detailed information: The input equation set is:
| | | | A | + | | B | + | | C | + | | D | + | | E | + | | F | + | | G | + | | H | + | | I | + | | J | = | | 1 | | (1) |
| | 2 | A | + | | 2 | B | + | | 2 | C | + | | 2 | D | + | | 2 | E | + | | 2 | F | + | | 2 | G | + | | 2 | H | + | | 2 | I | + | | 2 | J | = | | 2 | | (2) |
| | 3 | A | + | | 3 | B | + | | 3 | C | + | | 3 | D | + | | 3 | E | + | | 3 | F | + | | 3 | G | + | | 3 | H | + | | 3 | I | + | | 3 | J | = | | 3 | | (3) |
| | 4 | A | + | | 5 | B | + | | 6 | D | + | | 7 | E | + | | 8 | F | + | | 9 | G | + | | 10 | H | + | | 11 | I | + | | 12 | J | = | | 5 | | (4) |
| | 12 | A | + | | 11 | B | + | | 10 | C | + | | 9 | D | + | | 8 | E | + | | 7 | F | + | | 6 | G | + | | 5 | I | + | | 4 | J | = | | 7 | | (5) |
| | | | | | Question solving process:
Multiply both sides of equation (1) by 2, the equation can be obtained: | | 2 | A | + | | 2 | B | + | | 2 | C | + | | 2 | D | + | | 2 | E | + | | 2 | F | + | | 2 | G | + | | 2 | H | + | | 2 | I | + | | 2 | J | = | | 2 | (11) | , then subtract both sides of equation (11) from both sides of equation (2), the equations are reduced to:
| | | | A | + | | B | + | | C | + | | D | + | | E | + | | F | + | | G | + | | H | + | | I | + | | J | = | | 1 | | (1) |
| | | 3 | A | + | | 3 | B | + | | 3 | C | + | | 3 | D | + | | 3 | E | + | | 3 | F | + | | 3 | G | + | | 3 | H | + | | 3 | I | + | | 3 | J | = | | 3 | | (3) |
| | 4 | A | + | | 5 | B | + | | 6 | D | + | | 7 | E | + | | 8 | F | + | | 9 | G | + | | 10 | H | + | | 11 | I | + | | 12 | J | = | | 5 | | (4) |
| | 12 | A | + | | 11 | B | + | | 10 | C | + | | 9 | D | + | | 8 | E | + | | 7 | F | + | | 6 | G | + | | 5 | I | + | | 4 | J | = | | 7 | | (5) |
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Multiply both sides of equation (1) by 3, the equation can be obtained: | | 3 | A | + | | 3 | B | + | | 3 | C | + | | 3 | D | + | | 3 | E | + | | 3 | F | + | | 3 | G | + | | 3 | H | + | | 3 | I | + | | 3 | J | = | | 3 | (12) | , then subtract both sides of equation (12) from both sides of equation (3), the equations are reduced to:
| | | | A | + | | B | + | | C | + | | D | + | | E | + | | F | + | | G | + | | H | + | | I | + | | J | = | | 1 | | (1) |
| | | | 4 | A | + | | 5 | B | + | | 6 | D | + | | 7 | E | + | | 8 | F | + | | 9 | G | + | | 10 | H | + | | 11 | I | + | | 12 | J | = | | 5 | | (4) |
| | 12 | A | + | | 11 | B | + | | 10 | C | + | | 9 | D | + | | 8 | E | + | | 7 | F | + | | 6 | G | + | | 5 | I | + | | 4 | J | = | | 7 | | (5) |
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Multiply both sides of equation (1) by 4, the equation can be obtained: | | 4 | A | + | | 4 | B | + | | 4 | C | + | | 4 | D | + | | 4 | E | + | | 4 | F | + | | 4 | G | + | | 4 | H | + | | 4 | I | + | | 4 | J | = | | 4 | (13) | , then subtract both sides of equation (13) from both sides of equation (4), the equations are reduced to:
| | | | A | + | | B | + | | C | + | | D | + | | E | + | | F | + | | G | + | | H | + | | I | + | | J | = | | 1 | | (1) |
