Mathematics
         
语言：中文    Language:English
Equations   
Fold
Unary equation
Multivariate equation
Math OP  
Unfold
Linear algebra      
Unfold
Derivative function
Function image
Hot issues
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:
 20A + 19G + J = 1    (1)
 20B + 19H + K = 25    (2)
 20C + 19I + L = 20    (3)
 3D + 8G + J = 3    (4)
 24D + 4G + J = 14    (5)
 3E + 8H + K = 18    (6)
 24E + 4H + K = 15    (7)
 3F + 8I + L = 17    (8)
 24F + 4I + L = 11    (9)
 18A + 15D + 11G = 12    (10)
 18B + 15E + 11H = 18    (11)
 18C + 15F + 11I = 8    (12)
Question solving process:

Multiply both sides of equation (1) by 9
Divide the two sides of equation (1) by 10, the equation can be obtained:
         18A + 
171
10
G + 
9
10
J = 
9
10
    (13)
, then subtract both sides of equation (13) from both sides of equation (10), the equations are reduced to:
 20A + 19G + J = 1    (1)
 20B + 19H + K = 25    (2)
 20C + 19I + L = 20    (3)
 3D + 8G + J = 3    (4)
 24D + 4G + J = 14    (5)
 3E + 8H + K = 18    (6)
 24E + 4H + K = 15    (7)
 3F + 8I + L = 17    (8)
 24F + 4I + L = 11    (9)
 15D - 
61
10
G - 
9
10
J = 
111
10
    (10)
 18B + 15E + 11H = 18    (11)
 18C + 15F + 11I = 8    (12)

Multiply both sides of equation (2) by 9
Divide the two sides of equation (2) by 10, the equation can be obtained:
         18B + 
171
10
H + 
9
10
K = 
45
2
    (14)
, then subtract both sides of equation (14) from both sides of equation (11), the equations are reduced to:
 20A + 19G + J = 1    (1)
 20B + 19H + K = 25    (2)
 20C + 19I + L = 20    (3)
 3D + 8G + J = 3    (4)
 24D + 4G + J = 14    (5)
 3E + 8H + K = 18    (6)
 24E + 4H + K = 15    (7)
 3F + 8I + L = 17    (8)
 24F + 4I + L = 11    (9)
 15D - 
61
10
G - 
9
10
J = 
111
10
    (10)
 15E - 
61
10
H - 
9
10
K = - 
9
2
    (11)
 18C + 15F + 11I = 8    (12)

Multiply both sides of equation (3) by 9
Divide the two sides of equation (3) by 10, the equation can be obtained:
         18C + 
171
10
I + 
9
10
L = 18    (15)
, then subtract both sides of equation (15) from both sides of equation (12), the equations are reduced to:
 20A + 19G + J = 1    (1)
 20B + 19H + K = 25    (2)
 20C + 19I + L = 20    (3)
 3D + 8G + J = 3    (4)
 24D + 4G + J = 14    (5)
 3E + 8H + K = 18    (6)
 24E + 4H + K = 15    (7)
 3F + 8I + L = 17    (8)
 24F + 4I + L = 11    (9)
 15D - 
61
10
G - 
9
10
J = 
111
10
    (10)
 15E - 
61
10
H - 
9
10
K = - 
9
2
    (11)
 15F - 
61
10
I - 
9
10
L = -10    (12)

Multiply both sides of equation (4) by 8, the equation can be obtained:
         24D + 64G + 8J = 24    (16)
, then subtract both sides of equation (16) from both sides of equation (5), the equations are reduced to:
 20A + 19G + J = 1    (1)
 20B + 19H + K = 25    (2)
 20C + 19I + L = 20    (3)
 3D + 8G + J = 3    (4)
-60G -7J = -10    (5)
 3E + 8H + K = 18    (6)
 24E + 4H + K = 15    (7)
 3F + 8I + L = 17    (8)
 24F + 4I + L = 11    (9)
 15D - 
61
10
G - 
9
10
J = 
111
10
    (10)
 15E - 
61
10
H - 
9
10
K = - 
9
2
    (11)
 15F - 
61
10
I - 
9
10
L = -10    (12)

