Step 1: Convert all decimals to fractions

$$\begin{array}{ cccc }350~ x&+~~400~ y&+~~250~ z&~=~64800\\60~ x&+~~50~ y&+~~30~ z&~=~9000\\3~ x&+~~4~ y&+~~~~ z&~=~468\end{array}$$

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

$$\begin{array}{ cccc }7~ x&+~~8~ y&+~~5~ z&~=~1296\\60~ x&+~~50~ y&+~~30~ z&~=~9000\\3~ x&+~~4~ y&+~~~~ z&~=~468\end{array}$$

Step 3: Divide the second equation by $ 10 $. The result is:

$$\begin{array}{ cccc }7~ x&+~~8~ y&+~~5~ z&~=~1296\\6~ x&+~~5~ y&+~~3~ z&~=~900\\3~ x&+~~4~ y&+~~~~ z&~=~468\end{array}$$

Step 4: Multiply the first equation by $ -\frac{ 6 }{ 7 } $ and add the result to the second equation. The result is:

$$\begin{array}{ cccc }7~ x&+~~8~ y&+~~5~ z&~=~1296\\&-~~~\dfrac{ 13 }{ 7 }~ y&-~~~\dfrac{ 9 }{ 7 }~ z&~=~-\dfrac{ 1476 }{ 7 }\\3~ x&+~~4~ y&+~~~~ z&~=~468\end{array}$$

Step 5: Multiply the first equation by $ -\frac{ 3 }{ 7 } $ and add the result to the third equation. The result is:

$$\begin{array}{ cccc }7~ x&+~~8~ y&+~~5~ z&~=~1296\\&-~~~\dfrac{ 13 }{ 7 }~ y&-~~~\dfrac{ 9 }{ 7 }~ z&~=~-\dfrac{ 1476 }{ 7 }\\&\dfrac{ 4 }{ 7 }~ y&-~~~\dfrac{ 8 }{ 7 }~ z&~=~-\dfrac{ 612 }{ 7 }\end{array}$$

Step 6: Get rid of the fractions by multiplying the second equation by 7 and the third equation by 7. The result is:

$$\begin{array}{ cccc }7~ x&+~~8~ y&+~~5~ z&~=~1296\\&-~~~13~ y&-~~~9~ z&~=~-1476\\&4~ y&-~~~8~ z&~=~-612\end{array}$$

Step 7: Divide the third equation by $ 4 $. The result is:

$$\begin{array}{ cccc }7~ x&+~~8~ y&+~~5~ z&~=~1296\\&-~~~13~ y&-~~~9~ z&~=~-1476\\&~~ y&-~~~2~ z&~=~-153\end{array}$$

Step 8: Multiply the second equation by $ \frac{ 1 }{ 13 } $ and add the result to the third equation. The result is:

$$\begin{array}{ cccc }7~ x&+~~8~ y&+~~5~ z&~=~1296\\&-~~~13~ y&-~~~9~ z&~=~-1476\\&&-~~~\dfrac{ 35 }{ 13 }~ z&~=~-\dfrac{ 3465 }{ 13 }\end{array}$$

Step 9: Get rid of the fractions by multiplying the third equation by 13. The result is:

$$\begin{array}{ cccc }7~ x&+~~8~ y&+~~5~ z&~=~1296\\&-~~~13~ y&-~~~9~ z&~=~-1476\\&&-~~~35~ z&~=~-3465\end{array}$$

Step 10: Solve for $ z $.

$$ \begin{aligned} -35 ~ z & = -3465 \\ z & = 99 \end{aligned} $$

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

$$ \begin{aligned}-13y-9z &= -1476\\-13y-9\cdot 99 &= -1476\\y &= 45 \end{aligned}$$

Step 12: Solve for $ x $ by substituting $ y = 45 $ and $ z = 99 $ into the first equation.

System 3x3 calculator