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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:
 
3
25
A + 
3
25
B - 
41
50
C + 
41
50
D + 
41
50
E = -6000    (1)
 
4
5
A + 
4
5
B + 
4
5
C - 
4
5
D + 
4
5
E = 0    (2)
 
4
5
A + 
4
5
B + 
4
5
C + 
4
5
D - 
4
5
E = 0    (3)
- 
4
5
A + 
4
5
B + 
4
5
C + 
4
5
D + 
4
5
E = 28000    (4)
 
4
5
A - 
4
5
B + 
4
5
C + 
4
5
D + 
4
5
E = 1000    (5)
Question solving process:

Multiply both sides of equation (1) by 20
Divide the two sides of equation (1) by 3, the equation can be obtained:
         
4
5
A + 
4
5
B - 
82
15
C + 
82
15
D + 
82
15
E = -40000    (6)
, then subtract both sides of equation (6) from both sides of equation (2), the equations are reduced to:
 
3
25
A + 
3
25
B - 
41
50
C + 
41
50
D + 
41
50
E = -6000    (1)
 
94
15
C - 
94
15
D - 
14
3
E = 40000    (2)
 
4
5
A + 
4
5
B + 
4
5
C + 
4
5
D - 
4
5
E = 0    (3)
- 
4
5
A + 
4
5
B + 
4
5
C + 
4
5
D + 
4
5
E = 28000    (4)
 
4
5
A - 
4
5
B + 
4
5
C + 
4
5
D + 
4
5
E = 1000    (5)

Multiply both sides of equation (1) by 20
Divide the two sides of equation (1) by 3, the equation can be obtained:
         
4
5
A + 
4
5
B - 
82
15
C + 
82
15
D + 
82
15
E = -40000    (7)
, then subtract both sides of equation (7) from both sides of equation (3), the equations are reduced to:
 
3
25
A + 
3
25
B - 
41
50
C + 
41
50
D + 
41
50
E = -6000    (1)
 
94
15
C - 
94
15
D - 
14
3
E = 40000    (2)
 
94
15
C - 
14
3
D - 
94
15
E = 40000    (3)
- 
4
5
A + 
4
5
B + 
4
5
C + 
4
5
D + 
4
5
E = 28000    (4)
 
4
5
A - 
4
5
B + 
4
5
C + 
4
5
D + 
4
5
E = 1000    (5)

Multiply both sides of equation (1) by 20
Divide the two sides of equation (1) by 3, the equation can be obtained:
         
4
5
A + 
4
5
B - 
82
15
C + 
82
15
D + 
82
15
E = -40000    (8)
, then add the two sides of equation (8) to both sides of equation (4), the equations are reduced to:
 
3
25
A + 
3
25
B - 
41
50
C + 
41
50
D + 
41
50
E = -6000    (1)
 
94
15
C - 
94
15
D - 
14
3
E = 40000    (2)
 
94
15
C - 
14
3
D - 
94
15
E = 40000    (3)
 
8
5
B - 
14
3
C + 
94
15
D + 
94
15
E = -12000    (4)
 
4
5
A - 
4
5
B + 
4
5
C + 
4
5
D + 
4
5
E = 1000    (5)

Multiply both sides of equation (1) by 20
Divide the two sides of equation (1) by 3, the equation can be obtained:
         
4
5
A + 
4
5
B - 
82
15
C + 
82
15
D + 
82
15
E = -40000    (9)
, then subtract both sides of equation (9) from both sides of equation (5), the equations are reduced to:
 
3
25
A + 
3
25
B - 
41
50
C + 
41
50
D + 
41
50
E = -6000    (1)
 
94
15
C - 
94
15
D - 
14
3
E = 40000    (2)
 
94
15
C - 
14
3
D - 
94
15
E = 40000    (3)
 
8
5
B - 
14
3
C + 
94
15
D + 
94
15
E = -12000    (4)
- 
8
5
B + 
94
15
C - 
14
3
D - 
14
3
E = 41000    (5)

交After the exchange of equation (2) and equation (4)， the equation system becomes:
 
3
25
A + 
3
25
B - 
41
50
C + 
41
50
D + 
41
50
E = -6000    (1)
 
8
5
B - 
14
3
C + 
94
15
D + 
94
15
E = -12000    (2)
 
94
15
C - 
14
3
D - 
94
15
E = 40000    (3)
 
