detailed information: The input equation set is:
| | | | 4 | A | + | | 2 | B | + | | 3 | C | + | | 4 | D | + | | 5 | E | = | | 9 | | (1) |
| | 5 | A | | -2 | B | + | | 3 | C | | -6 | D | + | | 5 | E | = | | 7 | | (2) |
| | 2 | A | + | | 3 | B | + | | 7 | C | + | | 2 | D | + | | 3 | E | = | | 8 | | (3) |
| | | Question solving process:
Multiply both sides of equation (1) by 5 Divide the two sides of equation (1) by 4, the equation can be obtained: | | 5 | A | + | | | 5 2 | B | + | | | 15 4 | C | + | | 5 | D | + | | | 25 4 | E | = | | | 45 4 | (6) | , then subtract both sides of equation (6) from both sides of equation (2), the equations are reduced to:
| | | | 4 | A | + | | 2 | B | + | | 3 | C | + | | 4 | D | + | | 5 | E | = | | 9 | | (1) |
| - | 9 2 | B | | - | 3 4 | C | | -11 | D | | - | 5 4 | E | = | | - | 17 4 | | (2) |
| | 2 | A | + | | 3 | B | + | | 7 | C | + | | 2 | D | + | | 3 | E | = | | 8 | | (3) |
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Divide the two sides of equation (1) by 2, the equation can be obtained: | | 2 | A | + | | B | + | | | 3 2 | C | + | | 2 | D | + | | | 5 2 | E | = | | | 9 2 | (7) | , then subtract both sides of equation (7) from both sides of equation (3), the equations are reduced to:
| | | | 4 | A | + | | 2 | B | + | | 3 | C | + | | 4 | D | + | | 5 | E | = | | 9 | | (1) |
| - | 9 2 | B | | - | 3 4 | C | | -11 | D | | - | 5 4 | E | = | | - | 17 4 | | (2) |
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Multiply both sides of equation (2) by 4 Divide the two sides of equation (2) by 9, the equation can be obtained: | -2 | B | | - | 1 3 | C | | - | 44 9 | D | | - | 5 9 | E | = | | - | 17 9 | (8) | , then add the two sides of equation (8) to both sides of equation (3), the equations are reduced to:
| | | | 4 | A | + | | 2 | B | + | | 3 | C | + | | 4 | D | + | | 5 | E | = | | 9 | | (1) |
| - | 9 2 | B | | - | 3 4 | C | | -11 | D | | - | 5 4 | E | = | | - | 17 4 | | (2) |
| | | 31 6 | C | | - | 44 9 | D | | - | 1 18 | E | = | | | 29 18 | | (3) |
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Multiply both sides of equation (3) by 9 Divide both sides of equation (3) by 62, get the equation: | | | 3 4 | C | | - | 22 31 | D | | - | 1 124 | E | = | | | 29 124 | (9) | , then add the two sides of equation (9) to both sides of equation (2), get the equation:
| | | | 4 | A | + | | 2 | B | + | | 3 | C | + | | 4 | D | + | | 5 | E | = | | 9 | | (1) |
| - | 9 2 | B | | - | 363 31 | D | | - | 39 31 | E | = | | - | 249 62 | | (2) |
| | | 31 6 | C | | - | 44 9 | D | | - | 1 18 | E | = | | | 29 18 | | (3) |
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Multiply both sides of equation (3) by 18 Divide both sides of equation (3) by 31, get the equation: | | 3 | C | | - | 88 31 | D | | - | 1 31 | E | = | | | 29 31 | (10) | , then subtract both sides of equation (10) from both sides of equation (1), get the equation:
| | | | 4 | A | + | | 2 | B | + | | | 212 31 | D | + | | | 156 31 | E | = | | | 250 31 | | (1) |
| - | 9 2 | B | | - | 363 31 | D | | - | 39 31 | E | = | | - | 249 62 | | (2) |
| | | 31 6 | C | | - | 44 9 | D | | - | 1 18 | E | = | | | 29 18 | | (3) |
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Multiply both sides of equation (2) by 4 Divide both sides of equation (2) by 9, get the equation: | -2 | B | | - | 484 93 | D | | - | 52 93 | E | = | | - | 166 93 | (11) | , then add the two sides of equation (11) to both sides of equation (1), get the equation:
| | | | 4 | A | + | | | 152 93 | D | + | | | 416 93 | E | = | | | 584 93 | | (1) |
| - | 9 2 | B | | - | 363 31 | D | | - | 39 31 | E | = | | - | 249 62 | | (2) |
| | C | | - | 88 93 | D | | - | 1 93 | E | = | | | 29 93 | | (3) |
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The coefficient of the unknown number is reduced to 1, and the equations are reduced to:
| | | | A | + | | | 38 93 | D | + | | | 104 93 | E | = | | | 146 93 | | (1) |
| | B | + | | | 242 93 | D | + | | | 26 93 | E | = | | | 83 93 | | (2) |
| | C | | - | 88 93 | D | | - | 1 93 | E | = | | | 29 93 | | (3) |
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Therefore, the solution of the equation set is:
| | | A = | | | 146 93 | - | | | 38 93 | D | - | | | 104 93 | E |
| B = | | | 83 93 | - | | | 242 93 | D | - | | | 26 93 | E |
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Convert the solution of the equation set to decimals:
| | | A = | | 1.569892 | - | | | 38 93 | D | - | | | 104 93 | E |
| B = | | 0.892473 | - | | | 242 93 | D | - | | | 26 93 | E |
| C = | | 0.311828 | + | | | 88 93 | D | + | | | 1 93 | E |
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Where: D, E 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》 |