Step 1: Convert all decimals to fractions

$$\begin{array}{ cccc }~~ x&+~~2~ y&-~~~~~ z&~=~2\\~~ x&-~~~~~ y&+~~2~ z&~=~5\\2~ x&+~~2~ y&+~~2~ z&~=~12\end{array}$$

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

$$\begin{array}{ cccc }~~ x&+~~2~ y&-~~~~~ z&~=~2\\~~ x&-~~~~~ y&+~~2~ z&~=~5\\~~ x&+~~~~ y&+~~~~ z&~=~6\end{array}$$

Step 3: Multiply the first equation by $ -1 $ and add the result to the second equation. The result is:

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

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

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

Step 5: Divide the second equation by $ 3 $. The result is:

$$\begin{array}{ cccc }~~ x&+~~2~ y&-~~~~~ z&~=~2\\&-~~~~~ y&+~~~~ z&~=~1\\&-~~~~~ y&+~~2~ z&~=~4\end{array}$$

Step 6: Multiply the second equation by $ -1 $ and add the result to the third equation. The result is:

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

Step 7: Solve for $ z $.

$$ \begin{aligned} 1 ~ z & = 3 \\ z & = 3 \end{aligned} $$

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

$$ \begin{aligned}-y+z &= 1\\-y+3 &= 1\\y &= 2 \end{aligned}$$

Step 9: Solve for $ x $ by substituting $ y = 2 $ and $ z = 3 $ into the first equation.

System 3x3 calculator