Solve {c}{6+3x-2y-z=0}{19y-12x-3z=0}{-140+19z-5y-10x=0}

$$\left. \begin{array} { c } { 6 + 3 x - 2 y - z = 0 } \\ { 19 y - 12 x - 3 z = 0 } \\ { - 140 + 19 z - 5 y - 10 x = 0 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{89}{17} = 5\frac{4}{17} \approx 5.235294118$$

$$y = \frac{87}{17} = 5\frac{2}{17} \approx 5.117647059$$

$$z = \frac{195}{17} = 11\frac{8}{17} \approx 11.470588235$$

Short Steps Using Substitution

Solve 6+3x-2y-z=0 for z.

$$z=6+3x-2y$$

Substitute 6+3x-2y for z in the second and third equation.

$$19y-12x-3\left(6+3x-2y\right)=0 -140+19\left(6+3x-2y\right)-5y-10x=0$$

Solve these equations for y and x respectively.

$$y=\frac{18}{25}+\frac{21}{25}x x=\frac{26}{47}+\frac{43}{47}y$$

Substitute \frac{18}{25}+\frac{21}{25}x for y in the equation x=\frac{26}{47}+\frac{43}{47}y.

$$x=\frac{26}{47}+\frac{43}{47}\left(\frac{18}{25}+\frac{21}{25}x\right)$$

Solve x=\frac{26}{47}+\frac{43}{47}\left(\frac{18}{25}+\frac{21}{25}x\right) for x.

$$x=\frac{89}{17}$$

Substitute \frac{89}{17} for x in the equation y=\frac{18}{25}+\frac{21}{25}x.

$$y=\frac{18}{25}+\frac{21}{25}\times \frac{89}{17}$$

Calculate y from y=\frac{18}{25}+\frac{21}{25}\times \frac{89}{17}.

$$y=\frac{87}{17}$$

Substitute \frac{87}{17} for y and \frac{89}{17} for x in the equation z=6+3x-2y.

$$z=6+3\times \frac{89}{17}-2\times \frac{87}{17}$$

Calculate z from z=6+3\times \frac{89}{17}-2\times \frac{87}{17}.

$$z=\frac{195}{17}$$

The system is now solved.

$$x=\frac{89}{17} y=\frac{87}{17} z=\frac{195}{17}$$