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:
-2A -2B -1C + 
1
2
E + 
1
2
F = -1    (1)
 A + B + C -1D -1E -1F = 0    (2)
-2A -8B -2C + E + F = -1    (3)
-1A -2B -4C + E = -1    (4)
 
1
2
A + B + C -1E = -1    (5)
 
1
2
A + B -1F = -1    (6)
Question solving process:

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

Subtract both sides of equation (1) from both sides of equation (3) ,the equations are reduced to:
-2A -2B -1C + 
1
2
E + 
1
2
F = -1    (1)
 
1
2
C -1D - 
3
4
E - 
3
4
F = - 
1
2
    (2)
-6B -1C + 
1
2
E + 
1
2
F = 0    (3)
-1A -2B -4C + E = -1    (4)
 
1
2
A + B + C -1E = -1    (5)
 
1
2
A + B -1F = -1    (6)

Divide the two sides of equation (1) by 2, the equation can be obtained:
        -1A -1B - 
1
2
C + 
1
4
E + 
1
4
F = - 
1
2
    (8)
, then subtract both sides of equation (8) from both sides of equation (4), the equations are reduced to:
-2A -2B -1C + 
1
2
E + 
1
2
F = -1    (1)
 
1
2
C -1D - 
3
4
E - 
3
4
F = - 
1
2
    (2)
-6B -1C + 
1
2
E + 
1
2
F = 0    (3)
-1B - 
7
2
C + 
3
4
E - 
1
4
F = - 
1
2
    (4)
 
1
2
A + B + C -1E = -1    (5)
 
1
2
A + B -1F = -1    (6)

Divide the two sides of equation (1) by 4, the equation can be obtained:
        - 
1
2
A - 
1
2
B - 
1
4
C + 
1
8
E + 
1
8
F = - 
1
4
    (9)
, then add the two sides of equation (9) to both sides of equation (5), the equations are reduced to:
-2A -2B -1C + 
1
2
E + 
1
2
F = -1    (1)
 
1
2
C -1D - 
3
4
E - 
3
4
F = - 
1
2
    (2)
-6B -1C + 
1
2
E + 
1
2
F = 0    (3)
-1B - 
7
2
C + 
3
4
E - 
1
4
F = - 
1
2
    (4)
 
1
2
B + 
3
4
C - 
7
8
E + 
1
8
F = - 
5
4
    (5)
 
1
2
A + B -1F = -1    (6)

Divide the two sides of equation (1) by 4, the equation can be obtained:
        - 
1
2
A - 
1
2
B - 
1
4
C + 
1
8
E + 
1
8
F = - 
1
4
    (10)
, then add the two sides of equation (10) to both sides of equation (6), the equations are reduced to:
-2A -2B -1C + 
1
2
E + 
1
2
F = -1    (1)
 
1
2
C -1D - 
3
4
E - 
3
4
F = - 
1
2
    (2)
-6B -1C + 
1
2
E + 
1
2
F = 0    (3)
-1B - 
7
2
C + 
3
4
E - 
1
4
F = - 
1
2
    (4)
 
1
2
B + 
3
4
C - 
7
8
E + 
1
8
F = - 
5
4
    (5)
 
1
2
B - 
1
4
C + 
1
8
E - 
7
8
F = - 
5
4
    (6)

交After the exchange of equation (2) and equation (3)， the equation system becomes:
-2A -2B -1C + 
1
2
E + 
1
2
F = -1    (1)
-6B -1C + 
1
2
E + 
1
2
F = 0    (2)
 
1
2
C -1D - 
3
4
E - 
3
4
F = - 
1
2
    (3)
-1B - 
7
2
C + 
3
4
E - 
1
4
F = - 
1
2
    (4)
 
1
2
B + 
3
4
C - 
7
8
E + 
1
8
F = - 
5
4
    (5)
 
1
2
B - 
1
4
C + 
1
8
E - 
7
8
F = - 
5
4
    (6)

Divide the two sides of equation (2) by 6, the equation can be obtained:
        -1B - 
1
6
C + 
1
12
E + 
1
12
F = 0    (11)
, then subtract both sides of equation (11) from both sides of equation (4), the equations are reduced to:
-2A -2B -1C + 
1
2
E + 
1
2
F = -1    (1)
-6B -1C + 
1
2
E + 
1
2
F = 0    (2)
 
1
2
C -1D - 
3
4
E - 
3
4
F = - 
1
2
    (3)
- 
10
3
C + 
2
3
E - 
1
3
F = - 
1
2
    (4)
 
