Mathematics
         
语言:中文    Language:English
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 -2B -3E + 2F -2G = -4    (1)
 2A + B -2C + 3F -2G = 0    (2)
 A -2B + C + D -1E + 2F -1G = 1    (3)
-2A -2B + C + D + 2E -3F -3G = -1    (4)
 2A + 3B + 2C -2D + E = 1    (5)
-2B -2C + 3D -2E + F + 3G = -4    (6)
-3A -2B -2C -3D + 3E + F + G = -1    (7)
Question solving process:

Multiply both sides of equation (1) by 2, the equation can be obtained:
         2A -4B -6E + 4F -4G = -8    (8)
, then subtract both sides of equation (8) from both sides of equation (2), the equations are reduced to:
 A -2B -3E + 2F -2G = -4    (1)
 5B -2C + 6E -1F + 2G = 8    (2)
 A -2B + C + D -1E + 2F -1G = 1    (3)
-2A -2B + C + D + 2E -3F -3G = -1    (4)
 2A + 3B + 2C -2D + E = 1    (5)
-2B -2C + 3D -2E + F + 3G = -4    (6)
-3A -2B -2C -3D + 3E + F + G = -1    (7)

Subtract both sides of equation (1) from both sides of equation (3) ,the equations are reduced to:
 A -2B -3E + 2F -2G = -4    (1)
 5B -2C + 6E -1F + 2G = 8    (2)
 C + D + 2E + G = 5    (3)
-2A -2B + C + D + 2E -3F -3G = -1    (4)
 2A + 3B + 2C -2D + E = 1    (5)
-2B -2C + 3D -2E + F + 3G = -4    (6)
-3A -2B -2C -3D + 3E + F + G = -1    (7)

Multiply both sides of equation (1) by 2, the equation can be obtained:
         2A -4B -6E + 4F -4G = -8    (9)
, then add the two sides of equation (9) to both sides of equation (4), the equations are reduced to:
 A -2B -3E + 2F -2G = -4    (1)
 5B -2C + 6E -1F + 2G = 8    (2)
 C + D + 2E + G = 5    (3)
-6B + C + D -4E + F -7G = -9    (4)
 2A + 3B + 2C -2D + E = 1    (5)
-2B -2C + 3D -2E + F + 3G = -4    (6)
-3A -2B -2C -3D + 3E + F + G = -1    (7)

Multiply both sides of equation (1) by 2, the equation can be obtained:
         2A -4B -6E + 4F -4G = -8    (10)
, then subtract both sides of equation (10) from both sides of equation (5), the equations are reduced to:
 A -2B -3E + 2F -2G = -4    (1)
 5B -2C + 6E -1F + 2G = 8    (2)
 C + D + 2E + G = 5    (3)
-6B + C + D -4E + F -7G = -9    (4)
 7B + 2C -2D + 7E -4F + 4G = 9    (5)
-2B -2C + 3D -2E + F + 3G = -4    (6)
-3A -2B -2C -3D + 3E + F + G = -1    (7)

Multiply both sides of equation (1) by 3, the equation can be obtained:
         3A -6B -9E + 6F -6G = -12    (11)
, then add the two sides of equation (11) to both sides of equation (7), the equations are reduced to:
 A -2B -3E + 2F -2G = -4    (1)
 5B -2C + 6E -1F + 2G = 8    (2)
 C + D + 2E + G = 5    (3)
-6B + C + D -4E + F -7G = -9    (4)
 7B + 2C -2D + 7E -4F + 4G = 9    (5)
-2B -2C + 3D -2E + F + 3G = -4    (6)
-8B -2C -3D -6E + 7F -5G = -13    (7)

Multiply both sides of equation (2) by 6
Divide the two sides of equation (2) by 5, the equation can be obtained:
         6B 
12
5
C + 
36
5
E 
6
5
F + 
12
5
G = 
48
5
    (12)
, then add the two sides of equation (12) to both sides of equation (4), the equations are reduced to:
 A -2B -3E + 2F -2G = -4    (1)
 5B -2C + 6E -1F + 2G = 8    (2)
 C + D + 2E + G = 5    (3)
7
5
C + D + 
16
5
E 
1
5
F 
23
5
G = 
3
5
    (4)
 7B + 2C -2D + 7E -4F + 4G = 9    (5)
-2B -2C + 3D -2E + F + 3G = -4    (6)
-8B -2C -3D -6E + 7F -5G = -13    (7)

