Step 1: Convert all decimals to fractions

$$\begin{array}{ cccc }3~ x&+~~4~ y&+~~2~ z&~=~1160\\2~ x&+~~4~ y&+~~~~ z&~=~800\\6~ x&+~~12~ y&+~~~~ z&~=~1920\end{array}$$

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

$$\begin{array}{ cccc }3~ x&+~~4~ y&+~~2~ z&~=~1160\\&\dfrac{ 4 }{ 3 }~ y&-~~~\dfrac{ 1 }{ 3 }~ z&~=~\dfrac{ 80 }{ 3 }\\6~ x&+~~12~ y&+~~~~ z&~=~1920\end{array}$$

Step 3: Multiply the first equation by $ -2 $ and add the result to the third equation. The result is:

$$\begin{array}{ cccc }3~ x&+~~4~ y&+~~2~ z&~=~1160\\&\dfrac{ 4 }{ 3 }~ y&-~~~\dfrac{ 1 }{ 3 }~ z&~=~\dfrac{ 80 }{ 3 }\\&4~ y&-~~~3~ z&~=~-400\end{array}$$

Step 4: Get rid of the fractions by multiplying the second equation by 3. The result is:

$$\begin{array}{ cccc }3~ x&+~~4~ y&+~~2~ z&~=~1160\\&4~ y&-~~~~~ z&~=~80\\&4~ y&-~~~3~ z&~=~-400\end{array}$$

Step 5: Multiply the second equation by $ -1 $ and add the result to the third equation. The result is:

$$\begin{array}{ cccc }3~ x&+~~4~ y&+~~2~ z&~=~1160\\&4~ y&-~~~~~ z&~=~80\\&&-~~~2~ z&~=~-480\end{array}$$

Step 6: Divide the third equation by $ 2 $. The result is:

$$\begin{array}{ cccc }3~ x&+~~4~ y&+~~2~ z&~=~1160\\&4~ y&-~~~~~ z&~=~80\\&&-~~~~~ z&~=~-240\end{array}$$

Step 7: Solve for $ z $.

$$ \begin{aligned} -1 ~ z & = -240 \\ z & = 240 \end{aligned} $$

Step 8: Solve for $ y $ using the second equation:

$$ \begin{aligned}4y-z &= 80\\4y-240 &= 80\\y &= 80 \end{aligned}$$

Step 9: Solve for $ x $ by substituting $ y = 80 $ and $ z = 240 $ into the first equation.

System 3x3 calculator