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On line solution of multivariate equations：
    First set the elements of the equation (i.e. the number of unknowns), then click the "Next" button to enter the coefficients of each element of the equation set, and click the "Next" button to obtain the solution of the equation set.
    Note that the coefficients of each element of the equation system can only be numbers, not algebraic expressions (including mathematical functions).
    Current location：Equations > Multivariate equations > Answer
detailed information：
The input equation set is:
 A + B + C + D + E + F = 2    (1)
 32A + 16B + 8C + 4D + 2E + F = 4    (2)
 243A + 81B + 27C + 9D + 3E + F = 6    (3)
 1024A + 256B + 64C + 16D + 4E + F = 7    (4)
 3125A + 625B + 125C + 25D + 5E + F = 8    (5)
0 = 0    (6)
Question solving process:

Multiply both sides of equation (1) by 32, the equation can be obtained:
         32A + 32B + 32C + 32D + 32E + 32F = 64    (7)
, then subtract both sides of equation (7) from both sides of equation (2), the equations are reduced to:
 A + B + C + D + E + F = 2    (1)
-16B -24C -28D -30E -31F = -60    (2)
 243A + 81B + 27C + 9D + 3E + F = 6    (3)
 1024A + 256B + 64C + 16D + 4E + F = 7    (4)
 3125A + 625B + 125C + 25D + 5E + F = 8    (5)
0 = 0    (6)

Multiply both sides of equation (1) by 243, the equation can be obtained:
         243A + 243B + 243C + 243D + 243E + 243F = 486    (8)
, then subtract both sides of equation (8) from both sides of equation (3), the equations are reduced to:
 A + B + C + D + E + F = 2    (1)
-16B -24C -28D -30E -31F = -60    (2)
-162B -216C -234D -240E -242F = -480    (3)
 1024A + 256B + 64C + 16D + 4E + F = 7    (4)
 3125A + 625B + 125C + 25D + 5E + F = 8    (5)
0 = 0    (6)

Multiply both sides of equation (1) by 1024, the equation can be obtained:
         1024A + 1024B + 1024C + 1024D + 1024E + 1024F = 2048    (9)
, then subtract both sides of equation (9) from both sides of equation (4), the equations are reduced to:
 A + B + C + D + E + F = 2    (1)
-16B -24C -28D -30E -31F = -60    (2)
-162B -216C -234D -240E -242F = -480    (3)
-768B -960C -1008D -1020E -1023F = -2041    (4)
 3125A + 625B + 125C + 25D + 5E + F = 8    (5)
0 = 0    (6)

Multiply both sides of equation (1) by 3125, the equation can be obtained:
         3125A + 3125B + 3125C + 3125D + 3125E + 3125F = 6250    (10)
, then subtract both sides of equation (10) from both sides of equation (5), the equations are reduced to:
 A + B + C + D + E + F = 2    (1)
-16B -24C -28D -30E -31F = -60    (2)
-162B -216C -234D -240E -242F = -480    (3)
-768B -960C -1008D -1020E -1023F = -2041    (4)
-2500B -3000C -3100D -3120E -3124F = -6242    (5)
0 = 0    (6)

Multiply both sides of equation (2) by 81
Divide the two sides of equation (2) by 8, the equation can be obtained:
        -162B -243C - 
567
2
D - 
1215
4
E - 
2511
8
F = - 
1215
2
    (11)
, then subtract both sides of equation (11) from both sides of equation (3), the equations are reduced to:
 A + B + C + D + E + F = 2    (1)
-16B -24C -28D -30E -31F = -60    (2)
 27C + 
99
2
D + 
255
4
E + 
575
8
F = 
255
2
    (3)
-768B -960C -1008D -1020E -1023F = -2041    (4)
-2500B -3000C -3100D -3120E -3124F = -6242    (5)
0 = 0    (6)

Multiply both sides of equation (2) by 48, the equation can be obtained:
        -768B -1152C -1344D -1440E -1488F = -2880    (12)
, then subtract both sides of equation (12) from both sides of equation (4), the equations are reduced to:
 A + B + C + D + E + F = 2    (1)
-16B -24C -28D -30E -31F = -60    (2)
 27C + 
99
2
D + 
255
4
E + 
575
8
F = 
255
2
    (3)
 192C + 336D + 420E + 465F = 839    (4)
-2500B -3000C -3100D -3120E -3124F = -6242    (5)
0 = 0    (6)

