Step 1: Convert all decimals to fractions

$$\begin{array}{ cccc }-1~ x&-~~~5~ y&+~~~~ z&~=~17\\-5~ x&-~~~5~ y&+~~5~ z&~=~5\\2~ x&+~~5~ y&-~~~3~ z&~=~-10\end{array}$$

Step 2: Divide the second equation by $ 5 $. The result is:

$$\begin{array}{ cccc }-1~ x&-~~~5~ y&+~~~~ z&~=~17\\-1~ x&-~~~~~ y&+~~~~ z&~=~1\\2~ x&+~~5~ y&-~~~3~ z&~=~-10\end{array}$$

Step 3: Multiply the first equation by $ -1 $ and add the result to the second equation. The result is:

$$\begin{array}{ cccc }-1~ x&-~~~5~ y&+~~~~ z&~=~17\\&4~ y&&~=~-16\\2~ x&+~~5~ y&-~~~3~ z&~=~-10\end{array}$$

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

$$\begin{array}{ cccc }-1~ x&-~~~5~ y&+~~~~ z&~=~17\\&4~ y&&~=~-16\\&-~~~5~ y&-~~~~~ z&~=~24\end{array}$$

Step 5: Divide the second equation by $ 4 $. The result is:

$$\begin{array}{ cccc }-1~ x&-~~~5~ y&+~~~~ z&~=~17\\&~~ y&&~=~-4\\&-~~~5~ y&-~~~~~ z&~=~24\end{array}$$

Step 6: Swap Row 2 and Row 3. The result is:

$$\begin{array}{ cccc }-1~ x&-~~~5~ y&+~~~~ z&~=~17\\&-~~~5~ y&-~~~~~ z&~=~24\\&~~ y&&~=~-4\end{array}$$

Step 7: Solve for $ y $ using the third equation:

$$ \begin{aligned} 1 ~ y & = -4 \\ y & = -4 \end{aligned} $$

Step 8: Solve for $ z $ by substituting $ y = -4 $ into second equation :

$$ \begin{aligned}-5y-z &= 24\\-5\cdot \left( -4 \right)-z &= 24\\y &= -4 \end{aligned}$$

Step 9: Solve for $ x $ by substituting $ y = -4 $ and $ z = -4 $ into the first equation.

System 3x3 calculator