Multiply both sides of equation (2) by 7
Divide the two sides of equation (2) by 5, the equation can be obtained:
         7B 
14
5
C + 
42
5
E 
7
5
F + 
14
5
G = 
56
5
    (13)
, then subtract both sides of equation (13) from both sides of equation (5), the equations are reduced to:
 A -2B -3E + 2F -2G = -4    (1)
 5B -2C + 6E -1F + 2G = 8    (2)
 C + D + 2E + G = 5    (3)
7
5
C + D + 
16
5
E 
1
5
F 
23
5
G = 
3
5
    (4)
 
24
5
C -2D 
7
5
E 
13
5
F + 
6
5
G = 
11
5
    (5)
-2B -2C + 3D -2E + F + 3G = -4    (6)
-8B -2C -3D -6E + 7F -5G = -13    (7)

Multiply both sides of equation (2) by 2
Divide the two sides of equation (2) by 5, the equation can be obtained:
         2B 
4
5
C + 
12
5
E 
2
5
F + 
4
5
G = 
16
5
    (14)
, then add the two sides of equation (14) to both sides of equation (6), the equations are reduced to:
 A -2B -3E + 2F -2G = -4    (1)
 5B -2C + 6E -1F + 2G = 8    (2)
 C + D + 2E + G = 5    (3)
7
5
C + D + 
16
5
E 
1
5
F 
23
5
G = 
3
5
    (4)
 
24
5
C -2D 
7
5
E 
13
5
F + 
6
5
G = 
11
5
    (5)
14
5
C + 3D + 
2
5
E + 
3
5
F + 
19
5
G = 
4
5
    (6)
-8B -2C -3D -6E + 7F -5G = -13    (7)

Multiply both sides of equation (2) by 8
Divide the two sides of equation (2) by 5, the equation can be obtained:
         8B 
16
5
C + 
48
5
E 
8
5
F + 
16
5
G = 
64
5
    (15)
, then add the two sides of equation (15) to both sides of equation (7), the equations are reduced to:
 A -2B -3E + 2F -2G = -4    (1)
 5B -2C + 6E -1F + 2G = 8    (2)
 C + D + 2E + G = 5    (3)
7
5
C + D + 
16
5
E 
1
5
F 
23
5
G = 
3
5
    (4)
 
24
5
C -2D 
7
5
E 
13
5
F + 
6
5
G = 
11
5
    (5)
14
5
C + 3D + 
2
5
E + 
3
5
F + 
19
5
G = 
4
5
    (6)
26
5
C -3D + 
18
5
E + 
27
5
F 
9
5
G = 
1
5
    (7)

Multiply both sides of equation (3) by 7
Divide the two sides of equation (3) by 5, the equation can be obtained:
         
7
5
C + 
7
5
D + 
14
5
E + 
7
5
G = 7    (16)
, then add the two sides of equation (16) to both sides of equation (4), the equations are reduced to:
 A -2B -3E + 2F -2G = -4    (1)
 5B -2C + 6E -1F + 2G = 8    (2)
 C + D + 2E + G = 5    (3)
 
12
5
D + 6E 
1
5
F 
16
5
G = 
38
5
    (4)
 
24
5
C -2D 
7
5
E 
13
5
F + 
6
5
G = 
11
5
    (5)
14
5
C + 3D + 
2
5
E + 
3
5
F + 
19
5
G = 
4
5
    (6)
26
5
C -3D + 
18
5
E + 
27
5
F 
9
5
G = 
1
5
    (7)

Multiply both sides of equation (3) by 24
Divide the two sides of equation (3) by 5, the equation can be obtained:
         
24
5
C + 
24
5
D + 
48
5
E + 
24
5
G = 24    (17)
, then subtract both sides of equation (17) from both sides of equation (5), the equations are reduced to:
 A -2B -3E + 2F -2G = -4    (1)
 5B -2C + 6E -1F + 2G = 8    (2)
 C + D + 2E + G = 5    (3)
 