1
2
B + 
3
4
C - 
7
8
E + 
1
8
F = - 
5
4
    (5)
 
1
2
B - 
1
4
C + 
1
8
E - 
7
8
F = - 
5
4
    (6)

Divide the two sides of equation (2) by 12, the equation can be obtained:
        - 
1
2
B - 
1
12
C + 
1
24
E + 
1
24
F = 0    (12)
, then add the two sides of equation (12) to both sides of equation (5), the equations are reduced to:
-2A -2B -1C + 
1
2
E + 
1
2
F = -1    (1)
-6B -1C + 
1
2
E + 
1
2
F = 0    (2)
 
1
2
C -1D - 
3
4
E - 
3
4
F = - 
1
2
    (3)
- 
10
3
C + 
2
3
E - 
1
3
F = - 
1
2
    (4)
 
2
3
C - 
5
6
E + 
1
6
F = - 
5
4
    (5)
 
1
2
B - 
1
4
C + 
1
8
E - 
7
8
F = - 
5
4
    (6)

Divide the two sides of equation (2) by 12, the equation can be obtained:
        - 
1
2
B - 
1
12
C + 
1
24
E + 
1
24
F = 0    (13)
, then add the two sides of equation (13) to both sides of equation (6), the equations are reduced to:
-2A -2B -1C + 
1
2
E + 
1
2
F = -1    (1)
-6B -1C + 
1
2
E + 
1
2
F = 0    (2)
 
1
2
C -1D - 
3
4
E - 
3
4
F = - 
1
2
    (3)
- 
10
3
C + 
2
3
E - 
1
3
F = - 
1
2
    (4)
 
2
3
C - 
5
6
E + 
1
6
F = - 
5
4
    (5)
- 
1
3
C + 
1
6
E - 
5
6
F = - 
5
4
    (6)

Multiply both sides of equation (3) by 20
Divide the two sides of equation (3) by 3, the equation can be obtained:
         
10
3
C - 
20
3
D -5E -5F = - 
10
3
    (14)
, then add the two sides of equation (14) to both sides of equation (4), the equations are reduced to:
-2A -2B -1C + 
1
2
E + 
1
2
F = -1    (1)
-6B -1C + 
1
2
E + 
1
2
F = 0    (2)
 
1
2
C -1D - 
3
4
E - 
3
4
F = - 
1
2
    (3)
- 
20
3
D - 
13
3
E - 
16
3
F = - 
23
6
    (4)
 
2
3
C - 
5
6
E + 
1
6
F = - 
5
4
    (5)
- 
1
3
C + 
1
6
E - 
5
6
F = - 
5
4
    (6)

Multiply both sides of equation (3) by 4
Divide the two sides of equation (3) by 3, the equation can be obtained:
         
2
3
C - 
4
3
D -1E -1F = - 
2
3
    (15)
, then subtract both sides of equation (15) from both sides of equation (5), the equations are reduced to:
-2A -2B -1C + 
1
2
E + 
1
2
F = -1    (1)
-6B -1C + 
1
2
E + 
1
2
F = 0    (2)
 
1
2
C -1D - 
3
4
E - 
3
4
F = - 
1
2
    (3)
- 
20
3
D - 
13
3
E - 
16
3
F = - 
23
6
    (4)
 
4
3
D + 
1
6
E + 
7
6
F = - 
7
12
    (5)
- 
1
3
C + 
1
6
E - 
5
6
F = - 
5
4
    (6)

Multiply both sides of equation (3) by 2
Divide the two sides of equation (3) by 3, the equation can be obtained:
         
1
3
C - 
2
3
D - 
1
2
E - 
1
2
F = - 
1
3
    (16)
, then add the two sides of equation (16) to both sides of equation (6), the equations are reduced to:
-2A -2B -1C + 
1
2
E + 
1
2
F = -1    (1)
-6B -1C + 
1
2
E + 
1
2
F = 0    (2)
 
1
2
C -1D - 
3
4
E - 
3
4
F = - 
1
2
    (3)
- 
20
3
D - 
13
3
E - 
16
3
F = - 
23
6
    (4)
 
4
3
D + 
1
6
E + 
7
6
F = - 
7
12
    (5)
- 
2
3
D - 
1
3
E - 
4
3
F = - 
19
12
    (6)

Divide the two sides of equation (4) by 5, the equation can be obtained:
        - 
4
3
D - 
13
15
E - 
16
15
F = - 
23
30
    (17)
, then add the two sides of equation (17) to both sides of equation (5), the equations are reduced to:
-2A -2B -1C + 
1
2
E + 
1
2
F = -1    (1)
-6B -1C + 
1
2
E + 
1
2
F = 0    (2)
 
