Step 1: Convert all decimals to fractions

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

Step 2: Swap Row 1 and Row 2. The result is:

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

Step 3: Multiply the first equation by $ -5 $ and add the result to the second equation. The result is:

$$\begin{array}{ cccc }~~ x&&+~~2~ z&~=~4\\&2~ y&-~~~9~ z&~=~-16\\2~ x&+~~~~ y&-~~~~~ z&~=~-1\end{array}$$

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

$$\begin{array}{ cccc }~~ x&&+~~2~ z&~=~4\\&2~ y&-~~~9~ z&~=~-16\\&~~ y&-~~~5~ z&~=~-9\end{array}$$

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

$$\begin{array}{ cccc }~~ x&&+~~2~ z&~=~4\\&2~ y&-~~~9~ z&~=~-16\\&&-~~~\dfrac{ 1 }{ 2 }~ z&~=~-1\end{array}$$

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

$$\begin{array}{ cccc }~~ x&&+~~2~ z&~=~4\\&2~ y&-~~~9~ z&~=~-16\\&&-~~~~~ z&~=~-2\end{array}$$

Step 7: Solve for $ z $.

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

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

$$ \begin{aligned}2y-9z &= -16\\2y-9\cdot 2 &= -16\\y &= 1 \end{aligned}$$

Step 9: Solve for $ x $ by substituting $ y = 1 $ and $ z = 2 $ into the first equation.

System 3x3 calculator