Solve {l}{439942436x+191602z+158233491006y=4025322}{5084814600y+15823349100x+165100z=3467100}{2z+165100y+191600x=42}

$$\left. \begin{array} { l } { 439942436 x + 191602 z + 158233491006 y = 4025322 } \\ { 5084814600 y + 15823349100 x + 165100 z = 3467100 } \\ { 2 z + 165100 y + 191600 x = 42 } \end{array} \right.$$

Solve for x, z, y

$$x=-\frac{14354239872}{152109659099256521}\approx -0.000000094$$

$$y=-\frac{874776}{11700743007635117}\approx -7.47624317 \cdot 10^{-11}$$

$$z = \frac{3195678916029988941}{152109659099256521} = 21\frac{1376074945602112}{1.5210965909925651 \times 10^{17}} \approx 21.009046598$$

Short Steps Using Substitution

Solve 439942436x+191602z+158233491006y=4025322 for x.

$$x=\frac{87507}{9563966}-\frac{95801}{219971218}z-\frac{79116745503}{219971218}y$$

Substitute \frac{87507}{9563966}-\frac{95801}{219971218}z-\frac{79116745503}{219971218}y for x in the second and third equation.

$$5084814600y+15823349100\left(\frac{87507}{9563966}-\frac{95801}{219971218}z-\frac{79116745503}{219971218}y\right)+165100z=3467100 2z+165100y+191600\left(\frac{87507}{9563966}-\frac{95801}{219971218}z-\frac{79116745503}{219971218}y\right)=42$$

Solve these equations for z and y respectively.

$$z=\frac{8185915101}{389638801}-\frac{543814509082155615}{643293660451}y y=\frac{94096764111}{3780612797570750}-\frac{4478882291}{3780612797570750}z$$

Substitute \frac{8185915101}{389638801}-\frac{543814509082155615}{643293660451}y for z in the equation y=\frac{94096764111}{3780612797570750}-\frac{4478882291}{3780612797570750}z.

$$y=\frac{94096764111}{3780612797570750}-\frac{4478882291}{3780612797570750}\left(\frac{8185915101}{389638801}-\frac{543814509082155615}{643293660451}y\right)$$

Solve y=\frac{94096764111}{3780612797570750}-\frac{4478882291}{3780612797570750}\left(\frac{8185915101}{389638801}-\frac{543814509082155615}{643293660451}y\right) for y.

$$y=-\frac{874776}{11700743007635117}$$

Substitute -\frac{874776}{11700743007635117} for y in the equation z=\frac{8185915101}{389638801}-\frac{543814509082155615}{643293660451}y.

$$z=\frac{8185915101}{389638801}-\frac{543814509082155615}{643293660451}\left(-\frac{874776}{11700743007635117}\right)$$

Calculate z from z=\frac{8185915101}{389638801}-\frac{543814509082155615}{643293660451}\left(-\frac{874776}{11700743007635117}\right).

$$z=\frac{3195678916029988941}{152109659099256521}$$

Substitute \frac{3195678916029988941}{152109659099256521} for z and -\frac{874776}{11700743007635117} for y in the equation x=\frac{87507}{9563966}-\frac{95801}{219971218}z-\frac{79116745503}{219971218}y.

$$x=\frac{87507}{9563966}-\frac{95801}{219971218}\times \frac{3195678916029988941}{152109659099256521}-\frac{79116745503}{219971218}\left(-\frac{874776}{11700743007635117}\right)$$

Calculate x from x=\frac{87507}{9563966}-\frac{95801}{219971218}\times \frac{3195678916029988941}{152109659099256521}-\frac{79116745503}{219971218}\left(-\frac{874776}{11700743007635117}\right).

$$x=-\frac{14354239872}{152109659099256521}$$

The system is now solved.

$$x=-\frac{14354239872}{152109659099256521} z=\frac{3195678916029988941}{152109659099256521} y=-\frac{874776}{11700743007635117}$$