12
5
D + 6E 
1
5
F 
16
5
G = 
38
5
    (4)
34
5
D -11E 
13
5
F 
18
5
G = 
131
5
    (5)
14
5
C + 3D + 
2
5
E + 
3
5
F + 
19
5
G = 
4
5
    (6)
26
5
C -3D + 
18
5
E + 
27
5
F 
9
5
G = 
1
5
    (7)

Multiply both sides of equation (3) by 14
Divide the two sides of equation (3) by 5, the equation can be obtained:
         
14
5
C + 
14
5
D + 
28
5
E + 
14
5
G = 14    (18)
, then add the two sides of equation (18) to both sides of equation (6), the equations are reduced to:
 A -2B -3E + 2F -2G = -4    (1)
 5B -2C + 6E -1F + 2G = 8    (2)
 C + D + 2E + G = 5    (3)
 
12
5
D + 6E 
1
5
F 
16
5
G = 
38
5
    (4)
34
5
D -11E 
13
5
F 
18
5
G = 
131
5
    (5)
 
29
5
D + 6E + 
3
5
F + 
33
5
G = 
66
5
    (6)
26
5
C -3D + 
18
5
E + 
27
5
F 
9
5
G = 
1
5
    (7)

Multiply both sides of equation (3) by 26
Divide the two sides of equation (3) by 5, the equation can be obtained:
         
26
5
C + 
26
5
D + 
52
5
E + 
26
5
G = 26    (19)
, then add the two sides of equation (19) to both sides of equation (7), the equations are reduced to:
 A -2B -3E + 2F -2G = -4    (1)
 5B -2C + 6E -1F + 2G = 8    (2)
 C + D + 2E + G = 5    (3)
 
12
5
D + 6E 
1
5
F 
16
5
G = 
38
5
    (4)
34
5
D -11E 
13
5
F 
18
5
G = 
131
5
    (5)
 
29
5
D + 6E + 
3
5
F + 
33
5
G = 
66
5
    (6)
 
11
5
D + 14E + 
27
5
F + 
17
5
G = 
129
5
    (7)

Multiply both sides of equation (4) by 17
Divide the two sides of equation (4) by 6, the equation can be obtained:
         
34
5
D + 17E 
17
30
F 
136
15
G = 
323
15
    (20)
, then add the two sides of equation (20) to both sides of equation (5), the equations are reduced to:
 A -2B -3E + 2F -2G = -4    (1)
 5B -2C + 6E -1F + 2G = 8    (2)
 C + D + 2E + G = 5    (3)
 
12
5
D + 6E 
1
5
F 
16
5
G = 
38
5
    (4)
 6E 
19
6
F 
38
3
G = 
14
3
    (5)
 
29
5
D + 6E + 
3
5
F + 
33
5
G = 
66
5
    (6)
 
11
5
D + 14E + 
27
5
F + 
17
5
G = 
129
5
    (7)

Multiply both sides of equation (4) by 29
Divide the two sides of equation (4) by 12, the equation can be obtained:
         
29
5
D + 
29
2
E 
29
60
F 
116
15
G = 
551
30
    (21)
, then subtract both sides of equation (21) from both sides of equation (6), the equations are reduced to:
 A -2B -3E + 2F -2G = -4    (1)
 5B -2C + 6E -1F + 2G = 8    (2)
 C + D + 2E + G = 5    (3)
 
12
5
D + 6E 
1
5
F 
16
5
G = 
38
5
    (4)
 6E 
19
6
F 
38
3
G = 
14
3
    (5)
17
2
E + 
13
12
F + 
43
3
G = 
31
6
    (6)
 
11
5
D + 14E + 
27
5
F + 
17
5
G = 
129
5
    (7)

Multiply both sides of equation (4) by 11
Divide the two sides of equation (4) by 12, the equation can be obtained:
         
11
5
D + 
11
2
E 
11
60
F 
44
15
G = 
209
30
    (22)
, then subtract both sides of equation (22) from both sides of equation (7), the equations are reduced to:
 A -2B -3E + 2F -2G = -4    (1)
 5B -2C + 6E -1F + 2G = 8    (2)
 C + D + 2E + G = 5    (3)
 
