Step 1: Convert all decimals to fractions

$$\begin{array}{ cccc }900~ x&+~~600~ y&+~~300~ z&~=~126900\\\dfrac{ 2 }{ 3 }~ x&+~~\dfrac{ 5 }{ 9 }~ y&+~~\dfrac{ 1 }{ 3 }~ z&~=~106\\100~ x&+~~70~ y&+~~30~ z&~=~14040\end{array}$$

Step 2: Get rid of the fractions by multiplying the second equation by 9. The result is:

$$\begin{array}{ cccc }900~ x&+~~600~ y&+~~300~ z&~=~126900\\6~ x&+~~5~ y&+~~3~ z&~=~954\\100~ x&+~~70~ y&+~~30~ z&~=~14040\end{array}$$

Step 3: Divide the first equation by $ 300 $. The result is:

$$\begin{array}{ cccc }3~ x&+~~2~ y&+~~~~ z&~=~423\\6~ x&+~~5~ y&+~~3~ z&~=~954\\100~ x&+~~70~ y&+~~30~ z&~=~14040\end{array}$$

Step 4: Divide the third equation by $ 10 $. The result is:

$$\begin{array}{ cccc }3~ x&+~~2~ y&+~~~~ z&~=~423\\6~ x&+~~5~ y&+~~3~ z&~=~954\\10~ x&+~~7~ y&+~~3~ z&~=~1404\end{array}$$

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

$$\begin{array}{ cccc }3~ x&+~~2~ y&+~~~~ z&~=~423\\&~~ y&+~~~~ z&~=~108\\10~ x&+~~7~ y&+~~3~ z&~=~1404\end{array}$$

Step 6: Multiply the first equation by $ -\frac{ 10 }{ 3 } $ and add the result to the third equation. The result is:

$$\begin{array}{ cccc }3~ x&+~~2~ y&+~~~~ z&~=~423\\&~~ y&+~~~~ z&~=~108\\&\dfrac{ 1 }{ 3 }~ y&-~~~\dfrac{ 1 }{ 3 }~ z&~=~-6\end{array}$$

Step 7: Get rid of the fractions by multiplying the third equation by 3. The result is:

$$\begin{array}{ cccc }3~ x&+~~2~ y&+~~~~ z&~=~423\\&~~ y&+~~~~ z&~=~108\\&~~ y&-~~~~~ z&~=~-18\end{array}$$

Step 8: Multiply the second equation by $ -1 $ and add the result to the third equation. The result is:

$$\begin{array}{ cccc }3~ x&+~~2~ y&+~~~~ z&~=~423\\&~~ y&+~~~~ z&~=~108\\&&-~~~2~ z&~=~-126\end{array}$$

Step 9: Divide the third equation by $ 2 $. The result is:

$$\begin{array}{ cccc }3~ x&+~~2~ y&+~~~~ z&~=~423\\&~~ y&+~~~~ z&~=~108\\&&-~~~~~ z&~=~-63\end{array}$$

Step 10: Solve for $ z $.

$$ \begin{aligned} -1 ~ z & = -63 \\ z & = 63 \end{aligned} $$

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

$$ \begin{aligned}y+z &= 108\\y+63 &= 108\\y &= 45 \end{aligned}$$

Step 12: Solve for $ x $ by substituting $ y = 45 $ and $ z = 63 $ into the first equation.

System 3x3 calculator