detailed information: The input equation set is:
 | | | | | 504 5 | A | + | | 168 | B | + | | 3780 | C | + | | 30 | D | = | | 208320 | | (1) |
| | 275 | A | + | | | 165 2 | B | + | | 6000 | C | + | | 50 | D | = | | 345575 | | (2) |
| | | | 2179 5 | A | + | | | 573 2 | B | + | | 9780 | C | + | | 80 | D | = | | 574775 | | (4) |
| Question solving process:
Multiply both sides of equation (1) by 1375 Divide the two sides of equation (1) by 504, the equation can be obtained: | | 275 | A | + | | | 1375 3 | B | + | | | 20625 2 | C | + | | | 6875 84 | D | = | | | 1705000 3 | (5) | , then subtract both sides of equation (5) from both sides of equation (2), the equations are reduced to:
 | | | | | 504 5 | A | + | | 168 | B | + | | 3780 | C | + | | 30 | D | = | | 208320 | | (1) |
| - | 2255 6 | B | | - | 8625 2 | C | | - | 2675 84 | D | = | | - | 668275 3 | | (2) |
| | | | 2179 5 | A | + | | | 573 2 | B | + | | 9780 | C | + | | 80 | D | = | | 574775 | | (4) |
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Multiply both sides of equation (1) by 25 Divide the two sides of equation (1) by 42, the equation can be obtained: | | 60 | A | + | | 100 | B | + | | 2250 | C | + | | | 125 7 | D | = | | 124000 | (6) | , then subtract both sides of equation (6) from both sides of equation (3), the equations are reduced to:
 | | | | | 504 5 | A | + | | 168 | B | + | | 3780 | C | + | | 30 | D | = | | 208320 | | (1) |
| - | 2255 6 | B | | - | 8625 2 | C | | - | 2675 84 | D | = | | - | 668275 3 | | (2) |
| -64 | B | | -2250 | C | | - | 125 7 | D | = | | -103120 | | (3) |
| | | 2179 5 | A | + | | | 573 2 | B | + | | 9780 | C | + | | 80 | D | = | | 574775 | | (4) |
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Multiply both sides of equation (1) by 2179 Divide the two sides of equation (1) by 504, the equation can be obtained: | | | 2179 5 | A | + | | | 2179 3 | B | + | | | 32685 2 | C | + | | | 10895 84 | D | = | | | 2701960 3 | (7) | , then subtract both sides of equation (7) from both sides of equation (4), the equations are reduced to:
 | | | | | 504 5 | A | + | | 168 | B | + | | 3780 | C | + | | 30 | D | = | | 208320 | | (1) |
| - | 2255 6 | B | | - | 8625 2 | C | | - | 2675 84 | D | = | | - | 668275 3 | | (2) |
| -64 | B | | -2250 | C | | - | 125 7 | D | = | | -103120 | | (3) |
| - | 2639 6 | B | | - | 13125 2 | C | | - | 4175 84 | D | = | | - | 977635 3 | | (4) |
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Multiply both sides of equation (2) by 384 Divide the two sides of equation (2) by 2255, the equation can be obtained: | -64 | B | | - | 331200 451 | C | | - | 17120 3157 | D | = | | - | 17107840 451 | (8) | , then subtract both sides of equation (8) from both sides of equation (3), the equations are reduced to:
 | | | | | 504 5 | A | + | | 168 | B | + | | 3780 | C | + | | 30 | D | = | | 208320 | | (1) |
| - | 2255 6 | B | | - | 8625 2 | C | | - | 2675 84 | D | = | | - | 668275 3 | | (2) |
| - | 683550 451 | C | | - | 39255 3157 | D | = | | - | 29399280 451 | | (3) |
| - | 2639 6 | B | | - | 13125 2 | C | | - | 4175 84 | D | = | | - | 977635 3 | | (4) |
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Multiply both sides of equation (2) by 2639 Divide the two sides of equation (2) by 2255, the equation can be obtained: | - | 2639 6 | B | | - | 4552275 902 | C | | - | 201695 5412 | D | = | | - | 352715545 1353 | (9) | , then subtract both sides of equation (9) from both sides of equation (4), the equations are reduced to:
