Solve {l}{-2x+3y+z=4}{-4x+y+z=3}{5x-5y+z=2}

$$\left. \begin{array} { l } { - 2 x + 3 y + z = 4 } \\ { - 4 x + y + z = 3 } \\ { 5 x - 5 y + z = 2 } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{2}{15}\approx 0.133333333$$

$$y=\frac{11}{30}\approx 0.366666667$$

$$z = \frac{19}{6} = 3\frac{1}{6} \approx 3.166666667$$

Short Steps Using Substitution

Solve -2x+3y+z=4 for z.

$$z=2x-3y+4$$

Substitute 2x-3y+4 for z in the second and third equation.

$$-4x+y+2x-3y+4=3 5x-5y+2x-3y+4=2$$

Solve these equations for y and x respectively.

$$y=-x+\frac{1}{2} x=-\frac{2}{7}+\frac{8}{7}y$$

Substitute -x+\frac{1}{2} for y in the equation x=-\frac{2}{7}+\frac{8}{7}y.

$$x=-\frac{2}{7}+\frac{8}{7}\left(-x+\frac{1}{2}\right)$$

Solve x=-\frac{2}{7}+\frac{8}{7}\left(-x+\frac{1}{2}\right) for x.

$$x=\frac{2}{15}$$

Substitute \frac{2}{15} for x in the equation y=-x+\frac{1}{2}.

$$y=-\frac{2}{15}+\frac{1}{2}$$

Calculate y from y=-\frac{2}{15}+\frac{1}{2}.

$$y=\frac{11}{30}$$

Substitute \frac{11}{30} for y and \frac{2}{15} for x in the equation z=2x-3y+4.

$$z=2\times \frac{2}{15}-3\times \frac{11}{30}+4$$

Calculate z from z=2\times \frac{2}{15}-3\times \frac{11}{30}+4.

$$z=\frac{19}{6}$$

The system is now solved.

$$x=\frac{2}{15} y=\frac{11}{30} z=\frac{19}{6}$$