Solve {l}{3x+5y-4z=6000}{2x-3y+1z=5000}{3x+4y+6z=13000}

$$\left. \begin{array} { l } { 3 x + 5 y - 4 z = 6000 } \\ { 2 x - 3 y + 1 z = 5000 } \\ { 3 x + 4 y + 6 z = 13000 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{453000}{179} = 2530\frac{130}{179} \approx 2530.726256983$$

$$y = \frac{47000}{179} = 262\frac{102}{179} \approx 262.569832402$$

$$z = \frac{130000}{179} = 726\frac{46}{179} \approx 726.25698324$$

Short Steps Using Substitution

Solve 3x+5y-4z=6000 for x.

$$x=-\frac{5}{3}y+\frac{4}{3}z+2000$$

Substitute -\frac{5}{3}y+\frac{4}{3}z+2000 for x in the second and third equation.

$$2\left(-\frac{5}{3}y+\frac{4}{3}z+2000\right)-3y+1z=5000 3\left(-\frac{5}{3}y+\frac{4}{3}z+2000\right)+4y+6z=13000$$

Solve these equations for y and z respectively.

$$y=-\frac{3000}{19}+\frac{11}{19}z z=700+\frac{1}{10}y$$

Substitute -\frac{3000}{19}+\frac{11}{19}z for y in the equation z=700+\frac{1}{10}y.

$$z=700+\frac{1}{10}\left(-\frac{3000}{19}+\frac{11}{19}z\right)$$

Solve z=700+\frac{1}{10}\left(-\frac{3000}{19}+\frac{11}{19}z\right) for z.

$$z=\frac{130000}{179}$$

Substitute \frac{130000}{179} for z in the equation y=-\frac{3000}{19}+\frac{11}{19}z.

$$y=-\frac{3000}{19}+\frac{11}{19}\times \frac{130000}{179}$$

Calculate y from y=-\frac{3000}{19}+\frac{11}{19}\times \frac{130000}{179}.

$$y=\frac{47000}{179}$$

Substitute \frac{47000}{179} for y and \frac{130000}{179} for z in the equation x=-\frac{5}{3}y+\frac{4}{3}z+2000.

$$x=-\frac{5}{3}\times \frac{47000}{179}+\frac{4}{3}\times \frac{130000}{179}+2000$$

Calculate x from x=-\frac{5}{3}\times \frac{47000}{179}+\frac{4}{3}\times \frac{130000}{179}+2000.

$$x=\frac{453000}{179}$$

The system is now solved.

$$x=\frac{453000}{179} y=\frac{47000}{179} z=\frac{130000}{179}$$