Multiply both sides of equation (4) by 5, the equation can be obtained:
         15D + 40G + 5J = 15    (17)
, then subtract both sides of equation (17) from both sides of equation (10), the equations are reduced to:
 20A + 19G + J = 1    (1)
 20B + 19H + K = 25    (2)
 20C + 19I + L = 20    (3)
 3D + 8G + J = 3    (4)
-60G -7J = -10    (5)
 3E + 8H + K = 18    (6)
 24E + 4H + K = 15    (7)
 3F + 8I + L = 17    (8)
 24F + 4I + L = 11    (9)
- 
461
10
G - 
59
10
J = - 
39
10
    (10)
 15E - 
61
10
H - 
9
10
K = - 
9
2
    (11)
 15F - 
61
10
I - 
9
10
L = -10    (12)

交After the exchange of equation (5) and equation (6)， the equation system becomes:
 20A + 19G + J = 1    (1)
 20B + 19H + K = 25    (2)
 20C + 19I + L = 20    (3)
 3D + 8G + J = 3    (4)
 3E + 8H + K = 18    (5)
-60G -7J = -10    (6)
 24E + 4H + K = 15    (7)
 3F + 8I + L = 17    (8)
 24F + 4I + L = 11    (9)
- 
461
10
G - 
59
10
J = - 
39
10
    (10)
 15E - 
61
10
H - 
9
10
K = - 
9
2
    (11)
 15F - 
61
10
I - 
9
10
L = -10    (12)

Multiply both sides of equation (5) by 8, the equation can be obtained:
         24E + 64H + 8K = 144    (18)
, then subtract both sides of equation (18) from both sides of equation (7), the equations are reduced to:
 20A + 19G + J = 1    (1)
 20B + 19H + K = 25    (2)
 20C + 19I + L = 20    (3)
 3D + 8G + J = 3    (4)
 3E + 8H + K = 18    (5)
-60G -7J = -10    (6)
-60H -7K = -129    (7)
 3F + 8I + L = 17    (8)
 24F + 4I + L = 11    (9)
- 
461
10
G - 
59
10
J = - 
39
10
    (10)
 15E - 
61
10
H - 
9
10
K = - 
9
2
    (11)
 15F - 
61
10
I - 
9
10
L = -10    (12)

Multiply both sides of equation (5) by 5, the equation can be obtained:
         15E + 40H + 5K = 90    (19)
, then subtract both sides of equation (19) from both sides of equation (11), the equations are reduced to:
 20A + 19G + J = 1    (1)
 20B + 19H + K = 25    (2)
 20C + 19I + L = 20    (3)
 3D + 8G + J = 3    (4)
 3E + 8H + K = 18    (5)
-60G -7J = -10    (6)
-60H -7K = -129    (7)
 3F + 8I + L = 17    (8)
 24F + 4I + L = 11    (9)
- 
461
10
G - 
59
10
J = - 
39
10
    (10)
- 
461
10
H - 
59
10
K = - 
189
2
    (11)
 15F - 
61
10
I - 
9
10
L = -10    (12)

交After the exchange of equation (6) and equation (8)， the equation system becomes:
 20A + 19G + J = 1    (1)
 20B + 19H + K = 25    (2)
 20C + 19I + L = 20    (3)
 3D + 8G + J = 3    (4)
 3E + 8H + K = 18    (5)
 3F + 8I + L = 17    (6)
-60H -7K = -129    (7)
-60G -7J = -10    (8)
 24F + 4I + L = 11    (9)
- 
461
10
G - 
59
10
J = - 
39
10
    (10)
- 
461
10
H - 
59
10
K = - 
189
2
    (11)
 15F - 
61
10
I - 
9
10
L = -10    (12)

Multiply both sides of equation (6) by 8, the equation can be obtained:
         24F + 64I + 8L = 136    (20)
, then subtract both sides of equation (20) from both sides of equation (9), the equations are reduced to:
 20A + 19G + J = 1    (1)
 20B + 19H + K = 25    (2)
 20C + 19I + L = 20    (3)
 3D + 8G + J = 3    (4)
 3E + 8H + K = 18    (5)
 3F + 8I + L = 17    (6)
-60H -7K = -129    (7)
-60G -7J = -10    (8)
-60I -7L = -125    (9)
- 
461
10
G - 
59
10
J = - 
39
10
    (10)
- 
461
10
H - 
59
10
K = - 
189
2
    (11)
 15F - 
61
10
I - 
9
10
L = -10    (12)

