Step 1: Convert all decimals to fractions

$$\begin{array}{ cccc }300~ x&+~~200~ y&+~~200~ z&~=~0\\40~ x&+~~60~ y&+~~40~ z&~=~0\\120~ x&+~~150~ y&+~~100~ z&~=~0\end{array}$$

Step 2: Divide the first equation by $ 100 $. The result is:

$$\begin{array}{ cccc }3~ x&+~~2~ y&+~~2~ z&~=~0\\40~ x&+~~60~ y&+~~40~ z&~=~0\\120~ x&+~~150~ y&+~~100~ z&~=~0\end{array}$$

Step 3: Divide the second equation by $ 20 $. The result is:

$$\begin{array}{ cccc }3~ x&+~~2~ y&+~~2~ z&~=~0\\2~ x&+~~3~ y&+~~2~ z&~=~0\\120~ x&+~~150~ y&+~~100~ z&~=~0\end{array}$$

Step 4: Divide the third equation by $ 10 $. The result is:

$$\begin{array}{ cccc }3~ x&+~~2~ y&+~~2~ z&~=~0\\2~ x&+~~3~ y&+~~2~ z&~=~0\\12~ x&+~~15~ y&+~~10~ z&~=~0\end{array}$$

Step 5: Multiply the first equation by $ -\frac{ 2 }{ 3 } $ and add the result to the second equation. The result is:

$$\begin{array}{ cccc }3~ x&+~~2~ y&+~~2~ z&~=~0\\&\dfrac{ 5 }{ 3 }~ y&+~~\dfrac{ 2 }{ 3 }~ z&~=~0\\12~ x&+~~15~ y&+~~10~ z&~=~0\end{array}$$

Step 6: Multiply the first equation by $ -4 $ and add the result to the third equation. The result is:

$$\begin{array}{ cccc }3~ x&+~~2~ y&+~~2~ z&~=~0\\&\dfrac{ 5 }{ 3 }~ y&+~~\dfrac{ 2 }{ 3 }~ z&~=~0\\&7~ y&+~~2~ z&~=~0\end{array}$$

Step 7: Get rid of the fractions by multiplying the second equation by 3. The result is:

$$\begin{array}{ cccc }3~ x&+~~2~ y&+~~2~ z&~=~0\\&5~ y&+~~2~ z&~=~0\\&7~ y&+~~2~ z&~=~0\end{array}$$

Step 8: Multiply the second equation by $ -\frac{ 7 }{ 5 } $ and add the result to the third equation. The result is:

$$\begin{array}{ cccc }3~ x&+~~2~ y&+~~2~ z&~=~0\\&5~ y&+~~2~ z&~=~0\\&&-~~~\dfrac{ 4 }{ 5 }~ z&~=~0\end{array}$$

Step 9: Get rid of the fractions by multiplying the third equation by 5. The result is:

$$\begin{array}{ cccc }3~ x&+~~2~ y&+~~2~ z&~=~0\\&5~ y&+~~2~ z&~=~0\\&&-~~~4~ z&~=~0\end{array}$$

Step 10: Solve for $ z $.

$$ \begin{aligned} -4 ~ z & = 0 \\ z & = 0 \end{aligned} $$

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

$$ \begin{aligned}5y+2z &= 0\\5y+2\cdot 0 &= 0\\y &= 0 \end{aligned}$$

Step 12: Solve for $ x $ by substituting $ y = 0 $ and $ z = 0 $ into the first equation.

System 3x3 calculator