Solve {c}{2x+3y+z=11}{x-2y+3z=6}{-x+2y+z=-2}

$$\left.\begin{array} { c } { 2 x + 3 y + z = 11 } \\ { x - 2 y + 3 z = 6 } \\ { - x + 2 y + z = - 2 } \end{array} \right\}$$

Solve for x, y, z

$$x = \frac{29}{7} = 4\frac{1}{7} \approx 4.142857143$$

$$y=\frac{4}{7}\approx 0.571428571$$

$$z=1$$

Short Steps Using Substitution

Solve 2x+3y+z=11 for z.

$$z=-2x-3y+11$$

Substitute -2x-3y+11 for z in the second and third equation.

$$x-2y+3\left(-2x-3y+11\right)=6 -x+2y-2x-3y+11=-2$$

Solve these equations for y and x respectively.

$$y=-\frac{5}{11}x+\frac{27}{11} x=-\frac{1}{3}y+\frac{13}{3}$$

Substitute -\frac{5}{11}x+\frac{27}{11} for y in the equation x=-\frac{1}{3}y+\frac{13}{3}.

$$x=-\frac{1}{3}\left(-\frac{5}{11}x+\frac{27}{11}\right)+\frac{13}{3}$$

Solve x=-\frac{1}{3}\left(-\frac{5}{11}x+\frac{27}{11}\right)+\frac{13}{3} for x.

$$x=\frac{29}{7}$$

Substitute \frac{29}{7} for x in the equation y=-\frac{5}{11}x+\frac{27}{11}.

$$y=-\frac{5}{11}\times \frac{29}{7}+\frac{27}{11}$$

Calculate y from y=-\frac{5}{11}\times \frac{29}{7}+\frac{27}{11}.

$$y=\frac{4}{7}$$

Substitute \frac{4}{7} for y and \frac{29}{7} for x in the equation z=-2x-3y+11.

$$z=-2\times \frac{29}{7}-3\times \frac{4}{7}+11$$

Calculate z from z=-2\times \frac{29}{7}-3\times \frac{4}{7}+11.

$$z=1$$

The system is now solved.

$$x=\frac{29}{7} y=\frac{4}{7} z=1$$