Step 1: Convert all decimals to fractions

$$\begin{array}{ cccc }150~ x&+~~\dfrac{ 475 }{ 2 }~ y&+~~400~ z&~=~37000\\80~ x&+~~140~ y&+~~240~ z&~=~21760\\160~ x&+~~240~ y&+~~400~ z&~=~37440\end{array}$$

Step 2: Get rid of the fractions by multiplying the first equation by 2. The result is:

$$\begin{array}{ cccc }300~ x&+~~475~ y&+~~800~ z&~=~74000\\80~ x&+~~140~ y&+~~240~ z&~=~21760\\160~ x&+~~240~ y&+~~400~ z&~=~37440\end{array}$$

Step 3: Divide the first equation by $ 25 $. The result is:

$$\begin{array}{ cccc }12~ x&+~~19~ y&+~~32~ z&~=~2960\\80~ x&+~~140~ y&+~~240~ z&~=~21760\\160~ x&+~~240~ y&+~~400~ z&~=~37440\end{array}$$

Step 4: Divide the second equation by $ 20 $. The result is:

$$\begin{array}{ cccc }12~ x&+~~19~ y&+~~32~ z&~=~2960\\4~ x&+~~7~ y&+~~12~ z&~=~1088\\160~ x&+~~240~ y&+~~400~ z&~=~37440\end{array}$$

Step 5: Divide the third equation by $ 80 $. The result is:

$$\begin{array}{ cccc }12~ x&+~~19~ y&+~~32~ z&~=~2960\\4~ x&+~~7~ y&+~~12~ z&~=~1088\\2~ x&+~~3~ y&+~~5~ z&~=~468\end{array}$$

Step 6: Swap Row 1 and Row 3. The result is:

$$\begin{array}{ cccc }2~ x&+~~3~ y&+~~5~ z&~=~468\\4~ x&+~~7~ y&+~~12~ z&~=~1088\\12~ x&+~~19~ y&+~~32~ z&~=~2960\end{array}$$

Step 7: Multiply the first equation by $ -2 $ and add the result to the second equation. The result is:

$$\begin{array}{ cccc }2~ x&+~~3~ y&+~~5~ z&~=~468\\&~~ y&+~~2~ z&~=~152\\12~ x&+~~19~ y&+~~32~ z&~=~2960\end{array}$$

Step 8: Multiply the first equation by $ -6 $ and add the result to the third equation. The result is:

$$\begin{array}{ cccc }2~ x&+~~3~ y&+~~5~ z&~=~468\\&~~ y&+~~2~ z&~=~152\\&~~ y&+~~2~ z&~=~152\end{array}$$

Step 9: Multiply the second equation by $ -1 $ and add the result to the third equation. The result is:

$$\begin{array}{ cccc }2~ x&+~~3~ y&+~~5~ z&~=~468\\&~~ y&+~~2~ z&~=~152\\&&0&~=~0\end{array}$$

Step 10: All the elements in the last row are zero, so the variable $ z $ can take any value, let

$$ \color{blue}{ z = t } $$

Step 11: Supstitute $ \color{blue}{ z = t } $ into second equation and solve for $ y $:

$$ \color{red}{y} = \frac{ - 2 z + 152 } { 1 } = \color{red}{ - 2 t + 152 } $$

Step 12: Supstitute $ y = - 2 t + 152 $ and $ z = t $ into first equation to find $ x $:

$$ \color{green}{x} = \frac{ 468 - 3 \cdot y - 5 \cdot z } { 2 } = \color{green}{ \frac{ 1 }{ 2 } t + 6 }$$

System 3x3 calculator