detailed information: The input equation set is:
 | | | | 2 | A | + | | 16 | B | + | | 10 | C | + | | 6 | D | + | | 10 | E | + | | F | + | | 9 | G | + | | 12 | H | + | | 20 | I | = | | 42702 | | (1) |
| | 10 | E | + | | 6 | F | + | | 6 | G | + | | 6 | H | + | | J | + | | 5 | K | + | | 4 | L | + | | M | + | | 12 | N | = | | 31909 | | (2) |
| | 2 | A | + | | C | + | | 14 | F | + | | 18 | H | + | | 24 | I | + | | 18 | O | + | | 2 | P | + | | 10 | Q | + | | 10 | R | = | | 42562 | | (3) |
| | | | | | | | | | | | | | | | | Question solving process:
Subtract both sides of equation (1) from both sides of equation (3) ,the equations are reduced to:
 | | | | 2 | A | + | | 16 | B | + | | 10 | C | + | | 6 | D | + | | 10 | E | + | | F | + | | 9 | G | + | | 12 | H | + | | 20 | I | = | | 42702 | | (1) |
| | 10 | E | + | | 6 | F | + | | 6 | G | + | | 6 | H | + | | J | + | | 5 | K | + | | 4 | L | + | | M | + | | 12 | N | = | | 31909 | | (2) |
| -16 | B | | -9 | C | | -6 | D | | -10 | E | + | | 13 | F | | -9 | G | + | | 6 | H | + | | 4 | I | + | | 18 | O | + | | 2 | P | + | | 10 | Q | + | | 10 | R | = | | -140 | | (3) |
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交After the exchange of equation (2) and equation (3), the equation system becomes:
 | | | | 2 | A | + | | 16 | B | + | | 10 | C | + | | 6 | D | + | | 10 | E | + | | F | + | | 9 | G | + | | 12 | H | + | | 20 | I | = | | 42702 | | (1) |
| -16 | B | | -9 | C | | -6 | D | | -10 | E | + | | 13 | F | | -9 | G | + | | 6 | H | + | | 4 | I | + | | 18 | O | + | | 2 | P | + | | 10 | Q | + | | 10 | R | = | | -140 | | (2) |
| | 10 | E | + | | 6 | F | + | | 6 | G | + | | 6 | H | + | | J | + | | 5 | K | + | | 4 | L | + | | M | + | | 12 | N | = | | 31909 | | (3) |
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Add both sides of equation (3) to both sides of equation (2), get the equation:
 | | | | 2 | A | + | | 16 | B | + | | 10 | C | + | | 6 | D | + | | 10 | E | + | | F | + | | 9 | G | + | | 12 | H | + | | 20 | I | = | | 42702 | | (1) |
| -16 | B | | -9 | C | | -6 | D | + | | 19 | F | | -3 | G | + | | 12 | H | + | | 4 | I | + | | J | + | | 5 | K | + | | 4 | L | + | | M | + | | 12 | N | + | | 18 | O | + | | 2 | P | + | | 10 | Q | + | | 10 | R | = | | 31769 | | (2) |
| | 10 | E | + | | 6 | F | + | | 6 | G | + | | 6 | H | + | | J | + | | 5 | K | + | | 4 | L | + | | M | + | | 12 | N | = | | 31909 | | (3) |
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Subtract both sides of equation (3) from both sides of equation (1), get the equation:
 | | | | 2 | A | + | | 16 | B | + | | 10 | C | + | | 6 | D | | -5 | F | + | | 3 | G | + | | 6 | H | + | | 20 | I | | -1 | J | | -5 | K | | -4 | L | | -1 | M | | -12 | N | = | | 10793 | | (1) |
| -16 | B | | -9 | C | | -6 | D | + | | 19 | F | | -3 | G | + | | 12 | H | + | | 4 | I | + | | J | + | | 5 | K | + | | 4 | L | + | | M | + | | 12 | N | + | | 18 | O | + | | 2 | P | + | | 10 | Q | + | | 10 | R | = | | 31769 | | (2) |
| | 10 | E | + | | 6 | F | + | | 6 | G | + | | 6 | H | + | | J | + | | 5 | K | + | | 4 | L | + | | M | + | | 12 | N | = | | 31909 | | (3) |
