\frac{1}{3}y-\frac{121}{6}.

$$z=\frac{1}{3}\left(-2z+22\right)-\frac{121}{6}$$

Solve z=\frac{1}{3}\left(-2z+22\right)-\frac{121}{6} for z.

$$z=-\frac{77}{10}$$

Substitute -\frac{77}{10} for z in the equation y=-2z+22.

$$y=-2\left(-\frac{77}{10}\right)+22$$

Calculate y from y=-2\left(-\frac{77}{10}\right)+22.

$$y=\frac{187}{5}$$

Substitute \frac{187}{5} for y and -\frac{77}{10} for z in the equation x=-y-z+22.

$$x=-\frac{187}{5}-\left(-\frac{77}{10}\right)+22$$

Calculate x from x=-\frac{187}{5}-\left(-\frac{77}{10}\right)+22.

$$x=-\frac{77}{10}$$

The system is now solved.

$$x=-\frac{77}{10} y=\frac{187}{5} z=-\frac{77}{10}$$