Step 1: Convert all decimals to fractions

$$\begin{array}{ cccc }60~ x&+~~40~ y&+~~50~ z&~=~3900\\150~ x&+~~100~ y&+~~200~ z&~=~11250\\30~ x&+~~15~ y&+~~20~ z&~=~1675\end{array}$$

Step 2: Divide the first equation by $ 10 $. The result is:

$$\begin{array}{ cccc }6~ x&+~~4~ y&+~~5~ z&~=~390\\150~ x&+~~100~ y&+~~200~ z&~=~11250\\30~ x&+~~15~ y&+~~20~ z&~=~1675\end{array}$$

Step 3: Divide the second equation by $ 50 $. The result is:

$$\begin{array}{ cccc }6~ x&+~~4~ y&+~~5~ z&~=~390\\3~ x&+~~2~ y&+~~4~ z&~=~225\\30~ x&+~~15~ y&+~~20~ z&~=~1675\end{array}$$

Step 4: Divide the third equation by $ 5 $. The result is:

$$\begin{array}{ cccc }6~ x&+~~4~ y&+~~5~ z&~=~390\\3~ x&+~~2~ y&+~~4~ z&~=~225\\6~ x&+~~3~ y&+~~4~ z&~=~335\end{array}$$

Step 5: Swap Row 1 and Row 2. The result is:

$$\begin{array}{ cccc }3~ x&+~~2~ y&+~~4~ z&~=~225\\6~ x&+~~4~ y&+~~5~ z&~=~390\\6~ x&+~~3~ y&+~~4~ z&~=~335\end{array}$$

Step 6: Multiply the first equation by $ -2 $ and add the result to the second equation. The result is:

$$\begin{array}{ cccc }3~ x&+~~2~ y&+~~4~ z&~=~225\\&&-~~~3~ z&~=~-60\\6~ x&+~~3~ y&+~~4~ z&~=~335\end{array}$$

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

$$\begin{array}{ cccc }3~ x&+~~2~ y&+~~4~ z&~=~225\\&&-~~~3~ z&~=~-60\\&-~~~~~ y&-~~~4~ z&~=~-115\end{array}$$

Step 8: Divide the second equation by $ 3 $. The result is:

$$\begin{array}{ cccc }3~ x&+~~2~ y&+~~4~ z&~=~225\\&&-~~~~~ z&~=~-20\\&-~~~~~ y&-~~~4~ z&~=~-115\end{array}$$

Step 9: Swap Row 2 and Row 3. The result is:

$$\begin{array}{ cccc }3~ x&+~~2~ y&+~~4~ z&~=~225\\&-~~~~~ y&-~~~4~ z&~=~-115\\&&-~~~~~ z&~=~-20\end{array}$$

Step 10: Solve for $ z $.

$$ \begin{aligned} -1 ~ z & = -20 \\ z & = 20 \end{aligned} $$

Step 11: Solve for $ y $ using the second equation:

$$ \begin{aligned}-y-4z &= -115\\-y-4\cdot 20 &= -115\\y &= 35 \end{aligned}$$

Step 12: Solve for $ x $ by substituting $ y = 35 $ and $ z = 20 $ into the first equation.

System 3x3 calculator