Step 1: Convert all decimals to fractions

$$\begin{array}{ cccc }~~ x&+~~~~ y&-~~~3~ z&~=~1\\3~ x&+~~4~ y&-~~~~~ z&~=~7\\4~ x&-~~~5~ y&-~~~6~ z&~=~8\end{array}$$

Step 2: Multiply the first equation by $ -3 $ and add the result to the second equation. The result is:

$$\begin{array}{ cccc }~~ x&+~~~~ y&-~~~3~ z&~=~1\\&~~ y&+~~8~ z&~=~4\\4~ x&-~~~5~ y&-~~~6~ z&~=~8\end{array}$$

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

$$\begin{array}{ cccc }~~ x&+~~~~ y&-~~~3~ z&~=~1\\&~~ y&+~~8~ z&~=~4\\&-~~~9~ y&+~~6~ z&~=~4\end{array}$$

Step 4: Multiply the second equation by $ 9 $ and add the result to the third equation. The result is:

$$\begin{array}{ cccc }~~ x&+~~~~ y&-~~~3~ z&~=~1\\&~~ y&+~~8~ z&~=~4\\&&78~ z&~=~40\end{array}$$

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

$$\begin{array}{ cccc }~~ x&+~~~~ y&-~~~3~ z&~=~1\\&~~ y&+~~8~ z&~=~4\\&&39~ z&~=~20\end{array}$$

Step 6: Solve for $ z $.

$$ \begin{aligned} 39 ~ z & = 20 \\ z & = \dfrac{ 20 }{ 39 } \end{aligned} $$

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

$$ \begin{aligned}y+8z &= 4\\y+8\cdot \frac{ 20 }{ 39 } &= 4\\y &= -\dfrac{ 4 }{ 39 } \end{aligned}$$

Step 8: Solve for $ x $ by substituting $ y = -\dfrac{ 4 }{ 39 } $ and $ z = \dfrac{ 20 }{ 39 } $ into the first equation.

System 3x3 calculator