Step 1: Convert all decimals to fractions

$$\begin{array}{ cccc }16~ x&-~~~4~ y&+~~~~ z&~=~1\\81~ x&-~~~9~ y&+~~~~ z&~=~11\\~~ x&+~~~~ y&+~~~~ z&~=~11\end{array}$$

Step 2: Swap Row 1 and Row 3. The result is:

$$\begin{array}{ cccc }~~ x&+~~~~ y&+~~~~ z&~=~11\\81~ x&-~~~9~ y&+~~~~ z&~=~11\\16~ x&-~~~4~ y&+~~~~ z&~=~1\end{array}$$

Step 3: Multiply the first equation by $ -81 $ and add the result to the second equation. The result is:

$$\begin{array}{ cccc }~~ x&+~~~~ y&+~~~~ z&~=~11\\&-~~~90~ y&-~~~80~ z&~=~-880\\16~ x&-~~~4~ y&+~~~~ z&~=~1\end{array}$$

Step 4: Multiply the first equation by $ -16 $ and add the result to the third equation. The result is:

$$\begin{array}{ cccc }~~ x&+~~~~ y&+~~~~ z&~=~11\\&-~~~90~ y&-~~~80~ z&~=~-880\\&-~~~20~ y&-~~~15~ z&~=~-175\end{array}$$

Step 5: Divide the second equation by $ 10 $. The result is:

$$\begin{array}{ cccc }~~ x&+~~~~ y&+~~~~ z&~=~11\\&-~~~9~ y&-~~~8~ z&~=~-88\\&-~~~20~ y&-~~~15~ z&~=~-175\end{array}$$

Step 6: Divide the third equation by $ 5 $. The result is:

$$\begin{array}{ cccc }~~ x&+~~~~ y&+~~~~ z&~=~11\\&-~~~9~ y&-~~~8~ z&~=~-88\\&-~~~4~ y&-~~~3~ z&~=~-35\end{array}$$

Step 7: Multiply the second equation by $ -\frac{ 4 }{ 9 } $ and add the result to the third equation. The result is:

$$\begin{array}{ cccc }~~ x&+~~~~ y&+~~~~ z&~=~11\\&-~~~9~ y&-~~~8~ z&~=~-88\\&&\dfrac{ 5 }{ 9 }~ z&~=~\dfrac{ 37 }{ 9 }\end{array}$$

Step 8: Get rid of the fractions by multiplying the third equation by 9. The result is:

$$\begin{array}{ cccc }~~ x&+~~~~ y&+~~~~ z&~=~11\\&-~~~9~ y&-~~~8~ z&~=~-88\\&&5~ z&~=~37\end{array}$$

Step 9: Solve for $ z $.

$$ \begin{aligned} 5 ~ z & = 37 \\ z & = \dfrac{ 37 }{ 5 } \end{aligned} $$

Step 10: Solve for $ y $ using the second equation:

$$ \begin{aligned}-9y-8z &= -88\\-9y-8\cdot \frac{ 37 }{ 5 } &= -88\\y &= \dfrac{ 16 }{ 5 } \end{aligned}$$

Step 11: Solve for $ x $ by substituting $ y = \dfrac{ 16 }{ 5 } $ and $ z = \dfrac{ 37 }{ 5 } $ into the first equation.

System 3x3 calculator