Solve {l}{430000436x+191602z+158233491006y=4025322}{5084814600y+1580000100x+165100z=3467100}{50848z+165100y+191600x=42}

$$\left. \begin{array} { l } { 430000436 x + 191602 z + 158233491006 y = 4025322 } \\ { 5084814600 y + 1580000100 x + 165100 z = 3467100 } \\ { 50848 z + 165100 y + 191600 x = 42 } \end{array} \right.$$

Solve for x, z, y

$$x=\frac{67134938268772611321}{31491048460661376024950}\approx 0.002131874$$

$$y=\frac{618942473858907849}{31491048460661376024950}\approx 0.000019655$$

$$z=-\frac{9158761437526085763}{1259641938426455040998}\approx -0.007270925$$

Short Steps Using Substitution

Solve 430000436x+191602z+158233491006y=4025322 for x.

$$x=\frac{2012661}{215000218}-\frac{95801}{215000218}z-\frac{79116745503}{215000218}y$$

Substitute \frac{2012661}{215000218}-\frac{95801}{215000218}z-\frac{79116745503}{215000218}y for x in the second and third equation.

$$5084814600y+1580000100\left(\frac{2012661}{215000218}-\frac{95801}{215000218}z-\frac{79116745503}{215000218}y\right)+165100z=3467100 50848z+165100y+191600\left(\frac{2012661}{215000218}-\frac{95801}{215000218}z-\frac{79116745503}{215000218}y\right)=42$$

Solve these equations for z and y respectively.

$$z=\frac{24345773254383}{1158690535883}-\frac{1239112295589249675}{1158690535883}y y=\frac{94148959611}{3780817975595750}+\frac{1364246951658}{1890408987797875}z$$

Substitute \frac{24345773254383}{1158690535883}-\frac{1239112295589249675}{1158690535883}y for z in the equation y=\frac{94148959611}{3780817975595750}+\frac{1364246951658}{1890408987797875}z.

$$y=\frac{94148959611}{3780817975595750}+\frac{1364246951658}{1890408987797875}\left(\frac{24345773254383}{1158690535883}-\frac{1239112295589249675}{1158690535883}y\right)$$

Solve y=\frac{94148959611}{3780817975595750}+\frac{1364246951658}{1890408987797875}\left(\frac{24345773254383}{1158690535883}-\frac{1239112295589249675}{1158690535883}y\right) for y.

$$y=\frac{618942473858907849}{31491048460661376024950}$$

Substitute \frac{618942473858907849}{31491048460661376024950} for y in the equation z=\frac{24345773254383}{1158690535883}-\frac{1239112295589249675}{1158690535883}y.

$$z=\frac{24345773254383}{1158690535883}-\frac{1239112295589249675}{1158690535883}\times \frac{618942473858907849}{31491048460661376024950}$$

Calculate z from z=\frac{24345773254383}{1158690535883}-\frac{1239112295589249675}{1158690535883}\times \frac{618942473858907849}{31491048460661376024950}.

$$z=-\frac{9158761437526085763}{1259641938426455040998}$$

Substitute -\frac{9158761437526085763}{1259641938426455040998} for z and \frac{618942473858907849}{31491048460661376024950} for y in the equation x=\frac{2012661}{215000218}-\frac{95801}{215000218}z-\frac{79116745503}{215000218}y.

$$x=\frac{2012661}{215000218}-\frac{95801}{215000218}\left(-\frac{9158761437526085763}{1259641938426455040998}\right)-\frac{79116745503}{215000218}\times \frac{618942473858907849}{31491048460661376024950}$$

Calculate x from x=\frac{2012661}{215000218}-\frac{95801}{215000218}\left(-\frac{9158761437526085763}{1259641938426455040998}\right)-\frac{79116745503}{215000218}\times \frac{618942473858907849}{31491048460661376024950}.

$$x=\frac{67134938268772611321}{31491048460661376024950}$$

The system is now solved.

$$x=\frac{67134938268772611321}{31491048460661376024950} z=-\frac{9158761437526085763}{1259641938426455040998} y=\frac{618942473858907849}{31491048460661376024950}$$