Step 1: Convert all decimals to fractions

$$\begin{array}{ cccc }-3~ x&+~~~~ y&&~=~-8\\~~ x&-~~~12~ y&+~~3~ z&~=~0\\&3~ y&-~~~5~ z&~=~5\end{array}$$

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

$$\begin{array}{ cccc }~~ x&-~~~12~ y&+~~3~ z&~=~0\\-3~ x&+~~~~ y&&~=~-8\\&3~ y&-~~~5~ z&~=~5\end{array}$$

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

$$\begin{array}{ cccc }~~ x&-~~~12~ y&+~~3~ z&~=~0\\&-~~~35~ y&+~~9~ z&~=~-8\\&3~ y&-~~~5~ z&~=~5\end{array}$$

Step 4: Multiply the second equation by $ \frac{ 3 }{ 35 } $ and add the result to the third equation. The result is:

$$\begin{array}{ cccc }~~ x&-~~~12~ y&+~~3~ z&~=~0\\&-~~~35~ y&+~~9~ z&~=~-8\\&&-~~~\dfrac{ 148 }{ 35 }~ z&~=~\dfrac{ 151 }{ 35 }\end{array}$$

Step 5: Get rid of the fractions by multiplying the third equation by 35. The result is:

$$\begin{array}{ cccc }~~ x&-~~~12~ y&+~~3~ z&~=~0\\&-~~~35~ y&+~~9~ z&~=~-8\\&&-~~~148~ z&~=~151\end{array}$$

Step 6: Solve for $ z $.

$$ \begin{aligned} -148 ~ z & = 151 \\ z & = -\dfrac{ 151 }{ 148 } \end{aligned} $$

Step 7: Solve for $ y $ using the second equation:

$$ \begin{aligned}-35y+9z &= -8\\-35y+9\cdot \left( -\frac{ 151 }{ 148 } \right) &= -8\\y &= -\dfrac{ 5 }{ 148 } \end{aligned}$$

Step 8: Solve for $ x $ by substituting $ y = -\dfrac{ 5 }{ 148 } $ and $ z = -\dfrac{ 151 }{ 148 } $ into the first equation.

System 3x3 calculator