94
15
C - 
94
15
D - 
14
3
E = 40000    (4)
- 
8
5
B + 
94
15
C - 
14
3
D - 
14
3
E = 41000    (5)

Add both sides of equation (2) to both sides of equation (5) ,the equations are reduced to:
 
3
25
A + 
3
25
B - 
41
50
C + 
41
50
D + 
41
50
E = -6000    (1)
 
8
5
B - 
14
3
C + 
94
15
D + 
94
15
E = -12000    (2)
 
94
15
C - 
14
3
D - 
94
15
E = 40000    (3)
 
94
15
C - 
94
15
D - 
14
3
E = 40000    (4)
 
8
5
C + 
8
5
D + 
8
5
E = 29000    (5)

Subtract both sides of equation (3) from both sides of equation (4) ,the equations are reduced to:
 
3
25
A + 
3
25
B - 
41
50
C + 
41
50
D + 
41
50
E = -6000    (1)
 
8
5
B - 
14
3
C + 
94
15
D + 
94
15
E = -12000    (2)
 
94
15
C - 
14
3
D - 
94
15
E = 40000    (3)
- 
8
5
D + 
8
5
E = 0    (4)
 
8
5
C + 
8
5
D + 
8
5
E = 29000    (5)

Multiply both sides of equation (3) by 12
Divide the two sides of equation (3) by 47, the equation can be obtained:
         
8
5
C - 
56
47
D - 
8
5
E = 
480000
47
    (10)
, then subtract both sides of equation (10) from both sides of equation (5), the equations are reduced to:
 
3
25
A + 
3
25
B - 
41
50
C + 
41
50
D + 
41
50
E = -6000    (1)
 
8
5
B - 
14
3
C + 
94
15
D + 
94
15
E = -12000    (2)
 
94
15
C - 
14
3
D - 
94
15
E = 40000    (3)
- 
8
5
D + 
8
5
E = 0    (4)
 
656
235
D + 
16
5
E = 
883000
47
    (5)

Multiply both sides of equation (4) by 82
Divide the two sides of equation (4) by 47, the equation can be obtained:
        - 
656
235
D + 
656
235
E = 0    (11)
, then add the two sides of equation (11) to both sides of equation (5), the equations are reduced to:
 
3
25
A + 
3
25
B - 
41
50
C + 
41
50
D + 
41
50
E = -6000    (1)
 
8
5
B - 
14
3
C + 
94
15
D + 
94
15
E = -12000    (2)
 
94
15
C - 
14
3
D - 
94
15
E = 40000    (3)
- 
8
5
D + 
8
5
E = 0    (4)
 
1408
235
E = 
883000
47
    (5)

Multiply both sides of equation (5) by 47
Divide both sides of equation (5) by 176, get the equation：
         
8
5
E = 
110375
22
    (12)
, then subtract both sides of equation (12) from both sides of equation (4), get the equation：
 
3
25
A + 
3
25
B - 
41
50
C + 
41
50
D + 
41
50
E = -6000    (1)
 
8
5
B - 
14
3
C + 
94
15
D + 
94
15
E = -12000    (2)
 
94
15
C - 
14
3
D - 
94
15
E = 40000    (3)
- 
8
5
D = - 
110375
22
    (4)
 
1408
235
E = 
883000
47
    (5)

Multiply both sides of equation (5) by 2209
Divide both sides of equation (5) by 2112, get the equation：
         
4418
705
E = 
5187625
264
    (13)
, then add the two sides of equation (13) to both sides of equation (3), get the equation：
 
3
25
A + 
3
25
B - 
41
50
C + 
41
50
D + 
41
50
E = -6000    (1)
 
8
5
B - 
14
3
C + 
94
15
D + 
94
15
E = -12000    (2)
 
94
15
C - 
14
3
D = 
15747625
264
    (3)
- 
8
5
D = - 
110375
22
    (4)
 
1408
235
E = 
883000
47
    (5)

Multiply both sides of equation (5) by 2209
Divide both sides of equation (5) by 2112, get the equation：
         
4418
705
E = 
5187625
264
    (14)
, then subtract both sides of equation (14) from both sides of equation (2), get the equation：
 
3
25
A + 
3
25
B - 
41
50
C + 
41
50
D + 
41
50
E = -6000    (1)
 
8
5
B - 
14
3
C + 
94
15
D = - 
8355625
264
    (2)
 
94
15
C - 
14
3
D = 
15747625
264
    (3)
- 
8
5
D = - 
110375
22
    (4)
 
