Step 1: Convert all decimals to fractions

$$\begin{array}{ cccc }2~ x&+~~\dfrac{ 3 }{ 2 }~ y&+~~3~ z&~=~\dfrac{ 29 }{ 2 }\\~~ x&+~~~~ y&+~~4~ z&~=~12\\\dfrac{ 5 }{ 2 }~ x&+~~3~ y&+~~~~ z&~=~15\end{array}$$

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

$$\begin{array}{ cccc }4~ x&+~~3~ y&+~~6~ z&~=~29\\~~ x&+~~~~ y&+~~4~ z&~=~12\\5~ x&+~~6~ y&+~~2~ z&~=~30\end{array}$$

Step 3: Swap Row 1 and Row 2. The result is:

$$\begin{array}{ cccc }~~ x&+~~~~ y&+~~4~ z&~=~12\\4~ x&+~~3~ y&+~~6~ z&~=~29\\5~ x&+~~6~ y&+~~2~ z&~=~30\end{array}$$

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

$$\begin{array}{ cccc }~~ x&+~~~~ y&+~~4~ z&~=~12\\&-~~~~~ y&-~~~10~ z&~=~-19\\5~ x&+~~6~ y&+~~2~ z&~=~30\end{array}$$

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

$$\begin{array}{ cccc }~~ x&+~~~~ y&+~~4~ z&~=~12\\&-~~~~~ y&-~~~10~ z&~=~-19\\&~~ y&-~~~18~ z&~=~-30\end{array}$$

Step 6: Multiply the second equation by $ 1 $ and add the result to the third equation. The result is:

$$\begin{array}{ cccc }~~ x&+~~~~ y&+~~4~ z&~=~12\\&-~~~~~ y&-~~~10~ z&~=~-19\\&&-~~~28~ z&~=~-49\end{array}$$

Step 7: Divide the third equation by $ 7 $. The result is:

$$\begin{array}{ cccc }~~ x&+~~~~ y&+~~4~ z&~=~12\\&-~~~~~ y&-~~~10~ z&~=~-19\\&&-~~~4~ z&~=~-7\end{array}$$

Step 8: Solve for $ z $.

$$ \begin{aligned} -4 ~ z & = -7 \\ z & = \dfrac{ 7 }{ 4 } \end{aligned} $$

Step 9: Solve for $ y $ using the second equation:

$$ \begin{aligned}-y-10z &= -19\\-y-10\cdot \frac{ 7 }{ 4 } &= -19\\y &= \dfrac{ 3 }{ 2 } \end{aligned}$$

Step 10: Solve for $ x $ by substituting $ y = \dfrac{ 3 }{ 2 } $ and $ z = \dfrac{ 7 }{ 4 } $ into the first equation.

System 3x3 calculator