| | | | B | | -4 | C | + | | 2 | D | + | | 3 | E | + | | 4 | F | + | | 5 | G | + | | 6 | H | + | | 7 | I | + | | 8 | J | = | | 1 | | (4) |
| | 12 | A | + | | 11 | B | + | | 10 | C | + | | 9 | D | + | | 8 | E | + | | 7 | F | + | | 6 | G | + | | 5 | I | + | | 4 | J | = | | 7 | | (5) |
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Multiply both sides of equation (1) by 12, the equation can be obtained: | | 12 | A | + | | 12 | B | + | | 12 | C | + | | 12 | D | + | | 12 | E | + | | 12 | F | + | | 12 | G | + | | 12 | H | + | | 12 | I | + | | 12 | J | = | | 12 | (14) | , then subtract both sides of equation (14) from both sides of equation (5), the equations are reduced to:
| | | | A | + | | B | + | | C | + | | D | + | | E | + | | F | + | | G | + | | H | + | | I | + | | J | = | | 1 | | (1) |
| | | | B | | -4 | C | + | | 2 | D | + | | 3 | E | + | | 4 | F | + | | 5 | G | + | | 6 | H | + | | 7 | I | + | | 8 | J | = | | 1 | | (4) |
| -1 | B | | -2 | C | | -3 | D | | -4 | E | | -5 | F | | -6 | G | | -12 | H | | -7 | I | | -8 | J | = | | -5 | | (5) |
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交After the exchange of equation (2) and equation (4), the equation system becomes:
| | | | A | + | | B | + | | C | + | | D | + | | E | + | | F | + | | G | + | | H | + | | I | + | | J | = | | 1 | | (1) |
| | B | | -4 | C | + | | 2 | D | + | | 3 | E | + | | 4 | F | + | | 5 | G | + | | 6 | H | + | | 7 | I | + | | 8 | J | = | | 1 | | (2) |
| | | -1 | B | | -2 | C | | -3 | D | | -4 | E | | -5 | F | | -6 | G | | -12 | H | | -7 | I | | -8 | J | = | | -5 | | (5) |
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Add both sides of equation (2) to both sides of equation (5) ,the equations are reduced to:
| | | | A | + | | B | + | | C | + | | D | + | | E | + | | F | + | | G | + | | H | + | | I | + | | J | = | | 1 | | (1) |
| | B | | -4 | C | + | | 2 | D | + | | 3 | E | + | | 4 | F | + | | 5 | G | + | | 6 | H | + | | 7 | I | + | | 8 | J | = | | 1 | | (2) |
| | | -6 | C | | -1 | D | | -1 | E | | -1 | F | | -1 | G | | -6 | H | = | | -4 | | (5) |
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交After the exchange of equation (3) and equation (5), the equation system becomes:
| | | | A | + | | B | + | | C | + | | D | + | | E | + | | F | + | | G | + | | H | + | | I | + | | J | = | | 1 | | (1) |
| | B | | -4 | C | + | | 2 | D | + | | 3 | E | + | | 4 | F | + | | 5 | G | + | | 6 | H | + | | 7 | I | + | | 8 | J | = | | 1 | | (2) |
| -6 | C | | -1 | D | | -1 | E | | -1 | F | | -1 | G | | -6 | H | = | | -4 | | (3) |
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Multiply both sides of equation (3) by 2 Divide both sides of equation (3) by 3, get the equation: | -4 | C | | - | 2 3 | D | | - | 2 3 | E | | - | 2 3 | F | | - | 2 3 | G | | -4 | H | = | | - | 8 3 | (15) | , then subtract both sides of equation (15) from both sides of equation (2), get the equation:
| | | | A | + | | B | + | | C | + | | D | + | | E | + | | F | + | | G | + | | H | + | | I | + | | J | = | | 1 | | (1) |
