Solve {l}{2500x-700y+-500z=13400}{2200x-700y+300z=-4000}{4000x+800y-5000z=-5000}

$$\left. \begin{array} { l } { 2500 x - 700 y + - 500 z = 13400 } \\ { 2200 x - 700 y + 300 z = - 4000 } \\ { 4000 x + 800 y - 5000 z = - 5000 } \end{array} \right.$$

Solve for x, y, z

$$x = -\frac{31042}{1335} = -23\frac{337}{1335} \approx -23.252434457$$

$$y = -\frac{21473}{267} = -80\frac{113}{267} \approx -80.423220974$$

$$z = -\frac{13559}{445} = -30\frac{209}{445} \approx -30.469662921$$

Short Steps Using Substitution

Solve 2500x-700y-500z=13400 for x.

$$x=\frac{134}{25}+\frac{7}{25}y+\frac{1}{5}z$$

Substitute \frac{134}{25}+\frac{7}{25}y+\frac{1}{5}z for x in the second and third equation.

$$2200\left(\frac{134}{25}+\frac{7}{25}y+\frac{1}{5}z\right)-700y+300z=-4000 4000\left(\frac{134}{25}+\frac{7}{25}y+\frac{1}{5}z\right)+800y-5000z=-5000$$

Solve these equations for y and z respectively.

$$y=188+\frac{185}{21}z z=\frac{661}{105}+\frac{16}{35}y$$

Substitute 188+\frac{185}{21}z for y in the equation z=\frac{661}{105}+\frac{16}{35}y.

$$z=\frac{661}{105}+\frac{16}{35}\left(188+\frac{185}{21}z\right)$$

Solve z=\frac{661}{105}+\frac{16}{35}\left(188+\frac{185}{21}z\right) for z.

$$z=-\frac{13559}{445}$$

Substitute -\frac{13559}{445} for z in the equation y=188+\frac{185}{21}z.

$$y=188+\frac{185}{21}\left(-\frac{13559}{445}\right)$$

Calculate y from y=188+\frac{185}{21}\left(-\frac{13559}{445}\right).

$$y=-\frac{21473}{267}$$

Substitute -\frac{21473}{267} for y and -\frac{13559}{445} for z in the equation x=\frac{134}{25}+\frac{7}{25}y+\frac{1}{5}z.

$$x=\frac{134}{25}+\frac{7}{25}\left(-\frac{21473}{267}\right)+\frac{1}{5}\left(-\frac{13559}{445}\right)$$

Calculate x from x=\frac{134}{25}+\frac{7}{25}\left(-\frac{21473}{267}\right)+\frac{1}{5}\left(-\frac{13559}{445}\right).

$$x=-\frac{31042}{1335}$$

The system is now solved.

$$x=-\frac{31042}{1335} y=-\frac{21473}{267} z=-\frac{13559}{445}$$