Solve {l}{3x+4y+6z=18000}{3x+5y-4z=6000}{2x-3y+z=5000}

$$\left. \begin{array} { l } { 3 x + 4 y + 6 z = 18000 } \\ { 3 x + 5 y - 4 z = 6000 } \\ { 2 x - 3 y + z = 5000 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{488000}{179} = 2726\frac{46}{179} \approx 2726.25698324$$

$$y = \frac{102000}{179} = 569\frac{149}{179} \approx 569.832402235$$

$$z = \frac{225000}{179} = 1256\frac{176}{179} \approx 1256.983240223$$

Short Steps Using Substitution

Reorder the equations.

$$2x-3y+z=5000 3x+5y-4z=6000 3x+4y+6z=18000$$

Solve 2x-3y+z=5000 for z.

$$z=-2x+3y+5000$$

Substitute -2x+3y+5000 for z in the second and third equation.

$$3x+5y-4\left(-2x+3y+5000\right)=6000 3x+4y+6\left(-2x+3y+5000\right)=18000$$

Solve these equations for y and x respectively.

$$y=\frac{11}{7}x-\frac{26000}{7} x=\frac{4000}{3}+\frac{22}{9}y$$

Substitute \frac{11}{7}x-\frac{26000}{7} for y in the equation x=\frac{4000}{3}+\frac{22}{9}y.

$$x=\frac{4000}{3}+\frac{22}{9}\left(\frac{11}{7}x-\frac{26000}{7}\right)$$

Solve x=\frac{4000}{3}+\frac{22}{9}\left(\frac{11}{7}x-\frac{26000}{7}\right) for x.

$$x=\frac{488000}{179}$$

Substitute \frac{488000}{179} for x in the equation y=\frac{11}{7}x-\frac{26000}{7}.

$$y=\frac{11}{7}\times \frac{488000}{179}-\frac{26000}{7}$$

Calculate y from y=\frac{11}{7}\times \frac{488000}{179}-\frac{26000}{7}.

$$y=\frac{102000}{179}$$

Substitute \frac{102000}{179} for y and \frac{488000}{179} for x in the equation z=-2x+3y+5000.

$$z=-2\times \frac{488000}{179}+3\times \frac{102000}{179}+5000$$

Calculate z from z=-2\times \frac{488000}{179}+3\times \frac{102000}{179}+5000.

$$z=\frac{225000}{179}$$

The system is now solved.

$$x=\frac{488000}{179} y=\frac{102000}{179} z=\frac{225000}{179}$$