1408
235
E = 
883000
47
    (5)

Multiply both sides of equation (5) by 1927
Divide both sides of equation (5) by 14080, get the equation：
         
1927
2350
E = 
905075
352
    (15)
, then subtract both sides of equation (15) from both sides of equation (1), get the equation：
 
3
25
A + 
3
25
B - 
41
50
C + 
41
50
D = - 
3017075
352
    (1)
 
8
5
B - 
14
3
C + 
94
15
D = - 
8355625
264
    (2)
 
94
15
C - 
14
3
D = 
15747625
264
    (3)
- 
8
5
D = - 
110375
22
    (4)
 
1408
235
E = 
883000
47
    (5)

Multiply both sides of equation (4) by 35
Divide both sides of equation (4) by 12, get the equation：
        - 
14
3
D = - 
3863125
264
    (16)
, then subtract both sides of equation (16) from both sides of equation (3), get the equation：
 
3
25
A + 
3
25
B - 
41
50
C + 
41
50
D = - 
3017075
352
    (1)
 
8
5
B - 
14
3
C + 
94
15
D = - 
8355625
264
    (2)
 
94
15
C = 
9805375
132
    (3)
- 
8
5
D = - 
110375
22
    (4)
 E = 
551875
176
    (5)

Multiply both sides of equation (4) by 47
Divide both sides of equation (4) by 12, get the equation：
        - 
94
15
D = - 
5187625
264
    (17)
, then add the two sides of equation (17) to both sides of equation (2), get the equation：
 
3
25
A + 
3
25
B - 
41
50
C + 
41
50
D = - 
3017075
352
    (1)
 
8
5
B - 
14
3
C = - 
6771625
132
    (2)
 
94
15
C = 
9805375
132
    (3)
- 
8
5
D = - 
110375
22
    (4)
 E = 
551875
176
    (5)

Multiply both sides of equation (4) by 41
Divide both sides of equation (4) by 80, get the equation：
        - 
41
50
D = - 
905075
352
    (18)
, then add the two sides of equation (18) to both sides of equation (1), get the equation：
 
3
25
A + 
3
25
B - 
41
50
C = - 
1961075
176
    (1)
 
8
5
B - 
14
3
C = - 
6771625
132
    (2)
 
94
15
C = 
9805375
132
    (3)
- 
8
5
D = - 
110375
22
    (4)
 E = 
551875
176
    (5)

Multiply both sides of equation (3) by 35
Divide both sides of equation (3) by 47, get the equation：
         
14
3
C = 
7301875
132
    (19)
, then add the two sides of equation (19) to both sides of equation (2), get the equation：
 
3
25
A + 
3
25
B - 
41
50
C = - 
1961075
176
    (1)
 
8
5
B = 
88375
22
    (2)
 
94
15
C = 
9805375
132
    (3)
 D = 
551875
176
    (4)
 E = 
551875
176
    (5)

Multiply both sides of equation (3) by 123
Divide both sides of equation (3) by 940, get the equation：
         
41
50
C = 
1710725
176
    (20)
, then add the two sides of equation (20) to both sides of equation (1), get the equation：
 
3
25
A + 
3
25
B = - 
125175
88
    (1)
 
8
5
B = 
88375
22
    (2)
 
94
15
C = 
9805375
132
    (3)
 D = 
551875
176
    (4)
 E = 
551875
176
    (5)

Multiply both sides of equation (2) by 3
Divide both sides of equation (2) by 40, get the equation：
         
3
25
B = 
53025
176
    (21)
, then subtract both sides of equation (21) from both sides of equation (1), get the equation：
 
3
25
A = - 
303375
176
    (1)
 
8
5
B = 
88375
22
    (2)
 C = 
1043125
88
    (3)
 D = 
551875
176
    (4)
 E = 
551875
176
    (5)

The coefficient of the unknown number is reduced to 1, and the equations are reduced to：
 A = - 
2528125
176
    (1)
 B = 
441875
176
    (2)
 C = 
1043125
88
    (3)
 D = 
551875
176
    (4)
 E = 
551875
176
    (5)


Therefore, the solution of the equation set is:
A = - 
2528125
176
B = 
441875
176
C = 
1043125
88
D = 
551875
176
E = 
551875
176


Convert the solution of the equation set to decimals:
A = -14364.346591
B = 2510.653409
C = 11853.693182
D = 3135.653409
E = 3135.653409

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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