Step 1: Convert all decimals to fractions

$$\begin{array}{ cccc }3~ x&+~~2~ y&+~~~~ z&~=~0\\5~ x&+~~6~ y&+~~7~ z&~=~8\\-2~ x&+~~3~ y&+~~4~ z&~=~12\end{array}$$

Step 2: Multiply the first equation by $ -\frac{ 5 }{ 3 } $ and add the result to the second equation. The result is:

$$\begin{array}{ cccc }3~ x&+~~2~ y&+~~~~ z&~=~0\\&\dfrac{ 8 }{ 3 }~ y&+~~\dfrac{ 16 }{ 3 }~ z&~=~8\\-2~ x&+~~3~ y&+~~4~ z&~=~12\end{array}$$

Step 3: Multiply the first equation by $ \frac{ 2 }{ 3 } $ and add the result to the third equation. The result is:

$$\begin{array}{ cccc }3~ x&+~~2~ y&+~~~~ z&~=~0\\&\dfrac{ 8 }{ 3 }~ y&+~~\dfrac{ 16 }{ 3 }~ z&~=~8\\&\dfrac{ 13 }{ 3 }~ y&+~~\dfrac{ 14 }{ 3 }~ z&~=~12\end{array}$$

Step 4: Get rid of the fractions by multiplying the second equation by 3 and the third equation by 3. The result is:

$$\begin{array}{ cccc }3~ x&+~~2~ y&+~~~~ z&~=~0\\&8~ y&+~~16~ z&~=~24\\&13~ y&+~~14~ z&~=~36\end{array}$$

Step 5: Divide the second equation by $ 8 $. The result is:

$$\begin{array}{ cccc }3~ x&+~~2~ y&+~~~~ z&~=~0\\&~~ y&+~~2~ z&~=~3\\&13~ y&+~~14~ z&~=~36\end{array}$$

Step 6: Multiply the second equation by $ -13 $ and add the result to the third equation. The result is:

$$\begin{array}{ cccc }3~ x&+~~2~ y&+~~~~ z&~=~0\\&~~ y&+~~2~ z&~=~3\\&&-~~~12~ z&~=~-3\end{array}$$

Step 7: Solve for $ z $.

$$ \begin{aligned} -12 ~ z & = -3 \\ z & = \dfrac{ 1 }{ 4 } \end{aligned} $$

Step 8: Solve for $ y $ using the second equation:

$$ \begin{aligned}y+2z &= 3\\y+2\cdot \frac{ 1 }{ 4 } &= 3\\y &= \dfrac{ 5 }{ 2 } \end{aligned}$$

Step 9: Solve for $ x $ by substituting $ y = \dfrac{ 5 }{ 2 } $ and $ z = \dfrac{ 1 }{ 4 } $ into the first equation.

System 3x3 calculator