Step 1: Convert all decimals to fractions

$$\begin{array}{ cccc }3~ x&+~~~~ y&-~~~~~ z&~=~15\\~~ x&-~~~~~ y&+~~3~ z&~=~-19\\2~ x&+~~2~ y&+~~~~ z&~=~4\end{array}$$

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

$$\begin{array}{ cccc }~~ x&-~~~~~ y&+~~3~ z&~=~-19\\3~ x&+~~~~ y&-~~~~~ z&~=~15\\2~ x&+~~2~ y&+~~~~ z&~=~4\end{array}$$

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

$$\begin{array}{ cccc }~~ x&-~~~~~ y&+~~3~ z&~=~-19\\&4~ y&-~~~10~ z&~=~72\\2~ x&+~~2~ y&+~~~~ z&~=~4\end{array}$$

Step 4: Multiply the first equation by $ -2 $ and add the result to the third equation. The result is:

$$\begin{array}{ cccc }~~ x&-~~~~~ y&+~~3~ z&~=~-19\\&4~ y&-~~~10~ z&~=~72\\&4~ y&-~~~5~ z&~=~42\end{array}$$

Step 5: Divide the second equation by $ 2 $. The result is:

$$\begin{array}{ cccc }~~ x&-~~~~~ y&+~~3~ z&~=~-19\\&2~ y&-~~~5~ z&~=~36\\&4~ y&-~~~5~ z&~=~42\end{array}$$

Step 6: Multiply the second equation by $ -2 $ and add the result to the third equation. The result is:

$$\begin{array}{ cccc }~~ x&-~~~~~ y&+~~3~ z&~=~-19\\&2~ y&-~~~5~ z&~=~36\\&&5~ z&~=~-30\end{array}$$

Step 7: Divide the third equation by $ 5 $. The result is:

$$\begin{array}{ cccc }~~ x&-~~~~~ y&+~~3~ z&~=~-19\\&2~ y&-~~~5~ z&~=~36\\&&~~ z&~=~-6\end{array}$$

Step 8: Solve for $ z $.

$$ \begin{aligned} 1 ~ z & = -6 \\ z & = -6 \end{aligned} $$

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

$$ \begin{aligned}2y-5z &= 36\\2y-5\cdot \left( -6 \right) &= 36\\y &= 3 \end{aligned}$$

Step 10: Solve for $ x $ by substituting $ y = 3 $ and $ z = -6 $ into the first equation.

System 3x3 calculator