Step 1: Convert all decimals to fractions

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

Step 2: Divide the first equation by $ 20 $. The result is:

$$\begin{array}{ cccc }15~ x&+~~2~ y&+~~6~ z&~=~175\\200~ x&+~~60~ y&+~~150~ z&~=~660\\200~ x&+~~40~ y&+~~100~ z&~=~1790\end{array}$$

Step 3: Divide the second equation by $ 10 $. The result is:

$$\begin{array}{ cccc }15~ x&+~~2~ y&+~~6~ z&~=~175\\20~ x&+~~6~ y&+~~15~ z&~=~66\\200~ x&+~~40~ y&+~~100~ z&~=~1790\end{array}$$

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

$$\begin{array}{ cccc }15~ x&+~~2~ y&+~~6~ z&~=~175\\20~ x&+~~6~ y&+~~15~ z&~=~66\\20~ x&+~~4~ y&+~~10~ z&~=~179\end{array}$$

Step 5: Multiply the first equation by $ -\frac{ 4 }{ 3 } $ and add the result to the second equation. The result is:

$$\begin{array}{ cccc }15~ x&+~~2~ y&+~~6~ z&~=~175\\&\dfrac{ 10 }{ 3 }~ y&+~~7~ z&~=~-\dfrac{ 502 }{ 3 }\\20~ x&+~~4~ y&+~~10~ z&~=~179\end{array}$$

Step 6: Multiply the first equation by $ -\frac{ 4 }{ 3 } $ and add the result to the third equation. The result is:

$$\begin{array}{ cccc }15~ x&+~~2~ y&+~~6~ z&~=~175\\&\dfrac{ 10 }{ 3 }~ y&+~~7~ z&~=~-\dfrac{ 502 }{ 3 }\\&\dfrac{ 4 }{ 3 }~ y&+~~2~ z&~=~-\dfrac{ 163 }{ 3 }\end{array}$$

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

$$\begin{array}{ cccc }15~ x&+~~2~ y&+~~6~ z&~=~175\\&10~ y&+~~21~ z&~=~-502\\&4~ y&+~~6~ z&~=~-163\end{array}$$

Step 8: Multiply the second equation by $ -\frac{ 2 }{ 5 } $ and add the result to the third equation. The result is:

$$\begin{array}{ cccc }15~ x&+~~2~ y&+~~6~ z&~=~175\\&10~ y&+~~21~ z&~=~-502\\&&-~~~\dfrac{ 12 }{ 5 }~ z&~=~\dfrac{ 189 }{ 5 }\end{array}$$

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

$$\begin{array}{ cccc }15~ x&+~~2~ y&+~~6~ z&~=~175\\&10~ y&+~~21~ z&~=~-502\\&&-~~~12~ z&~=~189\end{array}$$

Step 10: Divide the third equation by $ 3 $. The result is:

$$\begin{array}{ cccc }15~ x&+~~2~ y&+~~6~ z&~=~175\\&10~ y&+~~21~ z&~=~-502\\&&-~~~4~ z&~=~63\end{array}$$

Step 11: Solve for $ z $.

$$ \begin{aligned} -4 ~ z & = 63 \\ z & = -\dfrac{ 63 }{ 4 } \end{aligned} $$

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

$$ \begin{aligned}10y+21z &= -502\\10y+21\cdot \left( -\frac{ 63 }{ 4 } \right) &= -502\\y &= -\dfrac{ 137 }{ 8 } \end{aligned}$$

Step 13: Solve for $ x $ by substituting $ y = -\dfrac{ 137 }{ 8 } $ and $ z = -\dfrac{ 63 }{ 4 } $ into the first equation.

System 3x3 calculator