Solve {l}{x+y-z=6}{x-y+2z=-7}{x+2y+z=4}

$$\left. \begin{array} { l } { x + y - z = 6 } \\ { x - y + 2 z = - 7 } \\ { x + 2 y + z = 4 } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{5}{7}\approx 0.714285714$$

$$y = \frac{20}{7} = 2\frac{6}{7} \approx 2.857142857$$

$$z = -\frac{17}{7} = -2\frac{3}{7} \approx -2.428571429$$

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.

$$-y+z+6-y+2z=-7 -y+z+6+2y+z=4$$

Solve these equations for y and z respectively.

$$y=\frac{3}{2}z+\frac{13}{2} z=-\frac{1}{2}y-1$$

Substitute \frac{3}{2}z+\frac{13}{2} for y in the equation z=-\frac{1}{2}y-1.

$$z=-\frac{1}{2}\left(\frac{3}{2}z+\frac{13}{2}\right)-1$$

Solve z=-\frac{1}{2}\left(\frac{3}{2}z+\frac{13}{2}\right)-1 for z.

$$z=-\frac{17}{7}$$

Substitute -\frac{17}{7} for z in the equation y=\frac{3}{2}z+\frac{13}{2}.

$$y=\frac{3}{2}\left(-\frac{17}{7}\right)+\frac{13}{2}$$

Calculate y from y=\frac{3}{2}\left(-\frac{17}{7}\right)+\frac{13}{2}.

$$y=\frac{20}{7}$$

Substitute \frac{20}{7} for y and -\frac{17}{7} for z in the equation x=-y+z+6.

$$x=-\frac{20}{7}-\frac{17}{7}+6$$

Calculate x from x=-\frac{20}{7}-\frac{17}{7}+6.

$$x=\frac{5}{7}$$

The system is now solved.

$$x=\frac{5}{7} y=\frac{20}{7} z=-\frac{17}{7}$$