Step 1: Convert all decimals to fractions

$$\begin{array}{ cccc }3~ x&-~~~\dfrac{ 2 }{ 5 }~ y&+~~9~ z&~=~\dfrac{ 417 }{ 250 }\\\dfrac{ 3 }{ 10 }~ x&+~~5~ y&-~~~8~ z&~=~-\dfrac{ 243 }{ 250 }\\5~ x&-~~~4~ y&-~~~8~ z&~=~\dfrac{ 9 }{ 5 }\end{array}$$

Step 2: Get rid of the fractions by multiplying the first equation by 250, the second equation by 250 and the third equation by 5. The result is:

$$\begin{array}{ cccc }750~ x&-~~~100~ y&+~~2250~ z&~=~417\\75~ x&+~~1250~ y&-~~~2000~ z&~=~-243\\25~ x&-~~~20~ y&-~~~40~ z&~=~9\end{array}$$

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

$$\begin{array}{ cccc }25~ x&-~~~20~ y&-~~~40~ z&~=~9\\75~ x&+~~1250~ y&-~~~2000~ z&~=~-243\\750~ x&-~~~100~ y&+~~2250~ z&~=~417\end{array}$$

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

$$\begin{array}{ cccc }25~ x&-~~~20~ y&-~~~40~ z&~=~9\\&1310~ y&-~~~1880~ z&~=~-270\\750~ x&-~~~100~ y&+~~2250~ z&~=~417\end{array}$$

Step 5: Multiply the first equation by $ -30 $ and add the result to the third equation. The result is:

$$\begin{array}{ cccc }25~ x&-~~~20~ y&-~~~40~ z&~=~9\\&1310~ y&-~~~1880~ z&~=~-270\\&500~ y&+~~3450~ z&~=~147\end{array}$$

Step 6: Divide the second equation by $ 10 $. The result is:

$$\begin{array}{ cccc }25~ x&-~~~20~ y&-~~~40~ z&~=~9\\&131~ y&-~~~188~ z&~=~-27\\&500~ y&+~~3450~ z&~=~147\end{array}$$

Step 7: Multiply the second equation by $ -\frac{ 500 }{ 131 } $ and add the result to the third equation. The result is:

$$\begin{array}{ cccc }25~ x&-~~~20~ y&-~~~40~ z&~=~9\\&131~ y&-~~~188~ z&~=~-27\\&&\dfrac{ 545950 }{ 131 }~ z&~=~\dfrac{ 32757 }{ 131 }\end{array}$$

Step 8: Get rid of the fractions by multiplying the third equation by 131. The result is:

$$\begin{array}{ cccc }25~ x&-~~~20~ y&-~~~40~ z&~=~9\\&131~ y&-~~~188~ z&~=~-27\\&&545950~ z&~=~32757\end{array}$$

Step 9: Divide the third equation by $ 10919 $. The result is:

$$\begin{array}{ cccc }25~ x&-~~~20~ y&-~~~40~ z&~=~9\\&131~ y&-~~~188~ z&~=~-27\\&&50~ z&~=~3\end{array}$$

Step 10: Solve for $ z $.

$$ \begin{aligned} 50 ~ z & = 3 \\ z & = \dfrac{ 3 }{ 50 } \end{aligned} $$

Step 11: Solve for $ y $ using the second equation:

$$ \begin{aligned}131y-188z &= -27\\131y-188\cdot \frac{ 3 }{ 50 } &= -27\\y &= -\dfrac{ 3 }{ 25 } \end{aligned}$$

Step 12: Solve for $ x $ by substituting $ y = -\dfrac{ 3 }{ 25 } $ and $ z = \dfrac{ 3 }{ 50 } $ into the first equation.

System 3x3 calculator