Step 1: Convert all decimals to fractions

$$\begin{array}{ cccc }-3~ x&+~~~~ y&&~=~-10\\~~ x&-~~~9~ y&+~~3~ z&~=~0\\&3~ y&-~~~5~ z&~=~5\end{array}$$

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

$$\begin{array}{ cccc }~~ x&-~~~9~ y&+~~3~ z&~=~0\\-3~ x&+~~~~ y&&~=~-10\\&3~ y&-~~~5~ z&~=~5\end{array}$$

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

$$\begin{array}{ cccc }~~ x&-~~~9~ y&+~~3~ z&~=~0\\&-~~~26~ y&+~~9~ z&~=~-10\\&3~ y&-~~~5~ z&~=~5\end{array}$$

Step 4: Multiply the second equation by $ \frac{ 3 }{ 26 } $ and add the result to the third equation. The result is:

$$\begin{array}{ cccc }~~ x&-~~~9~ y&+~~3~ z&~=~0\\&-~~~26~ y&+~~9~ z&~=~-10\\&&-~~~\dfrac{ 103 }{ 26 }~ z&~=~\dfrac{ 50 }{ 13 }\end{array}$$

Step 5: Get rid of the fractions by multiplying the third equation by 26. The result is:

$$\begin{array}{ cccc }~~ x&-~~~9~ y&+~~3~ z&~=~0\\&-~~~26~ y&+~~9~ z&~=~-10\\&&-~~~103~ z&~=~100\end{array}$$

Step 6: Solve for $ z $.

$$ \begin{aligned} -103 ~ z & = 100 \\ z & = -\dfrac{ 100 }{ 103 } \end{aligned} $$

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

$$ \begin{aligned}-26y+9z &= -10\\-26y+9\cdot \left( -\frac{ 100 }{ 103 } \right) &= -10\\y &= \dfrac{ 5 }{ 103 } \end{aligned}$$

Step 8: Solve for $ x $ by substituting $ y = \dfrac{ 5 }{ 103 } $ and $ z = -\dfrac{ 100 }{ 103 } $ into the first equation.

System 3x3 calculator