Step 1: Convert all decimals to fractions

$$\begin{array}{ cccc }4~ x&-~~~4~ y&+~~4~ z&~=~-4\\4~ x&+~~~~ y&-~~~2~ z&~=~5\\-3~ x&-~~~3~ y&-~~~4~ z&~=~-16\end{array}$$

Step 2: Divide the first equation by $ 4 $. The result is:

$$\begin{array}{ cccc }~~ x&-~~~~~ y&+~~~~ z&~=~-1\\4~ x&+~~~~ y&-~~~2~ z&~=~5\\-3~ x&-~~~3~ y&-~~~4~ z&~=~-16\end{array}$$

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

$$\begin{array}{ cccc }~~ x&-~~~~~ y&+~~~~ z&~=~-1\\&5~ y&-~~~6~ z&~=~9\\-3~ x&-~~~3~ y&-~~~4~ z&~=~-16\end{array}$$

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

$$\begin{array}{ cccc }~~ x&-~~~~~ y&+~~~~ z&~=~-1\\&5~ y&-~~~6~ z&~=~9\\&-~~~6~ y&-~~~~~ z&~=~-19\end{array}$$

Step 5: Multiply the second equation by $ \frac{ 6 }{ 5 } $ and add the result to the third equation. The result is:

$$\begin{array}{ cccc }~~ x&-~~~~~ y&+~~~~ z&~=~-1\\&5~ y&-~~~6~ z&~=~9\\&&-~~~\dfrac{ 41 }{ 5 }~ z&~=~-\dfrac{ 41 }{ 5 }\end{array}$$

Step 6: Get rid of the fractions by multiplying the third equation by 5. The result is:

$$\begin{array}{ cccc }~~ x&-~~~~~ y&+~~~~ z&~=~-1\\&5~ y&-~~~6~ z&~=~9\\&&-~~~41~ z&~=~-41\end{array}$$

Step 7: Divide the third equation by $ 41 $. The result is:

$$\begin{array}{ cccc }~~ x&-~~~~~ y&+~~~~ z&~=~-1\\&5~ y&-~~~6~ z&~=~9\\&&-~~~~~ z&~=~-1\end{array}$$

Step 8: Solve for $ z $.

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

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

$$ \begin{aligned}5y-6z &= 9\\5y-6\cdot 1 &= 9\\y &= 3 \end{aligned}$$

Step 10: Solve for $ x $ by substituting $ y = 3 $ and $ z = 1 $ into the first equation.

System 3x3 calculator