Solve {l}{75-x/15=x/30+x-y/7}{x-y/7+4=frac{y-z}{100}+frac{y-z}{29.4}}{frac{y-z}{100}+frac{y-z}{29.4}=4+frac{z}{8}}

$$\left\{ \begin{array} { l } { \frac { 75 - x } { 15 } = \frac { x } { 30 } + \frac { x - y } { 7 } } \\ { \frac { x - y } { 7 } + 4 = \frac { y - z } { 100 } + \frac { y - z } { 29.4 } } \\ { \frac { y - z } { 100 } + \frac { y - z } { 29.4 } = 4 + \frac { z } { 8 } } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{14466}{247} = 58\frac{140}{247} \approx 58.566801619$$

$$y = \frac{79736}{1235} = 64\frac{696}{1235} \approx 64.563562753$$

$$z = -\frac{8464}{1235} = -6\frac{1054}{1235} \approx -6.853441296$$

Short Steps Using Substitution

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

$$1050-14x=37x-30y 2800+100x-100y=\frac{647}{21}y-\frac{647}{21}z \frac{1294}{147}y-\frac{1294}{147}z=800+25z$$

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 2800+100x-100y=\frac{647}{21}y-\frac{647}{21}z.

$$2800+100\left(\frac{350}{17}+\frac{10}{17}y\right)-100y=\frac{647}{21}y-\frac{647}{21}z$$

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

$$y=\frac{1734600}{25699}+\frac{10999}{25699}z z=-\frac{117600}{4969}+\frac{1294}{4969}y$$

Substitute \frac{1734600}{25699}+\frac{10999}{25699}z for y in the equation z=-\frac{117600}{4969}+\frac{1294}{4969}y.

$$z=-\frac{117600}{4969}+\frac{1294}{4969}\left(\frac{1734600}{25699}+\frac{10999}{25699}z\right)$$

Solve z=-\frac{117600}{4969}+\frac{1294}{4969}\left(\frac{1734600}{25699}+\frac{10999}{25699}z\right) for z.

$$z=-\frac{8464}{1235}$$

Substitute -\frac{8464}{1235} for z in the equation y=\frac{1734600}{25699}+\frac{10999}{25699}z.

$$y=\frac{1734600}{25699}+\frac{10999}{25699}\left(-\frac{8464}{1235}\right)$$

Calculate y from y=\frac{1734600}{25699}+\frac{10999}{25699}\left(-\frac{8464}{1235}\right).

$$y=\frac{79736}{1235}$$

Substitute \frac{79736}{1235} for y in the equation x=\frac{350}{17}+\frac{10}{17}y.

$$x=\frac{350}{17}+\frac{10}{17}\times \frac{79736}{1235}$$

Calculate x from x=\frac{350}{17}+\frac{10}{17}\times \frac{79736}{1235}.

$$x=\frac{14466}{247}$$

The system is now solved.

$$x=\frac{14466}{247} y=\frac{79736}{1235} z=-\frac{8464}{1235}$$