Solve {l}{2x-y/3+z+3x/4=6x-1/6-1/3}{frac{2y+y}{2}-frac{4x-2y}{5}=frac{4z+3y-2x}{2}+1}{frac{2y-1}{4}+frac{2z+5}{6}=frac{4x+5}{12}}

$$\left. \begin{array} { l } { \frac { 2 x - y } { 3 } + \frac { z + 3 x } { 4 } = \frac { 6 x - 1 } { 6 } - \frac { 1 } { 3 } } \\ { \frac { 2 y + y } { 2 } - \frac { 4 x - 2 y } { 5 } = \frac { 4 z + 3 y - 2 x } { 2 } + 1 } \\ { \frac { 2 y - 1 } { 4 } + \frac { 2 z + 5 } { 6 } = \frac { 4 x + 5 } { 12 } } \end{array} \right.$$

Solve for x, y, z

$$x = -\frac{9}{7} = -1\frac{2}{7} \approx -1.285714286$$

$$y=-\frac{81}{119}\approx -0.680672269$$

$$z=-\frac{13}{17}\approx -0.764705882$$

Short Steps Using Substitution

Multiply each equation by the least common multiple of denominators in it. Simplify.

$$17x-4y+3z=-6+12x 19y-8x=10+20z+15y-10x 7+6y+4z=4x+5$$

Solve 17x-4y+3z=-6+12x for x.

$$x=\frac{4}{5}y-\frac{3}{5}z-\frac{6}{5}$$

Substitute \frac{4}{5}y-\frac{3}{5}z-\frac{6}{5} for x in the second and third equation.

$$19y-8\left(\frac{4}{5}y-\frac{3}{5}z-\frac{6}{5}\right)=10+20z+15y-10\left(\frac{4}{5}y-\frac{3}{5}z-\frac{6}{5}\right) 7+6y+4z=4\left(\frac{4}{5}y-\frac{3}{5}z-\frac{6}{5}\right)+5$$

Solve these equations for y and z respectively.

$$y=\frac{31}{14}+\frac{53}{14}z z=-\frac{17}{16}-\frac{7}{16}y$$

Substitute \frac{31}{14}+\frac{53}{14}z for y in the equation z=-\frac{17}{16}-\frac{7}{16}y.

$$z=-\frac{17}{16}-\frac{7}{16}\left(\frac{31}{14}+\frac{53}{14}z\right)$$

Solve z=-\frac{17}{16}-\frac{7}{16}\left(\frac{31}{14}+\frac{53}{14}z\right) for z.

$$z=-\frac{13}{17}$$

Substitute -\frac{13}{17} for z in the equation y=\frac{31}{14}+\frac{53}{14}z.

$$y=\frac{31}{14}+\frac{53}{14}\left(-\frac{13}{17}\right)$$

Calculate y from y=\frac{31}{14}+\frac{53}{14}\left(-\frac{13}{17}\right).

$$y=-\frac{81}{119}$$

Substitute -\frac{81}{119} for y and -\frac{13}{17} for z in the equation x=\frac{4}{5}y-\frac{3}{5}z-\frac{6}{5}.

$$x=\frac{4}{5}\left(-\frac{81}{119}\right)-\frac{3}{5}\left(-\frac{13}{17}\right)-\frac{6}{5}$$

Calculate x from x=\frac{4}{5}\left(-\frac{81}{119}\right)-\frac{3}{5}\left(-\frac{13}{17}\right)-\frac{6}{5}.

$$x=-\frac{9}{7}$$

The system is now solved.

$$x=-\frac{9}{7} y=-\frac{81}{119} z=-\frac{13}{17}$$