Multiply both sides of equation (6) by 5, the equation can be obtained:
         15F + 40I + 5L = 85    (21)
, then subtract both sides of equation (21) from both sides of equation (12), the equations are reduced to:
 20A + 19G + J = 1    (1)
 20B + 19H + K = 25    (2)
 20C + 19I + L = 20    (3)
 3D + 8G + J = 3    (4)
 3E + 8H + K = 18    (5)
 3F + 8I + L = 17    (6)
-60H -7K = -129    (7)
-60G -7J = -10    (8)
-60I -7L = -125    (9)
- 
461
10
G - 
59
10
J = - 
39
10
    (10)
- 
461
10
H - 
59
10
K = - 
189
2
    (11)
- 
461
10
I - 
59
10
L = -95    (12)

交After the exchange of equation (7) and equation (8)， the equation system becomes:
 20A + 19G + J = 1    (1)
 20B + 19H + K = 25    (2)
 20C + 19I + L = 20    (3)
 3D + 8G + J = 3    (4)
 3E + 8H + K = 18    (5)
 3F + 8I + L = 17    (6)
-60G -7J = -10    (7)
-60H -7K = -129    (8)
-60I -7L = -125    (9)
- 
461
10
G - 
59
10
J = - 
39
10
    (10)
- 
461
10
H - 
59
10
K = - 
189
2
    (11)
- 
461
10
I - 
59
10
L = -95    (12)

Multiply both sides of equation (7) by 461
Divide the two sides of equation (7) by 600, the equation can be obtained:
        - 
461
10
G - 
3227
600
J = - 
461
60
    (22)
, then subtract both sides of equation (22) from both sides of equation (10), the equations are reduced to:
 20A + 19G + J = 1    (1)
 20B + 19H + K = 25    (2)
 20C + 19I + L = 20    (3)
 3D + 8G + J = 3    (4)
 3E + 8H + K = 18    (5)
 3F + 8I + L = 17    (6)
-60G -7J = -10    (7)
-60H -7K = -129    (8)
-60I -7L = -125    (9)
- 
313
600
J = 
227
60
    (10)
- 
461
10
H - 
59
10
K = - 
189
2
    (11)
- 
461
10
I - 
59
10
L = -95    (12)

Multiply both sides of equation (8) by 461
Divide the two sides of equation (8) by 600, the equation can be obtained:
        - 
461
10
H - 
3227
600
K = - 
19823
200
    (23)
, then subtract both sides of equation (23) from both sides of equation (11), the equations are reduced to:
 20A + 19G + J = 1    (1)
 20B + 19H + K = 25    (2)
 20C + 19I + L = 20    (3)
 3D + 8G + J = 3    (4)
 3E + 8H + K = 18    (5)
 3F + 8I + L = 17    (6)
-60G -7J = -10    (7)
-60H -7K = -129    (8)
-60I -7L = -125    (9)
- 
313
600
J = 
227
60
    (10)
- 
313
600
K = 
923
200
    (11)
- 
461
10
I - 
59
10
L = -95    (12)

Multiply both sides of equation (9) by 461
Divide the two sides of equation (9) by 600, the equation can be obtained:
        - 
461
10
I - 
3227
600
L = - 
2305
24
    (24)
, then subtract both sides of equation (24) from both sides of equation (12), the equations are reduced to:
 20A + 19G + J = 1    (1)
 20B + 19H + K = 25    (2)
 20C + 19I + L = 20    (3)
 3D + 8G + J = 3    (4)
 3E + 8H + K = 18    (5)
 3F + 8I + L = 17    (6)
-60G -7J = -10    (7)
-60H -7K = -129    (8)
-60I -7L = -125    (9)
- 
313
600
J = 
227
60
    (10)
- 
313
600
K = 
923
200
    (11)
- 
313
600
L = 
25
24
    (12)