12
5
D + 6E 
1
5
F 
16
5
G = 
38
5
    (4)
 6E 
19
6
F 
38
3
G = 
14
3
    (5)
17
2
E + 
13
12
F + 
43
3
G = 
31
6
    (6)
 
17
2
E + 
67
12
F + 
19
3
G = 
113
6
    (7)

Multiply both sides of equation (5) by 17
Divide the two sides of equation (5) by 12, the equation can be obtained:
         
17
2
E 
323
72
F 
323
18
G = 
119
18
    (23)
, then add the two sides of equation (23) to both sides of equation (6), the equations are reduced to:
 A -2B -3E + 2F -2G = -4    (1)
 5B -2C + 6E -1F + 2G = 8    (2)
 C + D + 2E + G = 5    (3)
 
12
5
D + 6E 
1
5
F 
16
5
G = 
38
5
    (4)
 6E 
19
6
F 
38
3
G = 
14
3
    (5)
245
72
F 
65
18
G = 
106
9
    (6)
 
17
2
E + 
67
12
F + 
19
3
G = 
113
6
    (7)

Multiply both sides of equation (5) by 17
Divide the two sides of equation (5) by 12, the equation can be obtained:
         
17
2
E 
323
72
F 
323
18
G = 
119
18
    (24)
, then subtract both sides of equation (24) from both sides of equation (7), the equations are reduced to:
 A -2B -3E + 2F -2G = -4    (1)
 5B -2C + 6E -1F + 2G = 8    (2)
 C + D + 2E + G = 5    (3)
 
12
5
D + 6E 
1
5
F 
16
5
G = 
38
5
    (4)
 6E 
19
6
F 
38
3
G = 
14
3
    (5)
245
72
F 
65
18
G = 
106
9
    (6)
 
725
72
F + 
437
18
G = 
229
9
    (7)

Multiply both sides of equation (6) by 145
Divide the two sides of equation (6) by 49, the equation can be obtained:
        
725
72
F 
9425
882
G = 
15370
441
    (25)
, then add the two sides of equation (25) to both sides of equation (7), the equations are reduced to:
 A -2B -3E + 2F -2G = -4    (1)
 5B -2C + 6E -1F + 2G = 8    (2)
 C + D + 2E + G = 5    (3)
 
12
5
D + 6E 
1
5
F 
16
5
G = 
38
5
    (4)
 6E 
19
6
F 
38
3
G = 
14
3
    (5)
245
72
F 
65
18
G = 
106
9
    (6)
 
666
49
G = 
461
49
    (7)

Multiply both sides of equation (7) by 3185
Divide both sides of equation (7) by 11988, get the equation:
         
65
18
G = 
29965
11988
    (26)
, then add the two sides of equation (26) to both sides of equation (6), get the equation:
 A -2B -3E + 2F -2G = -4    (1)
 5B -2C + 6E -1F + 2G = 8    (2)
 C + D + 2E + G = 5    (3)
 
12
5
D + 6E 
1
5
F 
16
5
G = 
38
5
    (4)
 6E 
19
6
F 
38
3
G = 
14
3
    (5)
245
72
F = 
171157
11988
    (6)
 
666
49
G = 
461
49
    (7)

Multiply both sides of equation (7) by 931
Divide both sides of equation (7) by 999, get the equation:
         
38
3
G = 
8759
999
    (27)
, then add the two sides of equation (27) to both sides of equation (5), get the equation:
 A -2B -3E + 2F -2G = -4    (1)
 5B -2C + 6E -1F + 2G = 8    (2)
 C + D + 2E + G = 5    (3)
 
12
5
D + 6E 
1
5
F 
16
5
G = 
38
5
    (4)
 6E 
19
6
F = 
13421
999
    (5)
245
72
F = 
171157
11988
    (6)
 
666
49
G = 
461
49
    (7)

Multiply both sides of equation (7) by 392
Divide both sides of equation (7) by 1665, get the equation:
         
16
5
G = 
3688
1665
    (28)
, then add the two sides of equation (28) to both sides of equation (4), get the equation:
 A -2B -3E + 2F -2G = -4    (1)
 5B -2C + 6E -1F + 2G = 8    (2)
 C + D + 2E + G = 5    (3)
 