1
2
C -1D - 
3
4
E - 
3
4
F = - 
1
2
    (3)
- 
20
3
D - 
13
3
E - 
16
3
F = - 
23
6
    (4)
- 
7
10
E + 
1
10
F = - 
27
20
    (5)
- 
2
3
D - 
1
3
E - 
4
3
F = - 
19
12
    (6)

Divide the two sides of equation (4) by 10, the equation can be obtained:
        - 
2
3
D - 
13
30
E - 
8
15
F = - 
23
60
    (18)
, then subtract both sides of equation (18) from both sides of equation (6), the equations are reduced to:
-2A -2B -1C + 
1
2
E + 
1
2
F = -1    (1)
-6B -1C + 
1
2
E + 
1
2
F = 0    (2)
 
1
2
C -1D - 
3
4
E - 
3
4
F = - 
1
2
    (3)
- 
20
3
D - 
13
3
E - 
16
3
F = - 
23
6
    (4)
- 
7
10
E + 
1
10
F = - 
27
20
    (5)
 
1
10
E - 
4
5
F = - 
6
5
    (6)

Divide the two sides of equation (5) by 7, the equation can be obtained:
        - 
1
10
E + 
1
70
F = - 
27
140
    (19)
, then add the two sides of equation (19) to both sides of equation (6), the equations are reduced to:
-2A -2B -1C + 
1
2
E + 
1
2
F = -1    (1)
-6B -1C + 
1
2
E + 
1
2
F = 0    (2)
 
1
2
C -1D - 
3
4
E - 
3
4
F = - 
1
2
    (3)
- 
20
3
D - 
13
3
E - 
16
3
F = - 
23
6
    (4)
- 
7
10
E + 
1
10
F = - 
27
20
    (5)
- 
11
14
F = - 
39
28
    (6)

Multiply both sides of equation (6) by 7
Divide both sides of equation (6) by 55, get the equation：
        - 
1
10
F = - 
39
220
    (20)
, then add the two sides of equation (20) to both sides of equation (5), get the equation：
-2A -2B -1C + 
1
2
E + 
1
2
F = -1    (1)
-6B -1C + 
1
2
E + 
1
2
F = 0    (2)
 
1
2
C -1D - 
3
4
E - 
3
4
F = - 
1
2
    (3)
- 
20
3
D - 
13
3
E - 
16
3
F = - 
23
6
    (4)
- 
7
10
E = - 
84
55
    (5)
- 
11
14
F = - 
39
28
    (6)

Multiply both sides of equation (6) by 224
Divide both sides of equation (6) by 33, get the equation：
        - 
16
3
F = - 
104
11
    (21)
, then subtract both sides of equation (21) from both sides of equation (4), get the equation：
-2A -2B -1C + 
1
2
E + 
1
2
F = -1    (1)
-6B -1C + 
1
2
E + 
1
2
F = 0    (2)
 
1
2
C -1D - 
3
4
E - 
3
4
F = - 
1
2
    (3)
- 
20
3
D - 
13
3
E = 
371
66
    (4)
- 
7
10
E = - 
84
55
    (5)
- 
11
14
F = - 
39
28
    (6)

Multiply both sides of equation (6) by 21
Divide both sides of equation (6) by 22, get the equation：
        - 
3
4
F = - 
117
88
    (22)
, then subtract both sides of equation (22) from both sides of equation (3), get the equation：
-2A -2B -1C + 
1
2
E + 
1
2
F = -1    (1)
-6B -1C + 
1
2
E + 
1
2
F = 0    (2)
 
1
2
C -1D - 
3
4
E = 
73
88
    (3)
- 
20
3
D - 
13
3
E = 
371
66
    (4)
- 
7
10
E = - 
84
55
    (5)
- 
11
14
F = - 
39
28
    (6)

Multiply both sides of equation (6) by 7
Divide both sides of equation (6) by 11, get the equation：
        - 
1
2
F = - 
39
44
    (23)
, then add the two sides of equation (23) to both sides of equation (2), get the equation：
-2A -2B -1C + 
1
2
E + 
1
2
F = -1    (1)
-6B -1C + 
1
2
E = - 
39
44
    (2)
 
1
2
C -1D - 
3
4
E = 
73
88
    (3)
- 
20
3
D - 
13
3
E = 
371
66
    (4)
- 
7
10
E = - 
84
55
    (5)
- 
11
14
F = - 
39
28
    (6)