Multiply both sides of equation (12) by 4200
Divide both sides of equation (12) by 313, get the equation：
        - 
2191
313
L = 
4375
313
    (25)
, then subtract both sides of equation (25) from both sides of equation (9), get the equation：
 20A + 19G + J = 1    (1)
 20B + 19H + K = 25    (2)
 20C + 19I + L = 20    (3)
 3D + 8G + J = 3    (4)
 3E + 8H + K = 18    (5)
 3F + 8I + L = 17    (6)
-60G -7J = -10    (7)
-60H -7K = -129    (8)
-60I = - 
43500
313
    (9)
- 
313
600
J = 
227
60
    (10)
- 
313
600
K = 
923
200
    (11)
- 
313
600
L = 
25
24
    (12)

Multiply both sides of equation (12) by 600
Divide both sides of equation (12) by 313, get the equation：
        - 
313
313
L = 
625
313
    (26)
, then add the two sides of equation (26) to both sides of equation (6), get the equation：
 20A + 19G + J = 1    (1)
 20B + 19H + K = 25    (2)
 20C + 19I + L = 20    (3)
 3D + 8G + J = 3    (4)
 3E + 8H + K = 18    (5)
 3F + 8I = 
5946
313
    (6)
-60G -7J = -10    (7)
-60H -7K = -129    (8)
-60I = - 
43500
313
    (9)
- 
313
600
J = 
227
60
    (10)
- 
313
600
K = 
923
200
    (11)
- 
313
600
L = 
25
24
    (12)

Multiply both sides of equation (12) by 600
Divide both sides of equation (12) by 313, get the equation：
        - 
313
313
L = 
625
313
    (27)
, then add the two sides of equation (27) to both sides of equation (3), get the equation：
 20A + 19G + J = 1    (1)
 20B + 19H + K = 25    (2)
 20C + 19I = 
6885
313
    (3)
 3D + 8G + J = 3    (4)
 3E + 8H + K = 18    (5)
 3F + 8I = 
5946
313
    (6)
-60G -7J = -10    (7)
-60H -7K = -129    (8)
-60I = - 
43500
313
    (9)
- 
313
600
J = 
227
60
    (10)
- 
313
600
K = 
923
200
    (11)
- 
313
600
L = 
25
24
    (12)

Multiply both sides of equation (11) by 4200
Divide both sides of equation (11) by 313, get the equation：
        - 
2191
313
K = 
19383
313
    (28)
, then subtract both sides of equation (28) from both sides of equation (8), get the equation：
 20A + 19G + J = 1    (1)
 20B + 19H + K = 25    (2)
 20C + 19I = 
6885
313
    (3)
 3D + 8G + J = 3    (4)
 3E + 8H + K = 18    (5)
 3F + 8I = 
5946
313
    (6)
-60G -7J = -10    (7)
-60H = - 
59760
313
    (8)
-60I = - 
43500
313
    (9)
- 
313
600
J = 
227
60
    (10)
- 
313
600
K = 
923
200
    (11)
 L = - 
625
313
    (12)

Multiply both sides of equation (11) by 600
Divide both sides of equation (11) by 313, get the equation：
        - 
313
313
K = 
2769
313
    (29)
, then add the two sides of equation (29) to both sides of equation (5), get the equation：
 20A + 19G + J = 1    (1)
 20B + 19H + K = 25    (2)
 20C + 19I = 
6885
313
    (3)
 3D + 8G + J = 3    (4)
 3E + 8H = 
8403
313
    (5)
 3F + 8I = 
5946
313
    (6)
-60G -7J = -10    (7)
-60H = - 
59760
313
    (8)
-60I = - 
43500
313
    (9)
- 
313
600
J = 
227
60
    (10)
- 
313
600
K = 
923
200
    (11)
 L = - 
625
313
    (12)

