Step 1: Convert all decimals to fractions

$$\begin{array}{ cccc }100~ x&+~~150~ y&+~~50~ z&~=~40000\\100~ x&+~~100~ y&+~~200~ z&~=~50000\\4~ x&+~~3~ y&+~~6~ z&~=~1800\end{array}$$

Step 2: Divide the first equation by $ 50 $. The result is:

$$\begin{array}{ cccc }2~ x&+~~3~ y&+~~~~ z&~=~800\\100~ x&+~~100~ y&+~~200~ z&~=~50000\\4~ x&+~~3~ y&+~~6~ z&~=~1800\end{array}$$

Step 3: Divide the second equation by $ 100 $. The result is:

$$\begin{array}{ cccc }2~ x&+~~3~ y&+~~~~ z&~=~800\\~~ x&+~~~~ y&+~~2~ z&~=~500\\4~ x&+~~3~ y&+~~6~ z&~=~1800\end{array}$$

Step 4: Swap Row 1 and Row 2. The result is:

$$\begin{array}{ cccc }~~ x&+~~~~ y&+~~2~ z&~=~500\\2~ x&+~~3~ y&+~~~~ z&~=~800\\4~ x&+~~3~ y&+~~6~ z&~=~1800\end{array}$$

Step 5: Multiply the first equation by $ -2 $ and add the result to the second equation. The result is:

$$\begin{array}{ cccc }~~ x&+~~~~ y&+~~2~ z&~=~500\\&~~ y&-~~~3~ z&~=~-200\\4~ x&+~~3~ y&+~~6~ z&~=~1800\end{array}$$

Step 6: Multiply the first equation by $ -4 $ and add the result to the third equation. The result is:

$$\begin{array}{ cccc }~~ x&+~~~~ y&+~~2~ z&~=~500\\&~~ y&-~~~3~ z&~=~-200\\&-~~~~~ y&-~~~2~ z&~=~-200\end{array}$$

Step 7: Multiply the second equation by $ 1 $ and add the result to the third equation. The result is:

$$\begin{array}{ cccc }~~ x&+~~~~ y&+~~2~ z&~=~500\\&~~ y&-~~~3~ z&~=~-200\\&&-~~~5~ z&~=~-400\end{array}$$

Step 8: Divide the third equation by $ 5 $. The result is:

$$\begin{array}{ cccc }~~ x&+~~~~ y&+~~2~ z&~=~500\\&~~ y&-~~~3~ z&~=~-200\\&&-~~~~~ z&~=~-80\end{array}$$

Step 9: Solve for $ z $.

$$ \begin{aligned} -1 ~ z & = -80 \\ z & = 80 \end{aligned} $$

Step 10: Solve for $ y $ using the second equation:

$$ \begin{aligned}y-3z &= -200\\y-3\cdot 80 &= -200\\y &= 40 \end{aligned}$$

Step 11: Solve for $ x $ by substituting $ y = 40 $ and $ z = 80 $ into the first equation.

System 3x3 calculator