Step 1: Convert all decimals to fractions

$$\begin{array}{ cccc }\dfrac{ 1 }{ 10 }~ x&+~~\dfrac{ 3 }{ 20 }~ y&+~~\dfrac{ 1 }{ 20 }~ z&~=~40\\\dfrac{ 1 }{ 10 }~ x&+~~\dfrac{ 1 }{ 10 }~ y&+~~\dfrac{ 1 }{ 5 }~ z&~=~50\\4~ x&+~~3~ y&+~~6~ z&~=~1800\end{array}$$

Step 2: Get rid of the fractions by multiplying the first equation by 20 and the second equation by 10. 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 3: 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 4: 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 5: 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 6: 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 7: 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 8: Solve for $ z $.

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

Step 9: Solve for $ y $ using the second equation:

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

Step 10: Solve for $ x $ by substituting $ y = 40 $ and $ z = 80 $ into the first equation.

System 3x3 calculator