Step 1: Convert all decimals to fractions

$$\begin{array}{ cccc }150~ x&+~~250~ y&+~~350~ z&~=~44500\\80~ x&+~~140~ y&+~~240~ z&~=~27200\\160~ x&+~~240~ y&+~~400~ z&~=~46800\end{array}$$

Step 2: Divide the first equation by $ 50 $. The result is:

$$\begin{array}{ cccc }3~ x&+~~5~ y&+~~7~ z&~=~890\\80~ x&+~~140~ y&+~~240~ z&~=~27200\\160~ x&+~~240~ y&+~~400~ z&~=~46800\end{array}$$

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

$$\begin{array}{ cccc }3~ x&+~~5~ y&+~~7~ z&~=~890\\4~ x&+~~7~ y&+~~12~ z&~=~1360\\160~ x&+~~240~ y&+~~400~ z&~=~46800\end{array}$$

Step 4: Divide the third equation by $ 80 $. The result is:

$$\begin{array}{ cccc }3~ x&+~~5~ y&+~~7~ z&~=~890\\4~ x&+~~7~ y&+~~12~ z&~=~1360\\2~ x&+~~3~ y&+~~5~ z&~=~585\end{array}$$

Step 5: Multiply the first equation by $ -\frac{ 4 }{ 3 } $ and add the result to the second equation. The result is:

$$\begin{array}{ cccc }3~ x&+~~5~ y&+~~7~ z&~=~890\\&\dfrac{ 1 }{ 3 }~ y&+~~\dfrac{ 8 }{ 3 }~ z&~=~\dfrac{ 520 }{ 3 }\\2~ x&+~~3~ y&+~~5~ z&~=~585\end{array}$$

Step 6: Multiply the first equation by $ -\frac{ 2 }{ 3 } $ and add the result to the third equation. The result is:

$$\begin{array}{ cccc }3~ x&+~~5~ y&+~~7~ z&~=~890\\&\dfrac{ 1 }{ 3 }~ y&+~~\dfrac{ 8 }{ 3 }~ z&~=~\dfrac{ 520 }{ 3 }\\&-~~~\dfrac{ 1 }{ 3 }~ y&+~~\dfrac{ 1 }{ 3 }~ z&~=~-\dfrac{ 25 }{ 3 }\end{array}$$

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

$$\begin{array}{ cccc }3~ x&+~~5~ y&+~~7~ z&~=~890\\&~~ y&+~~8~ z&~=~520\\&-~~~~~ y&+~~~~ z&~=~-25\end{array}$$

Step 8: Multiply the second equation by $ 1 $ and add the result to the third equation. The result is:

$$\begin{array}{ cccc }3~ x&+~~5~ y&+~~7~ z&~=~890\\&~~ y&+~~8~ z&~=~520\\&&9~ z&~=~495\end{array}$$

Step 9: Divide the third equation by $ 9 $. The result is:

$$\begin{array}{ cccc }3~ x&+~~5~ y&+~~7~ z&~=~890\\&~~ y&+~~8~ z&~=~520\\&&~~ z&~=~55\end{array}$$

Step 10: Solve for $ z $.

$$ \begin{aligned} 1 ~ z & = 55 \\ z & = 55 \end{aligned} $$

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

$$ \begin{aligned}y+8z &= 520\\y+8\cdot 55 &= 520\\y &= 80 \end{aligned}$$

Step 12: Solve for $ x $ by substituting $ y = 80 $ and $ z = 55 $ into the first equation.

System 3x3 calculator