Step 1: Convert all decimals to fractions

$$\begin{array}{ cccc }2~ x&+~~2~ y&+~~4~ z&~=~82\\&4~ y&+~~6~ z&~=~119\\6~ x&+~~4~ y&+~~2~ z&~=~92\end{array}$$

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

$$\begin{array}{ cccc }~~ x&+~~~~ y&+~~2~ z&~=~41\\&4~ y&+~~6~ z&~=~119\\6~ x&+~~4~ y&+~~2~ z&~=~92\end{array}$$

Step 3: Divide the third equation by $ 2 $. The result is:

$$\begin{array}{ cccc }~~ x&+~~~~ y&+~~2~ z&~=~41\\&4~ y&+~~6~ z&~=~119\\3~ x&+~~2~ y&+~~~~ z&~=~46\end{array}$$

Step 4: Multiply the first equation by $ -3 $ and add the result to the third equation. The result is:

$$\begin{array}{ cccc }~~ x&+~~~~ y&+~~2~ z&~=~41\\&4~ y&+~~6~ z&~=~119\\&-~~~~~ y&-~~~5~ z&~=~-77\end{array}$$

Step 5: Multiply the second equation by $ \frac{ 1 }{ 4 } $ and add the result to the third equation. The result is:

$$\begin{array}{ cccc }~~ x&+~~~~ y&+~~2~ z&~=~41\\&4~ y&+~~6~ z&~=~119\\&&-~~~\dfrac{ 7 }{ 2 }~ z&~=~-\dfrac{ 189 }{ 4 }\end{array}$$

Step 6: Get rid of the fractions by multiplying the third equation by 4. The result is:

$$\begin{array}{ cccc }~~ x&+~~~~ y&+~~2~ z&~=~41\\&4~ y&+~~6~ z&~=~119\\&&-~~~14~ z&~=~-189\end{array}$$

Step 7: Divide the third equation by $ 7 $. The result is:

$$\begin{array}{ cccc }~~ x&+~~~~ y&+~~2~ z&~=~41\\&4~ y&+~~6~ z&~=~119\\&&-~~~2~ z&~=~-27\end{array}$$

Step 8: Solve for $ z $.

$$ \begin{aligned} -2 ~ z & = -27 \\ z & = \dfrac{ 27 }{ 2 } \end{aligned} $$

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

$$ \begin{aligned}4y+6z &= 119\\4y+6\cdot \frac{ 27 }{ 2 } &= 119\\y &= \dfrac{ 19 }{ 2 } \end{aligned}$$

Step 10: Solve for $ x $ by substituting $ y = \dfrac{ 19 }{ 2 } $ and $ z = \dfrac{ 27 }{ 2 } $ into the first equation.

System 3x3 calculator