12
5
D + 6E 
1
5
F = 
8966
1665
    (4)
 6E 
19
6
F = 
13421
999
    (5)
245
72
F = 
171157
11988
    (6)
 
666
49
G = 
461
49
    (7)

Multiply both sides of equation (7) by 49
Divide both sides of equation (7) by 666, get the equation:
         G = 
461
666
    (29)
, then subtract both sides of equation (29) from both sides of equation (3), get the equation:
 A -2B -3E + 2F -2G = -4    (1)
 5B -2C + 6E -1F + 2G = 8    (2)
 C + D + 2E = 
3791
666
    (3)
 
12
5
D + 6E 
1
5
F = 
8966
1665
    (4)
 6E 
19
6
F = 
13421
999
    (5)
245
72
F = 
171157
11988
    (6)
 
666
49
G = 
461
49
    (7)

Multiply both sides of equation (7) by 49
Divide both sides of equation (7) by 333, get the equation:
         2G = 
461
333
    (30)
, then subtract both sides of equation (30) from both sides of equation (2), get the equation:
 A -2B -3E + 2F -2G = -4    (1)
 5B -2C + 6E -1F = 
3125
333
    (2)
 C + D + 2E = 
3791
666
    (3)
 
12
5
D + 6E 
1
5
F = 
8966
1665
    (4)
 6E 
19
6
F = 
13421
999
    (5)
245
72
F = 
171157
11988
    (6)
 
666
49
G = 
461
49
    (7)

Multiply both sides of equation (7) by 49
Divide both sides of equation (7) by 333, get the equation:
         2G = 
461
333
    (31)
, then add the two sides of equation (31) to both sides of equation (1), get the equation:
 A -2B -3E + 2F = 
1793
333
    (1)
 5B -2C + 6E -1F = 
3125
333
    (2)
 C + D + 2E = 
3791
666
    (3)
 
12
5
D + 6E 
1
5
F = 
8966
1665
    (4)
 6E 
19
6
F = 
13421
999
    (5)
245
72
F = 
171157
11988
    (6)
 
666
49
G = 
461
49
    (7)

Multiply both sides of equation (6) by 228
Divide both sides of equation (6) by 245, get the equation:
        
19
6
F = 
66367
4995
    (32)
, then subtract both sides of equation (32) from both sides of equation (5), get the equation:
 A -2B -3E + 2F = 
1793
333
    (1)
 5B -2C + 6E -1F = 
3125
333
    (2)
 C + D + 2E = 
3791
666
    (3)
 
12
5
D + 6E 
1
5
F = 
8966
1665
    (4)
 6E = 
82
555
    (5)
245
72
F = 
171157
11988
    (6)
 G = 
461
666
    (7)

Multiply both sides of equation (6) by 72
Divide both sides of equation (6) by 1225, get the equation:
        
1
5
F = 
6986
8325
    (33)
, then subtract both sides of equation (33) from both sides of equation (4), get the equation:
 A -2B -3E + 2F = 
1793
333
    (1)
 5B -2C + 6E -1F = 
3125
333
    (2)
 C + D + 2E = 
3791
666
    (3)
 
12
5
D + 6E = 
17272
2775
    (4)
 6E = 
82
555
    (5)
245
72
F = 
171157
11988
    (6)
 G = 
461
666
    (7)

Multiply both sides of equation (6) by 72
Divide both sides of equation (6) by 245, get the equation:
        -1F = 
6986
1665
    (34)
, then subtract both sides of equation (34) from both sides of equation (2), get the equation:
 A -2B -3E + 2F = 
1793
333
    (1)
 5B -2C + 6E = 
7537
555
    (2)
 C + D + 2E = 
3791
666
    (3)
 
12
5
D + 6E = 
17272
2775
    (4)
 6E = 
82
555
    (5)
245
72
F = 
171157
11988
    (6)
 G = 
461
666
    (7)

Multiply both sides of equation (6) by 144
Divide both sides of equation (6) by 245, get the equation:
        -2F = 
13972
1665
    (35)
, then add the two sides of equation (35) to both sides of equation (1), get the equation:
 A -2B -3E = 
22937
1665
    (1)
 5B -2C + 6E = 
7537
555
    (2)
 C + D + 2E = 
3791
666
    (3)
 
