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

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

Solve for x, y, z

$$x = \frac{55}{23} = 2\frac{9}{23} \approx 2.391304348$$

$$y=-\frac{13}{23}\approx -0.565217391$$

$$z = \frac{25}{23} = 1\frac{2}{23} \approx 1.086956522$$

Short Steps Using Substitution

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

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

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

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

Solve these equations for y and z respectively.

$$y=-\frac{11}{6}+\frac{7}{6}z z=\frac{6}{5}+\frac{1}{5}y$$

Substitute -\frac{11}{6}+\frac{7}{6}z for y in the equation z=\frac{6}{5}+\frac{1}{5}y.

$$z=\frac{6}{5}+\frac{1}{5}\left(-\frac{11}{6}+\frac{7}{6}z\right)$$

Solve z=\frac{6}{5}+\frac{1}{5}\left(-\frac{11}{6}+\frac{7}{6}z\right) for z.

$$z=\frac{25}{23}$$

Substitute \frac{25}{23} for z in the equation y=-\frac{11}{6}+\frac{7}{6}z.

$$y=-\frac{11}{6}+\frac{7}{6}\times \frac{25}{23}$$

Calculate y from y=-\frac{11}{6}+\frac{7}{6}\times \frac{25}{23}.

$$y=-\frac{13}{23}$$

Substitute -\frac{13}{23} for y and \frac{25}{23} for z in the equation x=-2y+3z-2.

$$x=-2\left(-\frac{13}{23}\right)+3\times \frac{25}{23}-2$$

Calculate x from x=-2\left(-\frac{13}{23}\right)+3\times \frac{25}{23}-2.

$$x=\frac{55}{23}$$

The system is now solved.

$$x=\frac{55}{23} y=-\frac{13}{23} z=\frac{25}{23}$$