Solve {l}{x+y-z=2}{-x+2y+3z=-1}{x-4y-2z=-7}

$$\left. \begin{array} { l } { x + y - z = 2 } \\ { - x + 2 y + 3 z = - 1 } \\ { x - 4 y - 2 z = - 7 } \end{array} \right.$$

Solve for x, y, z

$$x = -\frac{25}{7} = -3\frac{4}{7} \approx -3.571428571$$

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

$$z = -\frac{22}{7} = -3\frac{1}{7} \approx -3.142857143$$

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.

$$-\left(-y+z+2\right)+2y+3z=-1 -y+z+2-4y-2z=-7$$

Solve these equations for y and z respectively.

$$y=-\frac{2}{3}z+\frac{1}{3} z=-5y+9$$

Substitute -\frac{2}{3}z+\frac{1}{3} for y in the equation z=-5y+9.

$$z=-5\left(-\frac{2}{3}z+\frac{1}{3}\right)+9$$

Solve z=-5\left(-\frac{2}{3}z+\frac{1}{3}\right)+9 for z.

$$z=-\frac{22}{7}$$

Substitute -\frac{22}{7} for z in the equation y=-\frac{2}{3}z+\frac{1}{3}.

$$y=-\frac{2}{3}\left(-\frac{22}{7}\right)+\frac{1}{3}$$

Calculate y from y=-\frac{2}{3}\left(-\frac{22}{7}\right)+\frac{1}{3}.

$$y=\frac{17}{7}$$

Substitute \frac{17}{7} for y and -\frac{22}{7} for z in the equation x=-y+z+2.

$$x=-\frac{17}{7}-\frac{22}{7}+2$$

Calculate x from x=-\frac{17}{7}-\frac{22}{7}+2.

$$x=-\frac{25}{7}$$

The system is now solved.

$$x=-\frac{25}{7} y=\frac{17}{7} z=-\frac{22}{7}$$