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Add both sides of equation (2) to both sides of equation (1), get the equation:
 | | | | 2 | A | + | | C | + | | 14 | F | + | | 18 | H | + | | 24 | I | + | | 18 | O | + | | 2 | P | + | | 10 | Q | + | | 10 | R | = | | 42562 | | (1) |
| -16 | B | | -9 | C | | -6 | D | + | | 19 | F | | -3 | G | + | | 12 | H | + | | 4 | I | + | | J | + | | 5 | K | + | | 4 | L | + | | M | + | | 12 | N | + | | 18 | O | + | | 2 | P | + | | 10 | Q | + | | 10 | R | = | | 31769 | | (2) |
| | E | + | | | 3 5 | F | + | | | 3 5 | G | + | | | 3 5 | H | + | | | 1 10 | J | + | | | 1 2 | K | + | | | 2 5 | L | + | | | 1 10 | M | + | | | 6 5 | N | = | | | 31909 10 | | (3) |
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The coefficient of the unknown number is reduced to 1, and the equations are reduced to:
 | | | | A | + | | | 1 2 | C | + | | 7 | F | + | | 9 | H | + | | 12 | I | + | | 9 | O | + | | P | + | | 5 | Q | + | | 5 | R | = | | 21281 | | (1) |
| | B | + | | | 9 16 | C | + | | | 3 8 | D | | - | 19 16 | F | + | | | 3 16 | G | | - | 3 4 | H | | - | 1 4 | I | | - | 1 16 | J | | - | 5 16 | K | | - | 1 4 | L | | - | 1 16 | M | | - | 3 4 | N | | - | 9 8 | O | | - | 1 8 | P | | - | 5 8 | Q | | - | 5 8 | R | = | | - | 31769 16 | | (2) |
| | E | + | | | 3 5 | F | + | | | 3 5 | G | + | | | 3 5 | H | + | | | 1 10 | J | + | | | 1 2 | K | + | | | 2 5 | L | + | | | 1 10 | M | + | | | 6 5 | N | = | | | 31909 10 | | (3) |
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Therefore, the solution of the equation set is:
 | | | A = | | 21281 | - | | | 1 2 | C | - | | 7 | F | - | | 9 | H | - | | 12 | I | - | | 9 | O | - | | 1 | P | - | | 5 | Q | - | | 5 | R |
| B = | | - | 31769 16 | - | | | 9 16 | C | - | | | 3 8 | D | + | | | 19 16 | F | - | | | 3 16 | G | + | | | 3 4 | H | + | | | 1 4 | I | + | | | 1 16 | J | + | | | 5 16 | K | + | | | 1 4 | L | + | | | 1 16 | M | + | | | 3 4 | N | + | | | 9 8 | O | + | | | 1 8 | P | + | | | 5 8 | Q | + | | | 5 8 | R |
| E = | | | 31909 10 | - | | | 3 5 | F | - | | | 3 5 | G | - | | | 3 5 | H | - | | | 1 10 | J | - | | | 1 2 | K | - | | | 2 5 | L | - | | | 1 10 | M | - | | | 6 5 | N |
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Convert the solution of the equation set to decimals:
 | | | A = | | 21281 | - | | | 1 2 | C | - | | 7 | F | - | | 9 | H | - | | 12 | I | - | | 9 | O | - | | 1 | P | - | | 5 | Q | - | | 5 | R |
| B = | | -1985.562500 | - | | | 9 16 | C | - | | | 3 8 | D | + | | | 19 16 | F | - | | | 3 16 | G | + | | | 3 4 | H | + | | | 1 4 | I | + | | | 1 16 | J | + | | | 5 16 | K | + | | | 1 4 | L | + | | | 1 16 | M | + | | | 3 4 | N | + | | | 9 8 | O | + | | | 1 8 | P | + | | | 5 8 | Q | + | | | 5 8 | R |
| E = | | 3190.900000 | - | | | 3 5 | F | - | | | 3 5 | G | - | | | 3 5 | H | - | | | 1 10 | J | - | | | 1 2 | K | - | | | 2 5 | L | - | | | 1 10 | M | - | | | 6 5 | N |
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Where: C, D, F, G, H, I, J, K, L, M, N, O, P, Q, R, S 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》 |