| | B | + | | | 8 3 | D | + | | | 11 3 | E | + | | | 14 3 | F | + | | | 17 3 | G | + | | 10 | H | + | | 7 | I | + | | 8 | J | = | | | 11 3 | | (2) |
| -6 | C | | -1 | D | | -1 | E | | -1 | F | | -1 | G | | -6 | H | = | | -4 | | (3) |
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Divide both sides of equation (3) by 6, get the equation: | -1 | C | | - | 1 6 | D | | - | 1 6 | E | | - | 1 6 | F | | - | 1 6 | G | | -1 | H | = | | - | 2 3 | (16) | , then add the two sides of equation (16) to both sides of equation (1), get the equation:
| | | | A | + | | B | + | | | 5 6 | D | + | | | 5 6 | E | + | | | 5 6 | F | + | | | 5 6 | G | + | | I | + | | J | = | | | 1 3 | | (1) |
| | B | + | | | 8 3 | D | + | | | 11 3 | E | + | | | 14 3 | F | + | | | 17 3 | G | + | | 10 | H | + | | 7 | I | + | | 8 | J | = | | | 11 3 | | (2) |
| -6 | C | | -1 | D | | -1 | E | | -1 | F | | -1 | G | | -6 | H | = | | -4 | | (3) |
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Subtract both sides of equation (2) from both sides of equation (1), get the equation:
| | | | A | | - | 11 6 | D | | - | 17 6 | E | | - | 23 6 | F | | - | 29 6 | G | | -10 | H | | -6 | I | | -7 | J | = | | - | 10 3 | | (1) |
| | B | + | | | 8 3 | D | + | | | 11 3 | E | + | | | 14 3 | F | + | | | 17 3 | G | + | | 10 | H | + | | 7 | I | + | | 8 | J | = | | | 11 3 | | (2) |
| | C | + | | | 1 6 | D | + | | | 1 6 | E | + | | | 1 6 | F | + | | | 1 6 | G | + | | H | = | | | 2 3 | | (3) |
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The coefficient of the unknown number is reduced to 1, and the equations are reduced to:
| | | | A | | - | 11 6 | D | | - | 17 6 | E | | - | 23 6 | F | | - | 29 6 | G | | -10 | H | | -6 | I | | -7 | J | = | | - | 10 3 | | (1) |
| | B | + | | | 8 3 | D | + | | | 11 3 | E | + | | | 14 3 | F | + | | | 17 3 | G | + | | 10 | H | + | | 7 | I | + | | 8 | J | = | | | 11 3 | | (2) |
| | C | + | | | 1 6 | D | + | | | 1 6 | E | + | | | 1 6 | F | + | | | 1 6 | G | + | | H | = | | | 2 3 | | (3) |
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Therefore, the solution of the equation set is:
| | | A = | | - | 10 3 | + | | | 11 6 | D | + | | | 17 6 | E | + | | | 23 6 | F | + | | | 29 6 | G | + | | 10 | H | + | | 6 | I | + | | 7 | J |
| B = | | | 11 3 | - | | | 8 3 | D | - | | | 11 3 | E | - | | | 14 3 | F | - | | | 17 3 | G | - | | 10 | H | - | | 7 | I | - | | 8 | J |
| C = | | | 2 3 | - | | | 1 6 | D | - | | | 1 6 | E | - | | | 1 6 | F | - | | | 1 6 | G | - | | 1 | H |
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Convert the solution of the equation set to decimals:
| | | A = | | -3.333333 | + | | | 11 6 | D | + | | | 17 6 | E | + | | | 23 6 | F | + | | | 29 6 | G | + | | 10 | H | + | | 6 | I | + | | 7 | J |
| B = | | 3.666667 | - | | | 8 3 | D | - | | | 11 3 | E | - | | | 14 3 | F | - | | | 17 3 | G | - | | 10 | H | - | | 7 | I | - | | 8 | J |
| C = | | 0.666667 | - | | | 1 6 | D | - | | | 1 6 | E | - | | | 1 6 | F | - | | | 1 6 | G | - | | 1 | H |
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Where: D, E, F, G, H, I, J 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》 |