Multiply both sides of equation (11) by 600
Divide both sides of equation (11) by 313, get the equation：
        - 
313
313
K = 
2769
313
    (30)
, then add the two sides of equation (30) to both sides of equation (2), get the equation：
 20A + 19G + J = 1    (1)
 20B + 19H = 
10594
313
    (2)
 20C + 19I = 
6885
313
    (3)
 3D + 8G + J = 3    (4)
 3E + 8H = 
8403
313
    (5)
 3F + 8I = 
5946
313
    (6)
-60G -7J = -10    (7)
-60H = - 
59760
313
    (8)
-60I = - 
43500
313
    (9)
- 
313
600
J = 
227
60
    (10)
- 
313
600
K = 
923
200
    (11)
 L = - 
625
313
    (12)

Multiply both sides of equation (10) by 4200
Divide both sides of equation (10) by 313, get the equation：
        - 
2191
313
J = 
15890
313
    (31)
, then subtract both sides of equation (31) from both sides of equation (7), get the equation：
 20A + 19G + J = 1    (1)
 20B + 19H = 
10594
313
    (2)
 20C + 19I = 
6885
313
    (3)
 3D + 8G + J = 3    (4)
 3E + 8H = 
8403
313
    (5)
 3F + 8I = 
5946
313
    (6)
-60G = - 
19020
313
    (7)
-60H = - 
59760
313
    (8)
-60I = - 
43500
313
    (9)
- 
313
600
J = 
227
60
    (10)
 K = - 
2769
313
    (11)
 L = - 
625
313
    (12)

Multiply both sides of equation (10) by 600
Divide both sides of equation (10) by 313, get the equation：
        - 
313
313
J = 
2270
313
    (32)
, then add the two sides of equation (32) to both sides of equation (4), get the equation：
 20A + 19G + J = 1    (1)
 20B + 19H = 
10594
313
    (2)
 20C + 19I = 
6885
313
    (3)
 3D + 8G = 
3209
313
    (4)
 3E + 8H = 
8403
313
    (5)
 3F + 8I = 
5946
313
    (6)
-60G = - 
19020
313
    (7)
-60H = - 
59760
313
    (8)
-60I = - 
43500
313
    (9)
- 
313
600
J = 
227
60
    (10)
 K = - 
2769
313
    (11)
 L = - 
625
313
    (12)

Multiply both sides of equation (10) by 600
Divide both sides of equation (10) by 313, get the equation：
        - 
313
313
J = 
2270
313
    (33)
, then add the two sides of equation (33) to both sides of equation (1), get the equation：
 20A + 19G = 
2583
313
    (1)
 20B + 19H = 
10594
313
    (2)
 20C + 19I = 
6885
313
    (3)
 3D + 8G = 
3209
313
    (4)
 3E + 8H = 
8403
313
    (5)
 3F + 8I = 
5946
313
    (6)
-60G = - 
19020
313
    (7)
-60H = - 
59760
313
    (8)
-60I = - 
43500
313
    (9)
- 
313
600
J = 
227
60
    (10)
 K = - 
2769
313
    (11)
 L = - 
625
313
    (12)

Multiply both sides of equation (9) by 2
Divide both sides of equation (9) by 15, get the equation：
        -8I = - 
5800
313
    (34)
, then add the two sides of equation (34) to both sides of equation (6), get the equation：
 20A + 19G = 
2583
313
    (1)
 20B + 19H = 
10594
313
    (2)
 20C + 19I = 
6885
313
    (3)
 3D + 8G = 
3209
313
    (4)
 3E + 8H = 
8403
313
    (5)
 3F = 
45698
97969
    (6)
-60G = - 
19020
313
    (7)
-60H = - 
59760
313
    (8)
-60I = - 
43500
313
    (9)
 J = - 
2270
313
    (10)
 K = - 
2769
313
    (11)
 L = - 
625
313
    (12)

Multiply both sides of equation (9) by 19
Divide both sides of equation (9) by 60, get the equation：
        -19I = - 
13775
313
    (35)
, then add the two sides of equation (35) to both sides of equation (3), get the equation：
 20A + 19G = 
2583
313
    (1)
 20B + 19H = 
10594
313
    (2)
 20C = - 
6890
313
    (3)
 3D + 8G = 
3209
313
    (4)
 3E + 8H = 
8403
313
    (5)
 3F = 
45698
97969
    (6)
-60G = - 
19020
313
    (7)
-60H = - 
59760
313
    (8)
-60I = - 
43500
313
    (9)
 J = - 
2270
313
    (10)
 K = - 
2769
313
    (11)
 L = - 
625
313
    (12)

