Step 1: Convert all decimals to fractions

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

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

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

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

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

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

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

Step 5: Multiply the second equation by $ -\frac{ 1 }{ 2 } $ and add the result to the third equation. The result is:

$$\begin{array}{ cccc }~~ x&+~~3~ y&-~~~2~ z&~=~5\\&-~~~2~ y&+~~7~ z&~=~-2\\&&-~~~\dfrac{ 1 }{ 2 }~ z&~=~-1\end{array}$$

Step 6: Get rid of the fractions by multiplying the third equation by 2. The result is:

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

Step 7: Solve for $ z $.

$$ \begin{aligned} -1 ~ z & = -2 \\ z & = 2 \end{aligned} $$

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

$$ \begin{aligned}-2y+7z &= -2\\-2y+7\cdot 2 &= -2\\y &= 8 \end{aligned}$$

Step 9: Solve for $ x $ by substituting $ y = 8 $ and $ z = 2 $ into the first equation.

System 3x3 calculator