Step 1: Convert all decimals to fractions

$$\begin{array}{ cccc }5~ x&+~~7~ y&+~~8~ z&~=~493\\4~ x&+~~10~ y&+~~4~ z&~=~484\\~~ x&+~~7~ y&+~~5~ z&~=~374\end{array}$$

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

$$\begin{array}{ cccc }5~ x&+~~7~ y&+~~8~ z&~=~493\\2~ x&+~~5~ y&+~~2~ z&~=~242\\~~ x&+~~7~ y&+~~5~ z&~=~374\end{array}$$

Step 3: Swap Row 1 and Row 3. The result is:

$$\begin{array}{ cccc }~~ x&+~~7~ y&+~~5~ z&~=~374\\2~ x&+~~5~ y&+~~2~ z&~=~242\\5~ x&+~~7~ y&+~~8~ z&~=~493\end{array}$$

Step 4: Multiply the first equation by $ -2 $ and add the result to the second equation. The result is:

$$\begin{array}{ cccc }~~ x&+~~7~ y&+~~5~ z&~=~374\\&-~~~9~ y&-~~~8~ z&~=~-506\\5~ x&+~~7~ y&+~~8~ z&~=~493\end{array}$$

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

$$\begin{array}{ cccc }~~ x&+~~7~ y&+~~5~ z&~=~374\\&-~~~9~ y&-~~~8~ z&~=~-506\\&-~~~28~ y&-~~~17~ z&~=~-1377\end{array}$$

Step 6: Multiply the second equation by $ -\frac{ 28 }{ 9 } $ and add the result to the third equation. The result is:

$$\begin{array}{ cccc }~~ x&+~~7~ y&+~~5~ z&~=~374\\&-~~~9~ y&-~~~8~ z&~=~-506\\&&\dfrac{ 71 }{ 9 }~ z&~=~\dfrac{ 1775 }{ 9 }\end{array}$$

Step 7: Get rid of the fractions by multiplying the third equation by 9. The result is:

$$\begin{array}{ cccc }~~ x&+~~7~ y&+~~5~ z&~=~374\\&-~~~9~ y&-~~~8~ z&~=~-506\\&&71~ z&~=~1775\end{array}$$

Step 8: Divide the third equation by $ 71 $. The result is:

$$\begin{array}{ cccc }~~ x&+~~7~ y&+~~5~ z&~=~374\\&-~~~9~ y&-~~~8~ z&~=~-506\\&&~~ z&~=~25\end{array}$$

Step 9: Solve for $ z $.

$$ \begin{aligned} 1 ~ z & = 25 \\ z & = 25 \end{aligned} $$

Step 10: Solve for $ y $ using the second equation:

$$ \begin{aligned}-9y-8z &= -506\\-9y-8\cdot 25 &= -506\\y &= 34 \end{aligned}$$

Step 11: Solve for $ x $ by substituting $ y = 34 $ and $ z = 25 $ into the first equation.

System 3x3 calculator