Solve {l}{75-x/15000=x/30000+x-Y/7000}{x-Y/7000=frac{Y-z}{100000}+frac{Y-z}{22600}-0.004}{frac{z}{8000}+0.004=frac{Y-z}{1000000}+frac{Y-z}{22600}}

$$\left. \begin{array} { l } { \frac { 75 - x } { 15000 } = \frac { x } { 30000 } + \frac { x - Y } { 7000 } } \\ { \frac { x - Y } { 7000 } = \frac { Y - z } { 100000 } + \frac { Y - z } { 22600 } - 0.004 } \\ { \frac { z } { 8000 } + 0.004 = \frac { Y - z } { 1000000 } + \frac { Y - z } { 22600 } } \end{array} \right.$$

Solve for x, Y, z

$$x = \frac{2339210}{43019} = 54\frac{16184}{43019} \approx 54.376205863$$

$$z = -\frac{354008}{43019} = -8\frac{9856}{43019} \approx -8.229108069$$

$$Y = \frac{2470992}{43019} = 57\frac{18909}{43019} \approx 57.439549966$$

Short Steps Using Substitution

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

$$1050-14x=37x-30Y 11300x-11300Y=-316400+4291Y-4291z 452000+14125z=5113Y-5113z$$

Solve 1050-14x=37x-30Y for x.

$$x=\frac{350}{17}+\frac{10}{17}Y$$

Substitute \frac{350}{17}+\frac{10}{17}Y for x in the equation 11300x-11300Y=-316400+4291Y-4291z.

$$11300\left(\frac{350}{17}+\frac{10}{17}Y\right)-11300Y=-316400+4291Y-4291z$$

Solve the second equation for Y and the third equation for z.

$$Y=\frac{1333400}{21721}+\frac{10421}{21721}z z=-\frac{226000}{9619}+\frac{5113}{19238}Y$$

Substitute \frac{1333400}{21721}+\frac{10421}{21721}z for Y in the equation z=-\frac{226000}{9619}+\frac{5113}{19238}Y.

$$z=-\frac{226000}{9619}+\frac{5113}{19238}\left(\frac{1333400}{21721}+\frac{10421}{21721}z\right)$$

Solve z=-\frac{226000}{9619}+\frac{5113}{19238}\left(\frac{1333400}{21721}+\frac{10421}{21721}z\right) for z.

$$z=-\frac{354008}{43019}$$

Substitute -\frac{354008}{43019} for z in the equation Y=\frac{1333400}{21721}+\frac{10421}{21721}z.

$$Y=\frac{1333400}{21721}+\frac{10421}{21721}\left(-\frac{354008}{43019}\right)$$

Calculate Y from Y=\frac{1333400}{21721}+\frac{10421}{21721}\left(-\frac{354008}{43019}\right).

$$Y=\frac{2470992}{43019}$$

Substitute \frac{2470992}{43019} for Y in the equation x=\frac{350}{17}+\frac{10}{17}Y.

$$x=\frac{350}{17}+\frac{10}{17}\times \frac{2470992}{43019}$$

Calculate x from x=\frac{350}{17}+\frac{10}{17}\times \frac{2470992}{43019}.

$$x=\frac{2339210}{43019}$$

The system is now solved.

$$x=\frac{2339210}{43019} Y=\frac{2470992}{43019} z=-\frac{354008}{43019}$$