Solve {l}{x+y-z=6}{3x-2y+z=-5}{z+3y-2z=14}

$$\left. \begin{array} { l } { x + y - z = 6 } \\ { 3 x - 2 y + z = - 5 } \\ { z + 3 y - 2 z = 14 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{10}{7} = 1\frac{3}{7} \approx 1.428571429$$

$$y = \frac{33}{7} = 4\frac{5}{7} \approx 4.714285714$$

$$z=\frac{1}{7}\approx 0.142857143$$

Short Steps Using Substitution

Solve x+y-z=6 for x.

$$x=-y+z+6$$

Substitute -y+z+6 for x in the equation 3x-2y+z=-5.

$$3\left(-y+z+6\right)-2y+z=-5$$

Solve the second equation for y and the third equation for z.

$$y=\frac{4}{5}z+\frac{23}{5} z=3y-14$$

Substitute \frac{4}{5}z+\frac{23}{5} for y in the equation z=3y-14.

$$z=3\left(\frac{4}{5}z+\frac{23}{5}\right)-14$$

Solve z=3\left(\frac{4}{5}z+\frac{23}{5}\right)-14 for z.

$$z=\frac{1}{7}$$

Substitute \frac{1}{7} for z in the equation y=\frac{4}{5}z+\frac{23}{5}.

$$y=\frac{4}{5}\times \frac{1}{7}+\frac{23}{5}$$

Calculate y from y=\frac{4}{5}\times \frac{1}{7}+\frac{23}{5}.

$$y=\frac{33}{7}$$

Substitute \frac{33}{7} for y and \frac{1}{7} for z in the equation x=-y+z+6.

$$x=-\frac{33}{7}+\frac{1}{7}+6$$

Calculate x from x=-\frac{33}{7}+\frac{1}{7}+6.

$$x=\frac{10}{7}$$

The system is now solved.

$$x=\frac{10}{7} y=\frac{33}{7} z=\frac{1}{7}$$