Step 1: Convert all decimals to fractions

$$\begin{array}{ cccc }2~ x&+~~4~ y&+~~5~ z&~=~1000\\3~ x&+~~7~ y&+~~10~ z&~=~1600\\5~ x&+~~9~ y&+~~14~ z&~=~2400\end{array}$$

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

$$\begin{array}{ cccc }2~ x&+~~4~ y&+~~5~ z&~=~1000\\&~~ y&+~~\dfrac{ 5 }{ 2 }~ z&~=~100\\5~ x&+~~9~ y&+~~14~ z&~=~2400\end{array}$$

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

$$\begin{array}{ cccc }2~ x&+~~4~ y&+~~5~ z&~=~1000\\&~~ y&+~~\dfrac{ 5 }{ 2 }~ z&~=~100\\&-~~~~~ y&+~~\dfrac{ 3 }{ 2 }~ z&~=~-100\end{array}$$

Step 4: Get rid of the fractions by multiplying the second equation by 2 and the third equation by 2. The result is:

$$\begin{array}{ cccc }2~ x&+~~4~ y&+~~5~ z&~=~1000\\&2~ y&+~~5~ z&~=~200\\&-~~~2~ y&+~~3~ z&~=~-200\end{array}$$

Step 5: Multiply the second equation by $ 1 $ and add the result to the third equation. The result is:

$$\begin{array}{ cccc }2~ x&+~~4~ y&+~~5~ z&~=~1000\\&2~ y&+~~5~ z&~=~200\\&&8~ z&~=~0\end{array}$$

Step 6: Divide the third equation by $ 8 $. The result is:

$$\begin{array}{ cccc }2~ x&+~~4~ y&+~~5~ z&~=~1000\\&2~ y&+~~5~ z&~=~200\\&&~~ z&~=~0\end{array}$$

Step 7: Solve for $ z $.

$$ \begin{aligned} 1 ~ z & = 0 \\ z & = 0 \end{aligned} $$

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

$$ \begin{aligned}2y+5z &= 200\\2y+5\cdot 0 &= 200\\y &= 100 \end{aligned}$$

Step 9: Solve for $ x $ by substituting $ y = 100 $ and $ z = 0 $ into the first equation.

System 3x3 calculator