 | | | | | 504 5 | A | + | | 168 | B | + | | 3780 | C | + | | 30 | D | = | | 208320 | | (1) |
| - | 2255 6 | B | | - | 8625 2 | C | | - | 2675 84 | D | = | | - | 668275 3 | | (2) |
| - | 683550 451 | C | | - | 39255 3157 | D | = | | - | 29399280 451 | | (3) |
| - | 683550 451 | C | | - | 39255 3157 | D | = | | - | 29399280 451 | | (4) |
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Subtract both sides of equation (3) from both sides of equation (4) ,the equations are reduced to:
 | | | | | 504 5 | A | + | | 168 | B | + | | 3780 | C | + | | 30 | D | = | | 208320 | | (1) |
| - | 2255 6 | B | | - | 8625 2 | C | | - | 2675 84 | D | = | | - | 668275 3 | | (2) |
| - | 683550 451 | C | | - | 39255 3157 | D | = | | - | 29399280 451 | | (3) |
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Multiply both sides of equation (3) by 51865 Divide both sides of equation (3) by 18228, get the equation: | - | 8625 2 | C | | - | 1504775 42532 | D | = | | - | 281743100 1519 | (10) | , then subtract both sides of equation (10) from both sides of equation (2), get the equation:
 | | | | | 504 5 | A | + | | 168 | B | + | | 3780 | C | + | | 30 | D | = | | 208320 | | (1) |
| - | 2255 6 | B | + | | | 112750 31899 | D | = | | - | 169880425 4557 | | (2) |
| - | 683550 451 | C | | - | 39255 3157 | D | = | | - | 29399280 451 | | (3) |
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Multiply both sides of equation (3) by 2706 Divide both sides of equation (3) by 1085, get the equation: | -3780 | C | | - | 47106 1519 | D | = | | - | 35279136 217 | (11) | , then add the two sides of equation (11) to both sides of equation (1), get the equation:
 | | | | | 504 5 | A | + | | 168 | B | | - | 1536 1519 | D | = | | | 9926304 217 | | (1) |
| - | 2255 6 | B | + | | | 112750 31899 | D | = | | - | 169880425 4557 | | (2) |
| - | 683550 451 | C | | - | 39255 3157 | D | = | | - | 29399280 451 | | (3) |
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Multiply both sides of equation (2) by 1008 Divide both sides of equation (2) by 2255, get the equation: | -168 | B | + | | | 2400 1519 | D | = | | - | 3616080 217 | (12) | , then add the two sides of equation (12) to both sides of equation (1), get the equation:
 | | | | | 504 5 | A | + | | | 864 1519 | D | = | | | 6310224 217 | | (1) |
| - | 2255 6 | B | + | | | 112750 31899 | D | = | | - | 169880425 4557 | | (2) |
| | C | + | | | 2617 318990 | D | = | | | 979976 22785 | | (3) |
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The coefficient of the unknown number is reduced to 1, and the equations are reduced to:
 | | | | A | + | | | 60 10633 | D | = | | | 438210 1519 | | (1) |
| | B | | - | 100 10633 | D | = | | | 150670 1519 | | (2) |
| | C | + | | | 2617 318990 | D | = | | | 979976 22785 | | (3) |
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Therefore, the solution of the equation set is:
 | | | A = | | | 438210 1519 | - | | | 60 10633 | D |
| B = | | | 150670 1519 | + | | | 100 10633 | D |
| C = | | | 979976 22785 | - | | | 2617 318990 | D |
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Convert the solution of the equation set to decimals:
 | | | A = | | 288.485846 | - | | | 60 10633 | D |
| B = | | 99.190257 | + | | | 100 10633 | D |
| C = | | 43.009699 | - | | | 2617 318990 | D |
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Where: D are arbitrary constants. 解方程组的详细方法请参阅:《多元一次方程组的解法》 |