detailed information: The input equation set is:
| | | | 6 | A | + | | 6 | B | + | | 12 | C | + | | D | + | | 5 | E | + | | 6 | F | + | | 6 | G | = | | | 753 100 | | (1) |
| | 4 | A | + | | 4 | B | + | | 8 | C | + | | D | + | | 3 | E | + | | 4 | F | + | | 4 | G | = | | | 251 50 | | (2) |
| | 6 | A | + | | 5 | B | + | | 15 | C | + | | D | + | | 4 | E | + | | 5 | F | + | | 7 | G | = | | | 168 25 | | (3) |
| | 6 | A | + | | 7 | B | + | | 18 | C | + | | D | + | | 7 | E | + | | 6 | F | + | | 8 | G | = | | | 243 25 | | (4) |
| | 7 | A | + | | 7 | B | + | | 19 | C | + | | D | + | | 5 | E | + | | 7 | F | + | | 9 | G | = | | | 218 25 | | (5) |
| | | Question solving process:
Multiply both sides of equation (1) by 2 Divide the two sides of equation (1) by 3, the equation can be obtained: | | 4 | A | + | | 4 | B | + | | 8 | C | + | | | 2 3 | D | + | | | 10 3 | E | + | | 4 | F | + | | 4 | G | = | | | 251 50 | (8) | , then subtract both sides of equation (8) from both sides of equation (2), the equations are reduced to:
| | | | 6 | A | + | | 6 | B | + | | 12 | C | + | | D | + | | 5 | E | + | | 6 | F | + | | 6 | G | = | | | 753 100 | | (1) |
| | | 6 | A | + | | 5 | B | + | | 15 | C | + | | D | + | | 4 | E | + | | 5 | F | + | | 7 | G | = | | | 168 25 | | (3) |
| | 6 | A | + | | 7 | B | + | | 18 | C | + | | D | + | | 7 | E | + | | 6 | F | + | | 8 | G | = | | | 243 25 | | (4) |
| | 7 | A | + | | 7 | B | + | | 19 | C | + | | D | + | | 5 | E | + | | 7 | F | + | | 9 | G | = | | | 218 25 | | (5) |
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Subtract both sides of equation (1) from both sides of equation (3) ,the equations are reduced to:
| | | | 6 | A | + | | 6 | B | + | | 12 | C | + | | D | + | | 5 | E | + | | 6 | F | + | | 6 | G | = | | | 753 100 | | (1) |
| | -1 | B | + | | 3 | C | | -1 | E | | -1 | F | + | | G | = | | - | 81 100 | | (3) |
| | 6 | A | + | | 7 | B | + | | 18 | C | + | | D | + | | 7 | E | + | | 6 | F | + | | 8 | G | = | | | 243 25 | | (4) |
| | 7 | A | + | | 7 | B | + | | 19 | C | + | | D | + | | 5 | E | + | | 7 | F | + | | 9 | G | = | | | 218 25 | | (5) |
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Subtract both sides of equation (1) from both sides of equation (4) ,the equations are reduced to:
| | | | 6 | A | + | | 6 | B | + | | 12 | C | + | | D | + | | 5 | E | + | | 6 | F | + | | 6 | G | = | | | 753 100 | | (1) |
| | -1 | B | + | | 3 | C | | -1 | E | | -1 | F | + | | G | = | | - | 81 100 | | (3) |
| | B | + | | 6 | C | + | | 2 | E | + | | 2 | G | = | | | 219 100 | | (4) |
| | 7 | A | + | | 7 | B | + | | 19 | C | + | | D | + | | 5 | E | + | | 7 | F | + | | 9 | G | = | | | 218 25 | | (5) |
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Multiply both sides of equation (1) by 7 Divide the two sides of equation (1) by 6, the equation can be obtained: | | 7 | A | + | | 7 | B | + | | 14 | C | + | | | 7 6 | D | + | | | 35 6 | E | + | | 7 | F | + | | 7 | G | = | | | 1757 200 | (9) | , then subtract both sides of equation (9) from both sides of equation (5), the equations are reduced to:
| | | | 6 | A | + | | 6 | B | + | | 12 | C | + | | D | + | | 5 | E | + | | 6 | F | + | | 6 | G | = | | | 753 100 | | (1) |
| | -1 | B | + | | 3 | C | | -1 | E | | -1 | F | + | | G | = | | - | 81 100 | | (3) |