Multiply both sides of equation (8) by 2
Divide both sides of equation (8) by 15, get the equation：
        -8H = - 
7968
313
    (36)
, then add the two sides of equation (36) to both sides of equation (5), get the equation：
 20A + 19G = 
2583
313
    (1)
 20B + 19H = 
10594
313
    (2)
 20C = - 
6890
313
    (3)
 3D + 8G = 
3209
313
    (4)
 3E = 
435
313
    (5)
 3F = 
45698
97969
    (6)
-60G = - 
19020
313
    (7)
-60H = - 
59760
313
    (8)
 I = 
725
313
    (9)
 J = - 
2270
313
    (10)
 K = - 
2769
313
    (11)
 L = - 
625
313
    (12)

Multiply both sides of equation (8) by 19
Divide both sides of equation (8) by 60, get the equation：
        -19H = - 
18924
313
    (37)
, then add the two sides of equation (37) to both sides of equation (2), get the equation：
 20A + 19G = 
2583
313
    (1)
 20B = - 
8330
313
    (2)
 20C = - 
6890
313
    (3)
 3D + 8G = 
3209
313
    (4)
 3E = 
435
313
    (5)
 3F = 
45698
97969
    (6)
-60G = - 
19020
313
    (7)
-60H = - 
59760
313
    (8)
 I = 
725
313
    (9)
 J = - 
2270
313
    (10)
 K = - 
2769
313
    (11)
 L = - 
625
313
    (12)

Multiply both sides of equation (7) by 2
Divide both sides of equation (7) by 15, get the equation：
        -8G = - 
2536
313
    (38)
, then add the two sides of equation (38) to both sides of equation (4), get the equation：
 20A + 19G = 
2583
313
    (1)
 20B = - 
8330
313
    (2)
 20C = - 
6890
313
    (3)
 3D = 
673
313
    (4)
 3E = 
435
313
    (5)
 3F = 
45698
97969
    (6)
-60G = - 
19020
313
    (7)
 H = 
996
313
    (8)
 I = 
725
313
    (9)
 J = - 
2270
313
    (10)
 K = - 
2769
313
    (11)
 L = - 
625
313
    (12)

Multiply both sides of equation (7) by 19
Divide both sides of equation (7) by 60, get the equation：
        -19G = - 
6023
313
    (39)
, then add the two sides of equation (39) to both sides of equation (1), get the equation：
 20A = - 
3440
313
    (1)
 20B = - 
8330
313
    (2)
 20C = - 
6890
313
    (3)
 3D = 
673
313
    (4)
 3E = 
435
313
    (5)
 3F = 
45698
97969
    (6)
-60G = - 
19020
313
    (7)
 H = 
996
313
    (8)
 I = 
725
313
    (9)
 J = - 
2270
313
    (10)
 K = - 
2769
313
    (11)
 L = - 
625
313
    (12)

The coefficient of the unknown number is reduced to 1, and the equations are reduced to：
 A = - 
172
313
    (1)
 B = - 
833
626
    (2)
 C = - 
689
626
    (3)
 D = 
673
939
    (4)
 E = 
145
313
    (5)
 F = 
45698
293907
    (6)
 G = 
317
313
    (7)
 H = 
996
313
    (8)
 I = 
725
313
    (9)
 J = - 
2270
313
    (10)
 K = - 
2769
313
    (11)
 L = - 
625
313
    (12)


Therefore, the solution of the equation set is:
A = - 
172
313
B = - 
833
626
C = - 
689
626
D = 
673
939
E = 
145
313
F = 
45698
293907
G = 
317
313
H = 
996
313
I = 
725
313
J = - 
2270
313
K = - 
2769
313
L = - 
625
313


Convert the solution of the equation set to decimals:
A = -0.549521
B = -1.330671
C = -1.100639
D = 0.716720
E = 0.463259
F = 0.155485
G = 1.012780
H = 3.182109
I = 2.316294
J = -7.252396
K = -8.846645
L = -1.996805

解方程组的详细方法请参阅：《多元一次方程组的解法》
return







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