detailed information: The input equation set is:
| | | | A | + | | B | + | | C | + | | D | + | | E | = | | 0 | | (1) |
| | 1024 | A | + | | 256 | B | + | | 64 | C | + | | 16 | D | + | | 4 | E | = | | 0 | | (2) |
| | 243 | A | + | | 81 | B | + | | 27 | C | + | | 9 | D | + | | 3 | E | = | | 0 | | (3) |
| | 7776 | A | + | | 1296 | B | + | | 216 | C | + | | 36 | D | + | | 6 | E | = | | 0 | | (4) |
| | 3125 | A | + | | 625 | B | + | | 125 | C | + | | 25 | D | + | | 5 | E | = | | 0 | | (5) |
| Question solving process:
Multiply both sides of equation (1) by 1024, the equation can be obtained: | | 1024 | A | + | | 1024 | B | + | | 1024 | C | + | | 1024 | D | + | | 1024 | E | = | | 0 | (6) | , then subtract both sides of equation (6) from both sides of equation (2), the equations are reduced to:
| | | | A | + | | B | + | | C | + | | D | + | | E | = | | 0 | | (1) |
| -768 | B | | -960 | C | | -1008 | D | | -1020 | E | = | | 0 | | (2) |
| | 243 | A | + | | 81 | B | + | | 27 | C | + | | 9 | D | + | | 3 | E | = | | 0 | | (3) |
| | 7776 | A | + | | 1296 | B | + | | 216 | C | + | | 36 | D | + | | 6 | E | = | | 0 | | (4) |
| | 3125 | A | + | | 625 | B | + | | 125 | C | + | | 25 | D | + | | 5 | E | = | | 0 | | (5) |
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Multiply both sides of equation (1) by 243, the equation can be obtained: | | 243 | A | + | | 243 | B | + | | 243 | C | + | | 243 | D | + | | 243 | E | = | | 0 | (7) | , then subtract both sides of equation (7) from both sides of equation (3), the equations are reduced to:
| | | | A | + | | B | + | | C | + | | D | + | | E | = | | 0 | | (1) |
| -768 | B | | -960 | C | | -1008 | D | | -1020 | E | = | | 0 | | (2) |
| -162 | B | | -216 | C | | -234 | D | | -240 | E | = | | 0 | | (3) |
| | 7776 | A | + | | 1296 | B | + | | 216 | C | + | | 36 | D | + | | 6 | E | = | | 0 | | (4) |
| | 3125 | A | + | | 625 | B | + | | 125 | C | + | | 25 | D | + | | 5 | E | = | | 0 | | (5) |
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Multiply both sides of equation (1) by 7776, the equation can be obtained: | | 7776 | A | + | | 7776 | B | + | | 7776 | C | + | | 7776 | D | + | | 7776 | E | = | | 0 | (8) | , then subtract both sides of equation (8) from both sides of equation (4), the equations are reduced to:
| | | | A | + | | B | + | | C | + | | D | + | | E | = | | 0 | | (1) |
| -768 | B | | -960 | C | | -1008 | D | | -1020 | E | = | | 0 | | (2) |
| -162 | B | | -216 | C | | -234 | D | | -240 | E | = | | 0 | | (3) |
| -6480 | B | | -7560 | C | | -7740 | D | | -7770 | E | = | | 0 | | (4) |
| | 3125 | A | + | | 625 | B | + | | 125 | C | + | | 25 | D | + | | 5 | E | = | | 0 | | (5) |
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Multiply both sides of equation (1) by 3125, the equation can be obtained: | | 3125 | A | + | | 3125 | B | + | | 3125 | C | + | | 3125 | D | + | | 3125 | E | = | | 0 | (9) | , then subtract both sides of equation (9) from both sides of equation (5), the equations are reduced to:
| | | | A | + | | B | + | | C | + | | D | + | | E | = | | 0 | | (1) |
| -768 | B | | -960 | C | | -1008 | D | | -1020 | E | = | | 0 | | (2) |
| -162 | B | | -216 | C | | -234 | D | | -240 | E | = | | 0 | | (3) |
| -6480 | B | | -7560 | C | | -7740 | D | | -7770 | E | = | | 0 | | (4) |
| -2500 | B | | -3000 | C | | -3100 | D | | -3120 | E | = | | 0 | | (5) |
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Multiply both sides of equation (2) by 27 Divide the two sides of equation (2) by 128, the equation can be obtained: | -162 | B | | - | 405 2 | C | | - | 1701 8 | D | | - | 6885 32 | E | = | | 0 | (10) | , then subtract both sides of equation (10) from both sides of equation (3), the equations are reduced to:
| | | | A | + | | B | + | | C | + | | D | + | | E | = | | 0 | | (1) |
| -768 | B | | -960 | C | | -1008 | D | | -1020 | E | = | | 0 | | (2) |
| - | 27 2 | C | | - | 171 8 | D | | - | 795 32 | E | = | | 0 | | (3) |
| -6480 | B | | -7560 | C | | -7740 | D | | -7770 | E | = | | 0 | | (4) |
| -2500 | B | | -3000 | C | | -3100 | D | | -3120 | E | = | | 0 | | (5) |
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Multiply both sides of equation (2) by 135 Divide the two sides of equation (2) by 16, the equation can be obtained: | -6480 | B | | -8100 | C | | -8505 | D | | - | 34425 4 | E | = | | 0 | (11) | , then subtract both sides of equation (11) from both sides of equation (4), the equations are reduced to:
| | | | A | + | | B | + | | C | + | | D | + | | E | = | | 0 | | (1) |
| -768 | B | | -960 | C | | -1008 | D | | -1020 | E | = | | 0 | | (2) |
| - | 27 2 | C | | - | 171 8 | D | | - | 795 32 | E | = | | 0 | | (3) |
| | 540 | C | + | | 765 | D | + | | | 3345 4 | E | = | | 0 | | (4) |
| -2500 | B | | -3000 | C | | -3100 | D | | -3120 | E | = | | 0 | | (5) |
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Multiply both sides of equation (2) by 625 Divide the two sides of equation (2) by 192, the equation can be obtained: | -2500 | B | | -3125 | C | | - | 13125 4 | D | | - | 53125 16 | E | = | | 0 | (12) | , then subtract both sides of equation (12) from both sides of equation (5), the equations are reduced to:
| | | | A | + | | B | + | | C | + | | D | + | | E | = | | 0 | | (1) |
| -768 | B | | -960 | C | | -1008 | D | | -1020 | E | = | | 0 | | (2) |
| - | 27 2 | C | | - | 171 8 | D | | - | 795 32 | E | = | | 0 | | (3) |
| | 540 | C | + | | 765 | D | + | | | 3345 4 | E | = | | 0 | | (4) |
| | 125 | C | + | | | 725 4 | D | + | | | 3205 16 | E | = | | 0 | | (5) |
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Multiply both sides of equation (3) by 40, the equation can be obtained: | -540 | C | | -855 | D | | - | 3975 4 | E | = | | 0 | (13) | , then add the two sides of equation (13) to both sides of equation (4), the equations are reduced to:
| | | | A | + | | B | + | | C | + | | D | + | | E | = | | 0 | | (1) |
| -768 | B | | -960 | C | | -1008 | D | | -1020 | E | = | | 0 | | (2) |
| - | 27 2 | C | | - | 171 8 | D | | - | 795 32 | E | = | | 0 | | (3) |
| | | 125 | C | + | | | 725 4 | D | + | | | 3205 16 | E | = | | 0 | | (5) |
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Multiply both sides of equation (3) by 250 Divide the two sides of equation (3) by 27, the equation can be obtained: | -125 | C | | - | 2375 12 | D | | - | 33125 144 | E | = | | 0 | (14) | , then add the two sides of equation (14) to both sides of equation (5), the equations are reduced to:
| | | | A | + | | B | + | | C | + | | D | + | | E | = | | 0 | | (1) |
| -768 | B | | -960 | C | | -1008 | D | | -1020 | E | = | | 0 | | (2) |
| - | 27 2 | C | | - | 171 8 | D | | - | 795 32 | E | = | | 0 | | (3) |
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Multiply both sides of equation (4) by 5 Divide the two sides of equation (4) by 27, the equation can be obtained: , then subtract both sides of equation (15) from both sides of equation (5), the equations are reduced to:
| | | | A | + | | B | + | | C | + | | D | + | | E | = | | 0 | | (1) |
| -768 | B | | -960 | C | | -1008 | D | | -1020 | E | = | | 0 | | (2) |
| - | 27 2 | C | | - | 171 8 | D | | - | 795 32 | E | = | | 0 | | (3) |
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Multiply both sides of equation (5) by 567 Divide both sides of equation (5) by 2, get the equation:, then subtract both sides of equation (16) from both sides of equation (4), get the equation:
| | | | A | + | | B | + | | C | + | | D | + | | E | = | | 0 | | (1) |
| -768 | B | | -960 | C | | -1008 | D | | -1020 | E | = | | 0 | | (2) |
| - | 27 2 | C | | - | 171 8 | D | | - | 795 32 | E | = | | 0 | | (3) |
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Multiply both sides of equation (5) by 1431 Divide both sides of equation (5) by 32, get the equation:, then subtract both sides of equation (17) from both sides of equation (3), get the equation:
| | | | A | + | | B | + | | C | + | | D | + | | E | = | | 0 | | (1) |
| -768 | B | | -960 | C | | -1008 | D | | -1020 | E | = | | 0 | | (2) |
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Multiply both sides of equation (5) by 1836, get the equation:, then subtract both sides of equation (18) from both sides of equation (2), get the equation:
| | | | A | + | | B | + | | C | + | | D | + | | E | = | | 0 | | (1) |
| -768 | B | | -960 | C | | -1008 | D | = | | 0 | | (2) |
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Multiply both sides of equation (5) by 9 Divide both sides of equation (5) by 5, get the equation:, then add the two sides of equation (19) to both sides of equation (1), get the equation:
| | | | -768 | B | | -960 | C | | -1008 | D | = | | 0 | | (2) |
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Multiply both sides of equation (4) by 19 Divide both sides of equation (4) by 80, get the equation:, then subtract both sides of equation (20) from both sides of equation (3), get the equation:
| | | | -768 | B | | -960 | C | | -1008 | D | = | | 0 | | (2) |
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Multiply both sides of equation (4) by 56 Divide both sides of equation (4) by 5, get the equation:, then subtract both sides of equation (21) from both sides of equation (2), get the equation:
Divide both sides of equation (4) by 90, get the equation:, then add the two sides of equation (22) to both sides of equation (1), get the equation:
Multiply both sides of equation (3) by 640 Divide both sides of equation (3) by 9, get the equation:, then subtract both sides of equation (23) from both sides of equation (2), get the equation:
Multiply both sides of equation (3) by 2 Divide both sides of equation (3) by 27, get the equation:, then add the two sides of equation (24) to both sides of equation (1), get the equation:
Divide both sides of equation (2) by 768, get the equation:, then add the two sides of equation (25) to both sides of equation (1), get the equation:
The coefficient of the unknown number is reduced to 1, and the equations are reduced to:
Therefore, the solution of the equation set is:
解方程组的详细方法请参阅:《多元一次方程组的解法》 |