| | B | + | | 6 | C | + | | 2 | E | + | | 2 | G | = | | | 219 100 | | (4) |
| | 5 | C | | - | 1 6 | D | | - | 5 6 | E | + | | 2 | G | = | | - | 13 200 | | (5) |
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交After the exchange of equation (2) and equation (3), the equation system becomes:
| | | | 6 | A | + | | 6 | B | + | | 12 | C | + | | D | + | | 5 | E | + | | 6 | F | + | | 6 | G | = | | | 753 100 | | (1) |
| -1 | B | + | | 3 | C | | -1 | E | | -1 | F | + | | G | = | | - | 81 100 | | (2) |
| | | B | + | | 6 | C | + | | 2 | E | + | | 2 | G | = | | | 219 100 | | (4) |
| | 5 | C | | - | 1 6 | D | | - | 5 6 | E | + | | 2 | G | = | | - | 13 200 | | (5) |
| | |
Add both sides of equation (2) to both sides of equation (4) ,the equations are reduced to:
| | | | 6 | A | + | | 6 | B | + | | 12 | C | + | | D | + | | 5 | E | + | | 6 | F | + | | 6 | G | = | | | 753 100 | | (1) |
| -1 | B | + | | 3 | C | | -1 | E | | -1 | F | + | | G | = | | - | 81 100 | | (2) |
| | | 9 | C | + | | E | | -1 | F | + | | 3 | G | = | | | 69 50 | | (4) |
| | 5 | C | | - | 1 6 | D | | - | 5 6 | E | + | | 2 | G | = | | - | 13 200 | | (5) |
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交After the exchange of equation (3) and equation (4), the equation system becomes:
| | | | 6 | A | + | | 6 | B | + | | 12 | C | + | | D | + | | 5 | E | + | | 6 | F | + | | 6 | G | = | | | 753 100 | | (1) |
| -1 | B | + | | 3 | C | | -1 | E | | -1 | F | + | | G | = | | - | 81 100 | | (2) |
| | 9 | C | + | | E | | -1 | F | + | | 3 | G | = | | | 69 50 | | (3) |
| | | 5 | C | | - | 1 6 | D | | - | 5 6 | E | + | | 2 | G | = | | - | 13 200 | | (5) |
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Multiply both sides of equation (3) by 5 Divide the two sides of equation (3) by 9, the equation can be obtained: | | 5 | C | + | | | 5 9 | E | | - | 5 9 | F | + | | | 5 3 | G | = | | | 23 30 | (10) | , then subtract both sides of equation (10) from both sides of equation (5), the equations are reduced to:
| | | | 6 | A | + | | 6 | B | + | | 12 | C | + | | D | + | | 5 | E | + | | 6 | F | + | | 6 | G | = | | | 753 100 | | (1) |
| -1 | B | + | | 3 | C | | -1 | E | | -1 | F | + | | G | = | | - | 81 100 | | (2) |
| | 9 | C | + | | E | | -1 | F | + | | 3 | G | = | | | 69 50 | | (3) |
| | - | 1 6 | D | | - | 25 18 | E | + | | | 5 9 | F | + | | | 1 3 | G | = | | - | 499 600 | | (5) |
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Divide the two sides of equation (4) by 2, the equation can be obtained: , then add the two sides of equation (11) to both sides of equation (5), the equations are reduced to:
| | | | 6 | A | + | | 6 | B | + | | 12 | C | + | | D | + | | 5 | E | + | | 6 | F | + | | 6 | G | = | | | 753 100 | | (1) |
| -1 | B | + | | 3 | C | | -1 | E | | -1 | F | + | | G | = | | - | 81 100 | | (2) |
| | 9 | C | + | | E | | -1 | F | + | | 3 | G | = | | | 69 50 | | (3) |
| | - | 14 9 | E | + | | | 5 9 | F | + | | | 1 3 | G | = | | - | 499 600 | | (5) |
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Multiply both sides of equation (5) by 3 Divide both sides of equation (5) by 14, get the equation: | - | 1 3 | E | + | | | 5 42 | F | + | | | 1 14 | G | = | | - | 499 2800 | (12) | , then subtract both sides of equation (12) from both sides of equation (4), get the equation:
| | | | 6 | A | + | | 6 | B | + | | 12 | C | + | | D | + | | 5 | E | + | | 6 | F | + | | 6 | G | = | | | 753 100 | | (1) |
| -1 | B | + | | 3 | C | | -1 | E | | -1 | F | + | | G | = | | - | 81 100 | | (2) |
| | 9 | C | + | | E | | -1 | F | + | | 3 | G | = | | | 69 50 | | (3) |
| | | 1 3 | D | | - | 5 42 | F | | - | 1 14 | G | = | | | 499 2800 | | (4) |
| - | 14 9 | E | + | | | 5 9 | F | + | | | 1 3 | G | = | | - | 499 600 | | (5) |
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Multiply both sides of equation (5) by 9 Divide both sides of equation (5) by 14, get the equation: | -1 | E | + | | | 5 14 | F | + | | | 3 14 | G | = | | - | 1497 2800 | (13) | , then add the two sides of equation (13) to both sides of equation (3), get the equation:
| | | | 6 | A | + | | 6 | B | + | | 12 | C | + | | D | + | | 5 | E | + | | 6 | F | + | | 6 | G | = | | | 753 100 | | (1) |
| -1 | B | + | | 3 | C | | -1 | E | | -1 | F | + | | G | = | | - | 81 100 | | (2) |
| | 9 | C | | - | 9 14 | F | + | | | 45 14 | G | = | | | 2367 2800 | | (3) |
| | | 1 3 | D | | - | 5 42 | F | | - | 1 14 | G | = | | | 499 2800 | | (4) |
| - | 14 9 | E | + | | | 5 9 | F | + | | | 1 3 | G | = | | - | 499 600 | | (5) |
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Multiply both sides of equation (5) by 9 Divide both sides of equation (5) by 14, get the equation: | -1 | E | + | | | 5 14 | F | + | | | 3 14 | G | = | | - | 1497 2800 | (14) | , then subtract both sides of equation (14) from both sides of equation (2), get the equation:
| | | | 6 | A | + | | 6 | B | + | | 12 | C | + | | D | + | | 5 | E | + | | 6 | F | + | | 6 | G | = | | | 753 100 | | (1) |
| -1 | B | + | | 3 | C | | - | 19 14 | F | + | | | 11 14 | G | = | | - | 771 2800 | | (2) |
| | 9 | C | | - | 9 14 | F | + | | | 45 14 | G | = | | | 2367 2800 | | (3) |
| | | 1 3 | D | | - | 5 42 | F | | - | 1 14 | G | = | | | 499 2800 | | (4) |
| - | 14 9 | E | + | | | 5 9 | F | + | | | 1 3 | G | = | | - | 499 600 | | (5) |
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Multiply both sides of equation (5) by 45 Divide both sides of equation (5) by 14, get the equation: | -5 | E | + | | | 25 14 | F | + | | | 15 14 | G | = | | - | 1497 560 | (15) | , then add the two sides of equation (15) to both sides of equation (1), get the equation:
| | | | 6 | A | + | | 6 | B | + | | 12 | C | + | | D | + | | | 109 14 | F | + | | | 99 14 | G | = | | | 13599 2800 | | (1) |
| -1 | B | + | | 3 | C | | - | 19 14 | F | + | | | 11 14 | G | = | | - | 771 2800 | | (2) |
| | 9 | C | | - | 9 14 | F | + | | | 45 14 | G | = | | | 2367 2800 | | (3) |
| | | 1 3 | D | | - | 5 42 | F | | - | 1 14 | G | = | | | 499 2800 | | (4) |
| - | 14 9 | E | + | | | 5 9 | F | + | | | 1 3 | G | = | | - | 499 600 | | (5) |
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Multiply both sides of equation (4) by 3, get the equation: | | D | | - | 5 14 | F | | - | 3 14 | G | = | | | 1497 2800 | (16) | , then subtract both sides of equation (16) from both sides of equation (1), get the equation:
| | | | 6 | A | + | | 6 | B | + | | 12 | C | + | | | 57 7 | F | + | | | 51 7 | G | = | | | 6051 1400 | | (1) |
| -1 | B | + | | 3 | C | | - | 19 14 | F | + | | | 11 14 | G | = | | - | 771 2800 | | (2) |
| | 9 | C | | - | 9 14 | F | + | | | 45 14 | G | = | | | 2367 2800 | | (3) |