12
5
D + 6E = 
17272
2775
    (4)
 6E = 
82
555
    (5)
245
72
F = 
171157
11988
    (6)
 G = 
461
666
    (7)

Subtract both sides of equation (5) from both sides of equation (4), get the equation:
 A -2B -3E = 
22937
1665
    (1)
 5B -2C + 6E = 
7537
555
    (2)
 C + D + 2E = 
3791
666
    (3)
 
12
5
D = 
5894
925
    (4)
 6E = 
82
555
    (5)
 F = 
6986
1665
    (6)
 G = 
461
666
    (7)

Divide both sides of equation (5) by 3, get the equation:
         2E = 
82
1665
    (36)
, then subtract both sides of equation (36) from both sides of equation (3), get the equation:
 A -2B -3E = 
22937
1665
    (1)
 5B -2C + 6E = 
7537
555
    (2)
 C + D = 
6373
1110
    (3)
 
12
5
D = 
5894
925
    (4)
 6E = 
82
555
    (5)
 F = 
6986
1665
    (6)
 G = 
461
666
    (7)

Subtract both sides of equation (5) from both sides of equation (2), get the equation:
 A -2B -3E = 
22937
1665
    (1)
 5B -2C = 
7619
555
    (2)
 C + D = 
6373
1110
    (3)
 
12
5
D = 
5894
925
    (4)
 6E = 
82
555
    (5)
 F = 
6986
1665
    (6)
 G = 
461
666
    (7)

Divide both sides of equation (5) by 2, get the equation:
         3E = 
41
555
    (37)
, then add the two sides of equation (37) to both sides of equation (1), get the equation:
 A -2B = 
4612
333
    (1)
 5B -2C = 
7619
555
    (2)
 C + D = 
6373
1110
    (3)
 
12
5
D = 
5894
925
    (4)
 6E = 
82
555
    (5)
 F = 
6986
1665
    (6)
 G = 
461
666
    (7)

Multiply both sides of equation (4) by 5
Divide both sides of equation (4) by 12, get the equation:
         D = 
2947
1110
    (38)
, then subtract both sides of equation (38) from both sides of equation (3), get the equation:
 A -2B = 
4612
333
    (1)
 5B -2C = 
7619
555
    (2)
 C = 
571
185
    (3)
 
12
5
D = 
5894
925
    (4)
 E = 
41
1665
    (5)
 F = 
6986
1665
    (6)
 G = 
461
666
    (7)

Multiply both sides of equation (3) by 2, get the equation:
         2C = 
1142
185
    (39)
, then add the two sides of equation (39) to both sides of equation (2), get the equation:
 A -2B = 
4612
333
    (1)
 5B = 
2209
111
    (2)
 C = 
571
185
    (3)
 D = 
2947
1110
    (4)
 E = 
41
1665
    (5)
 F = 
6986
1665
    (6)
 G = 
461
666
    (7)

Multiply both sides of equation (2) by 2
Divide both sides of equation (2) by 5, get the equation:
         2B = 
4418
555
    (40)
, then add the two sides of equation (40) to both sides of equation (1), get the equation:
 A = 
9806
1665
    (1)
 5B = 
2209
111
    (2)
 C = 
571
185
    (3)
 D = 
2947
1110
    (4)
 E = 
41
1665
    (5)
 F = 
6986
1665
    (6)
 G = 
461
666
    (7)

The coefficient of the unknown number is reduced to 1, and the equations are reduced to:
 A = 
9806
1665
    (1)
 B = 
2209
555
    (2)
 C = 
571
185
    (3)
 D = 
2947
1110
    (4)
 E = 
41
1665
    (5)
 F = 
6986
1665
    (6)
 G = 
461
666
    (7)


Therefore, the solution of the equation set is:
A = 
9806
1665
B = 
2209
555
C = 
571
185
D = 
2947
1110
E = 
41
1665
F = 
6986
1665
G = 
461
666


Convert the solution of the equation set to decimals:
A = -5.889489
B = 3.980180
C = 3.086486
D = 2.654955
E = -0.024625
F = 4.195796
G = -0.692192

解方程组的详细方法请参阅:《多元一次方程组的解法》







  New addition:Lenders ToolBox module(Specific location:Math OP > Lenders ToolBox ),welcome。