Step 1: Convert all decimals to fractions

$$\begin{array}{ cccc }300~ x&+~~200~ y&+~~200~ z&~=~3500\\40~ x&+~~60~ y&+~~40~ z&~=~660\\120~ x&+~~150~ y&+~~100~ z&~=~1790\end{array}$$

Step 2: Divide the first equation by $ 100 $. The result is:

$$\begin{array}{ cccc }3~ x&+~~2~ y&+~~2~ z&~=~35\\40~ x&+~~60~ y&+~~40~ z&~=~660\\120~ x&+~~150~ y&+~~100~ z&~=~1790\end{array}$$

Step 3: Divide the second equation by $ 20 $. The result is:

$$\begin{array}{ cccc }3~ x&+~~2~ y&+~~2~ z&~=~35\\2~ x&+~~3~ y&+~~2~ z&~=~33\\120~ x&+~~150~ y&+~~100~ z&~=~1790\end{array}$$

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

$$\begin{array}{ cccc }3~ x&+~~2~ y&+~~2~ z&~=~35\\2~ x&+~~3~ y&+~~2~ z&~=~33\\12~ x&+~~15~ y&+~~10~ z&~=~179\end{array}$$

Step 5: Multiply the first equation by $ -\frac{ 2 }{ 3 } $ and add the result to the second equation. The result is:

$$\begin{array}{ cccc }3~ x&+~~2~ y&+~~2~ z&~=~35\\&\dfrac{ 5 }{ 3 }~ y&+~~\dfrac{ 2 }{ 3 }~ z&~=~\dfrac{ 29 }{ 3 }\\12~ x&+~~15~ y&+~~10~ z&~=~179\end{array}$$

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

$$\begin{array}{ cccc }3~ x&+~~2~ y&+~~2~ z&~=~35\\&\dfrac{ 5 }{ 3 }~ y&+~~\dfrac{ 2 }{ 3 }~ z&~=~\dfrac{ 29 }{ 3 }\\&7~ y&+~~2~ z&~=~39\end{array}$$

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

$$\begin{array}{ cccc }3~ x&+~~2~ y&+~~2~ z&~=~35\\&5~ y&+~~2~ z&~=~29\\&7~ y&+~~2~ z&~=~39\end{array}$$

Step 8: Multiply the second equation by $ -\frac{ 7 }{ 5 } $ and add the result to the third equation. The result is:

$$\begin{array}{ cccc }3~ x&+~~2~ y&+~~2~ z&~=~35\\&5~ y&+~~2~ z&~=~29\\&&-~~~\dfrac{ 4 }{ 5 }~ z&~=~-\dfrac{ 8 }{ 5 }\end{array}$$

Step 9: Get rid of the fractions by multiplying the third equation by 5. The result is:

$$\begin{array}{ cccc }3~ x&+~~2~ y&+~~2~ z&~=~35\\&5~ y&+~~2~ z&~=~29\\&&-~~~4~ z&~=~-8\end{array}$$

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

$$\begin{array}{ cccc }3~ x&+~~2~ y&+~~2~ z&~=~35\\&5~ y&+~~2~ z&~=~29\\&&-~~~~~ z&~=~-2\end{array}$$

Step 11: Solve for $ z $.

$$ \begin{aligned} -1 ~ z & = -2 \\ z & = 2 \end{aligned} $$

Step 12: Solve for $ y $ using the second equation:

$$ \begin{aligned}5y+2z &= 29\\5y+2\cdot 2 &= 29\\y &= 5 \end{aligned}$$

Step 13: Solve for $ x $ by substituting $ y = 5 $ and $ z = 2 $ into the first equation.

System 3x3 calculator