Multiply both sides of equation (2) by 625
Divide the two sides of equation (2) by 4, the equation can be obtained:
        -2500B -3750C -4375D - 
9375
2
E - 
19375
4
F = -9375    (13)
, then subtract both sides of equation (13) from both sides of equation (5), the equations are reduced to:
 A + B + C + D + E + F = 2    (1)
-16B -24C -28D -30E -31F = -60    (2)
 27C + 
99
2
D + 
255
4
E + 
575
8
F = 
255
2
    (3)
 192C + 336D + 420E + 465F = 839    (4)
 750C + 1275D + 
3135
2
E + 
6879
4
F = 3133    (5)
0 = 0    (6)

Multiply both sides of equation (3) by 64
Divide the two sides of equation (3) by 9, the equation can be obtained:
         192C + 352D + 
1360
3
E + 
4600
9
F = 
2720
3
    (14)
, then subtract both sides of equation (14) from both sides of equation (4), the equations are reduced to:
 A + B + C + D + E + F = 2    (1)
-16B -24C -28D -30E -31F = -60    (2)
 27C + 
99
2
D + 
255
4
E + 
575
8
F = 
255
2
    (3)
-16D - 
100
3
E - 
415
9
F = - 
203
3
    (4)
 750C + 1275D + 
3135
2
E + 
6879
4
F = 3133    (5)
0 = 0    (6)

Multiply both sides of equation (3) by 250
Divide the two sides of equation (3) by 9, the equation can be obtained:
         750C + 1375D + 
10625
6
E + 
71875
36
F = 
10625
3
    (15)
, then subtract both sides of equation (15) from both sides of equation (5), the equations are reduced to:
 A + B + C + D + E + F = 2    (1)
-16B -24C -28D -30E -31F = -60    (2)
 27C + 
99
2
D + 
255
4
E + 
575
8
F = 
255
2
    (3)
-16D - 
100
3
E - 
415
9
F = - 
203
3
    (4)
-100D - 
610
3
E - 
2491
9
F = - 
1226
3
    (5)
0 = 0    (6)

Multiply both sides of equation (4) by 25
Divide the two sides of equation (4) by 4, the equation can be obtained:
        -100D - 
625
3
E - 
10375
36
F = - 
5075
12
    (16)
, then subtract both sides of equation (16) from both sides of equation (5), the equations are reduced to:
 A + B + C + D + E + F = 2    (1)
-16B -24C -28D -30E -31F = -60    (2)
 27C + 
99
2
D + 
255
4
E + 
575
8
F = 
255
2
    (3)
-16D - 
100
3
E - 
415
9
F = - 
203
3
    (4)
 5E + 
137
12
F = 
57
4
    (5)
0 = 0    (6)

Multiply both sides of equation (5) by 20
Divide both sides of equation (5) by 3, get the equation：
         
100
3
E + 
685
9
F = 95    (17)
, then add the two sides of equation (17) to both sides of equation (4), get the equation：
 A + B + C + D + E + F = 2    (1)
-16B -24C -28D -30E -31F = -60    (2)
 27C + 
99
2
D + 
255
4
E + 
575
8
F = 
255
2
    (3)
-16D + 30F = 
82
3
    (4)
 5E + 
137
12
F = 
57
4
    (5)
0 = 0    (6)

Multiply both sides of equation (5) by 51
Divide both sides of equation (5) by 4, get the equation：
         
255
4
E + 
2329
16
F = 
2907
16
    (18)
, then subtract both sides of equation (18) from both sides of equation (3), get the equation：
 A + B + C + D + E + F = 2    (1)
-16B -24C -28D -30E -31F = -60    (2)
 27C + 
99
2
D - 
1179
16
F = - 
867
16
    (3)
-16D + 30F = 
82
3
    (4)
 5E + 
137
12
F = 
57
4
    (5)
0 = 0    (6)

Multiply both sides of equation (5) by 6, get the equation：
         30E + 
137
2
F = 
171
2
    (19)
, then add the two sides of equation (19) to both sides of equation (2), get the equation：
 A + B + C + D + E + F = 2    (1)
-16B -24C -28D + 
75
2
F = 
51
2
    (2)
 27C + 
99
2
D - 
1179
16
F = - 
867
16
    (3)
-16D + 30F = 
82
3
    (4)
 5E + 
137
12
F = 
57
4
    (5)
0 = 0    (6)

