Step 1: Convert all decimals to fractions

$$\begin{array}{ cccc }100~ x&+~~100~ y&+~~100~ z&~=~4500\\40~ x&+~~56~ y&+~~72~ z&~=~2424\\7~ x&+~~12~ y&+~~10~ z&~=~426\end{array}$$

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

$$\begin{array}{ cccc }~~ x&+~~~~ y&+~~~~ z&~=~45\\40~ x&+~~56~ y&+~~72~ z&~=~2424\\7~ x&+~~12~ y&+~~10~ z&~=~426\end{array}$$

Step 3: Divide the second equation by $ 8 $. The result is:

$$\begin{array}{ cccc }~~ x&+~~~~ y&+~~~~ z&~=~45\\5~ x&+~~7~ y&+~~9~ z&~=~303\\7~ x&+~~12~ y&+~~10~ z&~=~426\end{array}$$

Step 4: Multiply the first equation by $ -5 $ and add the result to the second equation. The result is:

$$\begin{array}{ cccc }~~ x&+~~~~ y&+~~~~ z&~=~45\\&2~ y&+~~4~ z&~=~78\\7~ x&+~~12~ y&+~~10~ z&~=~426\end{array}$$

Step 5: Multiply the first equation by $ -7 $ and add the result to the third equation. The result is:

$$\begin{array}{ cccc }~~ x&+~~~~ y&+~~~~ z&~=~45\\&2~ y&+~~4~ z&~=~78\\&5~ y&+~~3~ z&~=~111\end{array}$$

Step 6: Divide the second equation by $ 2 $. The result is:

$$\begin{array}{ cccc }~~ x&+~~~~ y&+~~~~ z&~=~45\\&~~ y&+~~2~ z&~=~39\\&5~ y&+~~3~ z&~=~111\end{array}$$

Step 7: Multiply the second equation by $ -5 $ and add the result to the third equation. The result is:

$$\begin{array}{ cccc }~~ x&+~~~~ y&+~~~~ z&~=~45\\&~~ y&+~~2~ z&~=~39\\&&-~~~7~ z&~=~-84\end{array}$$

Step 8: Solve for $ z $.

$$ \begin{aligned} -7 ~ z & = -84 \\ z & = 12 \end{aligned} $$

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

$$ \begin{aligned}y+2z &= 39\\y+2\cdot 12 &= 39\\y &= 15 \end{aligned}$$

Step 10: Solve for $ x $ by substituting $ y = 15 $ and $ z = 12 $ into the first equation.

System 3x3 calculator