Step 1: Convert all decimals to fractions

$$\begin{array}{ cccc }~~ x&+~~2~ y&+~~3~ z&~=~10\\2~ x&+~~6~ y&+~~12~ z&~=~4\\4~ x&-~~~8~ y&+~~4~ z&~=~8\end{array}$$

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

$$\begin{array}{ cccc }~~ x&+~~2~ y&+~~3~ z&~=~10\\~~ x&+~~3~ y&+~~6~ z&~=~2\\4~ x&-~~~8~ y&+~~4~ z&~=~8\end{array}$$

Step 3: Divide the third equation by $ 4 $. The result is:

$$\begin{array}{ cccc }~~ x&+~~2~ y&+~~3~ z&~=~10\\~~ x&+~~3~ y&+~~6~ z&~=~2\\~~ x&-~~~2~ y&+~~~~ z&~=~2\end{array}$$

Step 4: Multiply the first equation by $ -1 $ and add the result to the second equation. The result is:

$$\begin{array}{ cccc }~~ x&+~~2~ y&+~~3~ z&~=~10\\&~~ y&+~~3~ z&~=~-8\\~~ x&-~~~2~ y&+~~~~ z&~=~2\end{array}$$

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

$$\begin{array}{ cccc }~~ x&+~~2~ y&+~~3~ z&~=~10\\&~~ y&+~~3~ z&~=~-8\\&-~~~4~ y&-~~~2~ z&~=~-8\end{array}$$

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

$$\begin{array}{ cccc }~~ x&+~~2~ y&+~~3~ z&~=~10\\&~~ y&+~~3~ z&~=~-8\\&-~~~2~ y&-~~~~~ z&~=~-4\end{array}$$

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

$$\begin{array}{ cccc }~~ x&+~~2~ y&+~~3~ z&~=~10\\&~~ y&+~~3~ z&~=~-8\\&&5~ z&~=~-20\end{array}$$

Step 8: Solve for $ z $.

$$ \begin{aligned} 5 ~ z & = -20 \\ z & = -4 \end{aligned} $$

Step 9: Solve for $ y $ using the second equation:

$$ \begin{aligned}y+3z &= -8\\y+3\cdot \left( -4 \right) &= -8\\y &= 4 \end{aligned}$$

Step 10: Solve for $ x $ by substituting $ y = 4 $ and $ z = -4 $ into the first equation.

System 3x3 calculator