Divide both sides of equation (5) by 5, get the equation：
         E + 
137
60
F = 
57
20
    (20)
, then subtract both sides of equation (20) from both sides of equation (1), get the equation：
 A + B + C + D - 
77
60
F = - 
17
20
    (1)
-16B -24C -28D + 
75
2
F = 
51
2
    (2)
 27C + 
99
2
D - 
1179
16
F = - 
867
16
    (3)
-16D + 30F = 
82
3
    (4)
 5E + 
137
12
F = 
57
4
    (5)
0 = 0    (6)

Multiply both sides of equation (4) by 99
Divide both sides of equation (4) by 32, get the equation：
        - 
99
2
D + 
1485
16
F = 
1353
16
    (21)
, then add the two sides of equation (21) to both sides of equation (3), get the equation：
 A + B + C + D - 
77
60
F = - 
17
20
    (1)
-16B -24C -28D + 
75
2
F = 
51
2
    (2)
 27C + 
153
8
F = 
243
8
    (3)
-16D + 30F = 
82
3
    (4)
 E + 
137
60
F = 
57
20
    (5)
0 = 0    (6)

Multiply both sides of equation (4) by 7
Divide both sides of equation (4) by 4, get the equation：
        -28D + 
105
2
F = 
287
6
    (22)
, then subtract both sides of equation (22) from both sides of equation (2), get the equation：
 A + B + C + D - 
77
60
F = - 
17
20
    (1)
-16B -24C -15F = - 
67
3
    (2)
 27C + 
153
8
F = 
243
8
    (3)
-16D + 30F = 
82
3
    (4)
 E + 
137
60
F = 
57
20
    (5)
0 = 0    (6)

Divide both sides of equation (4) by 16, get the equation：
        -1D + 
15
8
F = 
41
24
    (23)
, then add the two sides of equation (23) to both sides of equation (1), get the equation：
 A + B + C + 
71
120
F = 
103
120
    (1)
-16B -24C -15F = - 
67
3
    (2)
 27C + 
153
8
F = 
243
8
    (3)
-16D + 30F = 
82
3
    (4)
 E + 
137
60
F = 
57
20
    (5)
0 = 0    (6)

Multiply both sides of equation (3) by 8
Divide both sides of equation (3) by 9, get the equation：
         24C + 17F = 27    (24)
, then add the two sides of equation (24) to both sides of equation (2), get the equation：
 A + B + C + 
71
120
F = 
103
120
    (1)
-16B + 2F = 
14
3
    (2)
 27C + 
153
8
F = 
243
8
    (3)
 D - 
15
8
F = - 
41
24
    (4)
 E + 
137
60
F = 
57
20
    (5)
0 = 0    (6)

Divide both sides of equation (3) by 27, get the equation：
         C + 
17
24
F = 
9
8
    (25)
, then subtract both sides of equation (25) from both sides of equation (1), get the equation：
 A + B - 
7
60
F = - 
4
15
    (1)
-16B + 2F = 
14
3
    (2)
 27C + 
153
8
F = 
243
8
    (3)
 D - 
15
8
F = - 
41
24
    (4)
 E + 
137
60
F = 
57
20
    (5)
0 = 0    (6)

Divide both sides of equation (2) by 16, get the equation：
        -1B + 
1
8
F = 
7
24
    (26)
, then add the two sides of equation (26) to both sides of equation (1), get the equation：
 A + 
1
120
F = 
1
40
    (1)
-16B + 2F = 
14
3
    (2)
 C + 
17
24
F = 
9
8
    (3)
 D - 
15
8
F = - 
41
24
    (4)
 E + 
137
60
F = 
57
20
    (5)
0 = 0    (6)

The coefficient of the unknown number is reduced to 1, and the equations are reduced to：
 A + 
1
120
F = 
1
40
    (1)
 B - 
1
8
F = - 
7
24
    (2)
 C + 
17
24
F = 
9
8
    (3)
 D - 
15
8
F = - 
41
24
    (4)
 E + 
137
60
F = 
57
20
    (5)
0 = 0    (6)


Therefore, the solution of the equation set is:
A = 
1
40
 - 
1
120
F
B = - 
7
24
 + 
1
8
F
C = 
9
8
 - 
17
24
F
D = - 
41
24
 + 
15
8
F
E = 
57
20
 - 
137
60
F


Convert the solution of the equation set to decimals:
A = 0.025000 - 
1
120
F
B = -0.291667 + 
1
8
F
C = 1.125000 - 
17
24
F
D = -1.708333 + 
15
8
F
E = 2.850000 - 
137
60
F

Where:  F 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》
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