Step 1: Convert all decimals to fractions

$$\begin{array}{ cccc }~~ x&-~~~3~ y&+~~2~ z&~=~1\\-1~ x&+~~4~ y&-~~~4~ z&~=~1\\5~ x&-~~~~~ y&+~~6~ z&~=~9\end{array}$$

Step 2: Multiply the first equation by $ 1 $ and add the result to the second equation. The result is:

$$\begin{array}{ cccc }~~ x&-~~~3~ y&+~~2~ z&~=~1\\&~~ y&-~~~2~ z&~=~2\\5~ x&-~~~~~ y&+~~6~ z&~=~9\end{array}$$

Step 3: Multiply the first equation by $ -5 $ and add the result to the third equation. The result is:

$$\begin{array}{ cccc }~~ x&-~~~3~ y&+~~2~ z&~=~1\\&~~ y&-~~~2~ z&~=~2\\&14~ y&-~~~4~ z&~=~4\end{array}$$

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

$$\begin{array}{ cccc }~~ x&-~~~3~ y&+~~2~ z&~=~1\\&~~ y&-~~~2~ z&~=~2\\&7~ y&-~~~2~ z&~=~2\end{array}$$

Step 5: Multiply the second equation by $ -7 $ and add the result to the third equation. The result is:

$$\begin{array}{ cccc }~~ x&-~~~3~ y&+~~2~ z&~=~1\\&~~ y&-~~~2~ z&~=~2\\&&12~ z&~=~-12\end{array}$$

Step 6: Solve for $ z $.

$$ \begin{aligned} 12 ~ z & = -12 \\ z & = -1 \end{aligned} $$

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

$$ \begin{aligned}y-2z &= 2\\y-2\cdot \left( -1 \right) &= 2\\y &= 0 \end{aligned}$$

Step 8: Solve for $ x $ by substituting $ y = 0 $ and $ z = -1 $ into the first equation.

System 3x3 calculator