Step 1: Convert all decimals to fractions

$$\begin{array}{ cccc }3~ x&+~~2~ y&-~~~\dfrac{ 2 }{ 5 }~ z&~=~\dfrac{ 1 }{ 10 }\\\dfrac{ 37 }{ 10 }~ x&-~~~\dfrac{ 1 }{ 5 }~ y&+~~\dfrac{ 1 }{ 20 }~ z&~=~\dfrac{ 41 }{ 100 }\\-2~ x&+~~\dfrac{ 19 }{ 5 }~ y&-~~~\dfrac{ 21 }{ 10 }~ z&~=~-\dfrac{ 163 }{ 50 }\end{array}$$

Step 2: Get rid of the fractions by multiplying the first equation by 10, the second equation by 100 and the third equation by 50. The result is:

$$\begin{array}{ cccc }30~ x&+~~20~ y&-~~~4~ z&~=~1\\370~ x&-~~~20~ y&+~~5~ z&~=~41\\-100~ x&+~~190~ y&-~~~105~ z&~=~-163\end{array}$$

Step 3: Multiply the first equation by $ -\frac{ 37 }{ 3 } $ and add the result to the second equation. The result is:

$$\begin{array}{ cccc }30~ x&+~~20~ y&-~~~4~ z&~=~1\\&-~~~\dfrac{ 800 }{ 3 }~ y&+~~\dfrac{ 163 }{ 3 }~ z&~=~\dfrac{ 86 }{ 3 }\\-100~ x&+~~190~ y&-~~~105~ z&~=~-163\end{array}$$

Step 4: Multiply the first equation by $ \frac{ 10 }{ 3 } $ and add the result to the third equation. The result is:

$$\begin{array}{ cccc }30~ x&+~~20~ y&-~~~4~ z&~=~1\\&-~~~\dfrac{ 800 }{ 3 }~ y&+~~\dfrac{ 163 }{ 3 }~ z&~=~\dfrac{ 86 }{ 3 }\\&\dfrac{ 770 }{ 3 }~ y&-~~~\dfrac{ 355 }{ 3 }~ z&~=~-\dfrac{ 479 }{ 3 }\end{array}$$

Step 5: Get rid of the fractions by multiplying the second equation by 3 and the third equation by 3. The result is:

$$\begin{array}{ cccc }30~ x&+~~20~ y&-~~~4~ z&~=~1\\&-~~~800~ y&+~~163~ z&~=~86\\&770~ y&-~~~355~ z&~=~-479\end{array}$$

Step 6: Multiply the second equation by $ \frac{ 77 }{ 80 } $ and add the result to the third equation. The result is:

$$\begin{array}{ cccc }30~ x&+~~20~ y&-~~~4~ z&~=~1\\&-~~~800~ y&+~~163~ z&~=~86\\&&-~~~\dfrac{ 15849 }{ 80 }~ z&~=~-\dfrac{ 15849 }{ 40 }\end{array}$$

Step 7: Get rid of the fractions by multiplying the third equation by 80. The result is:

$$\begin{array}{ cccc }30~ x&+~~20~ y&-~~~4~ z&~=~1\\&-~~~800~ y&+~~163~ z&~=~86\\&&-~~~15849~ z&~=~-31698\end{array}$$

Step 8: Divide the third equation by $ 15849 $. The result is:

$$\begin{array}{ cccc }30~ x&+~~20~ y&-~~~4~ z&~=~1\\&-~~~800~ y&+~~163~ z&~=~86\\&&-~~~~~ z&~=~-2\end{array}$$

Step 9: Solve for $ z $.

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

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

$$ \begin{aligned}-800y+163z &= 86\\-800y+163\cdot 2 &= 86\\y &= \dfrac{ 3 }{ 10 } \end{aligned}$$

Step 11: Solve for $ x $ by substituting $ y = \dfrac{ 3 }{ 10 } $ and $ z = 2 $ into the first equation.

System 3x3 calculator