Solve {l}{2x-y+z=3}{x-2y-3z=2}{x+2y-z=12}

$$\left. \begin{array} { l } { 2 x - y + z = 3 } \\ { x - 2 y - 3 z = 2 } \\ { x + 2 y - z = 12 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{43}{11} = 3\frac{10}{11} \approx 3.909090909$$

$$y = \frac{36}{11} = 3\frac{3}{11} \approx 3.272727273$$

$$z = -\frac{17}{11} = -1\frac{6}{11} \approx -1.545454545$$

Short Steps Using Substitution

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

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

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

$$x-2\left(2x+z-3\right)-3z=2 x+2\left(2x+z-3\right)-z=12$$

Solve these equations for x and z respectively.

$$x=-\frac{5}{3}z+\frac{4}{3} z=-5x+18$$

Substitute -\frac{5}{3}z+\frac{4}{3} for x in the equation z=-5x+18.

$$z=-5\left(-\frac{5}{3}z+\frac{4}{3}\right)+18$$

Solve z=-5\left(-\frac{5}{3}z+\frac{4}{3}\right)+18 for z.

$$z=-\frac{17}{11}$$

Substitute -\frac{17}{11} for z in the equation x=-\frac{5}{3}z+\frac{4}{3}.

$$x=-\frac{5}{3}\left(-\frac{17}{11}\right)+\frac{4}{3}$$

Calculate x from x=-\frac{5}{3}\left(-\frac{17}{11}\right)+\frac{4}{3}.

$$x=\frac{43}{11}$$

Substitute \frac{43}{11} for x and -\frac{17}{11} for z in the equation y=2x+z-3.

$$y=2\times \frac{43}{11}-\frac{17}{11}-3$$

Calculate y from y=2\times \frac{43}{11}-\frac{17}{11}-3.

$$y=\frac{36}{11}$$

The system is now solved.

$$x=\frac{43}{11} y=\frac{36}{11} z=-\frac{17}{11}$$