| | | 1 3 | D | | - | 5 42 | F | | - | 1 14 | G | = | | | 499 2800 | | (4) |
| | E | | - | 5 14 | F | | - | 3 14 | G | = | | | 1497 2800 | | (5) |
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Divide both sides of equation (3) by 3, get the equation: | | 3 | C | | - | 3 14 | F | + | | | 15 14 | G | = | | | 789 2800 | (17) | , then subtract both sides of equation (17) from both sides of equation (2), get the equation:
| | | | 6 | A | + | | 6 | B | + | | 12 | C | + | | | 57 7 | F | + | | | 51 7 | G | = | | | 6051 1400 | | (1) |
| -1 | B | | - | 8 7 | F | | - | 2 7 | G | = | | - | 39 70 | | (2) |
| | 9 | C | | - | 9 14 | F | + | | | 45 14 | G | = | | | 2367 2800 | | (3) |
| | D | | - | 5 14 | F | | - | 3 14 | G | = | | | 1497 2800 | | (4) |
| | E | | - | 5 14 | F | | - | 3 14 | G | = | | | 1497 2800 | | (5) |
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Multiply both sides of equation (3) by 4 Divide both sides of equation (3) by 3, get the equation: | | 12 | C | | - | 6 7 | F | + | | | 30 7 | G | = | | | 789 700 | (18) | , then subtract both sides of equation (18) from both sides of equation (1), get the equation:
| | | | 6 | A | + | | 6 | B | + | | 9 | F | + | | 3 | G | = | | | 639 200 | | (1) |
| -1 | B | | - | 8 7 | F | | - | 2 7 | G | = | | - | 39 70 | | (2) |
| | 9 | C | | - | 9 14 | F | + | | | 45 14 | G | = | | | 2367 2800 | | (3) |
| | D | | - | 5 14 | F | | - | 3 14 | G | = | | | 1497 2800 | | (4) |
| | E | | - | 5 14 | F | | - | 3 14 | G | = | | | 1497 2800 | | (5) |
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Multiply both sides of equation (2) by 6, get the equation: | -6 | B | | - | 48 7 | F | | - | 12 7 | G | = | | - | 117 35 | (19) | , then add the two sides of equation (19) to both sides of equation (1), get the equation:
| | | | 6 | A | + | | | 15 7 | F | + | | | 9 7 | G | = | | - | 207 1400 | | (1) |
| -1 | B | | - | 8 7 | F | | - | 2 7 | G | = | | - | 39 70 | | (2) |
| | C | | - | 1 14 | F | + | | | 5 14 | G | = | | | 263 2800 | | (3) |
| | D | | - | 5 14 | F | | - | 3 14 | G | = | | | 1497 2800 | | (4) |
| | E | | - | 5 14 | F | | - | 3 14 | G | = | | | 1497 2800 | | (5) |
| | |
The coefficient of the unknown number is reduced to 1, and the equations are reduced to:
| | | | A | + | | | 5 14 | F | + | | | 3 14 | G | = | | - | 69 2800 | | (1) |
| | | C | | - | 1 14 | F | + | | | 5 14 | G | = | | | 263 2800 | | (3) |
| | D | | - | 5 14 | F | | - | 3 14 | G | = | | | 1497 2800 | | (4) |
| | E | | - | 5 14 | F | | - | 3 14 | G | = | | | 1497 2800 | | (5) |
| | |
Therefore, the solution of the equation set is:
| | | A = | | - | 69 2800 | - | | | 5 14 | F | - | | | 3 14 | G |
| | C = | | | 263 2800 | + | | | 1 14 | F | - | | | 5 14 | G |
| D = | | | 1497 2800 | + | | | 5 14 | F | + | | | 3 14 | G |
| E = | | | 1497 2800 | + | | | 5 14 | F | + | | | 3 14 | G |
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Convert the solution of the equation set to decimals:
| | | A = | | -0.024643 | - | | | 5 14 | F | - | | | 3 14 | G |
| | C = | | 0.093929 | + | | | 1 14 | F | - | | | 5 14 | G |
| D = | | 0.534643 | + | | | 5 14 | F | + | | | 3 14 | G |
| E = | | 0.534643 | + | | | 5 14 | F | + | | | 3 14 | G |
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Where: F, G 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》 |