Multiply both sides of equation (6) by 7
Divide both sides of equation (6) by 11, get the equation：
        - 
1
2
F = - 
39
44
    (24)
, then add the two sides of equation (24) to both sides of equation (1), get the equation：
-2A -2B -1C + 
1
2
E = - 
83
44
    (1)
-6B -1C + 
1
2
E = - 
39
44
    (2)
 
1
2
C -1D - 
3
4
E = 
73
88
    (3)
- 
20
3
D - 
13
3
E = 
371
66
    (4)
- 
7
10
E = - 
84
55
    (5)
- 
11
14
F = - 
39
28
    (6)

Multiply both sides of equation (5) by 130
Divide both sides of equation (5) by 21, get the equation：
        - 
13
3
E = - 
104
11
    (25)
, then subtract both sides of equation (25) from both sides of equation (4), get the equation：
-2A -2B -1C + 
1
2
E = - 
83
44
    (1)
-6B -1C + 
1
2
E = - 
39
44
    (2)
 
1
2
C -1D - 
3
4
E = 
73
88
    (3)
- 
20
3
D = 
995
66
    (4)
- 
7
10
E = - 
84
55
    (5)
 F = 
39
22
    (6)

Multiply both sides of equation (5) by 15
Divide both sides of equation (5) by 14, get the equation：
        - 
3
4
E = - 
18
11
    (26)
, then subtract both sides of equation (26) from both sides of equation (3), get the equation：
-2A -2B -1C + 
1
2
E = - 
83
44
    (1)
-6B -1C + 
1
2
E = - 
39
44
    (2)
 
1
2
C -1D = 
217
88
    (3)
- 
20
3
D = 
995
66
    (4)
- 
7
10
E = - 
84
55
    (5)
 F = 
39
22
    (6)

Multiply both sides of equation (5) by 5
Divide both sides of equation (5) by 7, get the equation：
        - 
1
2
E = - 
12
11
    (27)
, then add the two sides of equation (27) to both sides of equation (2), get the equation：
-2A -2B -1C + 
1
2
E = - 
83
44
    (1)
-6B -1C = - 
87
44
    (2)
 
1
2
C -1D = 
217
88
    (3)
- 
20
3
D = 
995
66
    (4)
- 
7
10
E = - 
84
55
    (5)
 F = 
39
22
    (6)

Multiply both sides of equation (5) by 5
Divide both sides of equation (5) by 7, get the equation：
        - 
1
2
E = - 
12
11
    (28)
, then add the two sides of equation (28) to both sides of equation (1), get the equation：
-2A -2B -1C = - 
131
44
    (1)
-6B -1C = - 
87
44
    (2)
 
1
2
C -1D = 
217
88
    (3)
- 
20
3
D = 
995
66
    (4)
- 
7
10
E = - 
84
55
    (5)
 F = 
39
22
    (6)

Multiply both sides of equation (4) by 3
Divide both sides of equation (4) by 20, get the equation：
        -1D = 
199
88
    (29)
, then subtract both sides of equation (29) from both sides of equation (3), get the equation：
-2A -2B -1C = - 
131
44
    (1)
-6B -1C = - 
87
44
    (2)
 
1
2
C = 
9
44
    (3)
- 
20
3
D = 
995
66
    (4)
 E = 
24
11
    (5)
 F = 
39
22
    (6)

Multiply both sides of equation (3) by 2, get the equation：
         C = 
9
22
    (30)
, then add the two sides of equation (30) to both sides of equation (2), get the equation：
-2A -2B -1C = - 
131
44
    (1)
-6B = - 
69
44
    (2)
 
1
2
C = 
9
44
    (3)
 D = - 
199
88
    (4)
 E = 
24
11
    (5)
 F = 
39
22
    (6)

Multiply both sides of equation (3) by 2, get the equation：
         C = 
9
22
    (31)
, then add the two sides of equation (31) to both sides of equation (1), get the equation：
-2A -2B = - 
113
44
    (1)
-6B = - 
69
44
    (2)
 
1
2
C = 
9
44
    (3)
 D = - 
199
88
    (4)
 E = 
24
11
    (5)
 F = 
39
22
    (6)

Divide both sides of equation (2) by 3, get the equation：
        -2B = - 
23
44
    (32)
, then subtract both sides of equation (32) from both sides of equation (1), get the equation：
-2A = - 
45
22
    (1)
-6B = - 
69
44
    (2)
 C = 
9
22
    (3)
 D = - 
199
88
    (4)
 E = 
24
11
    (5)
 F = 
39
22
    (6)

The coefficient of the unknown number is reduced to 1, and the equations are reduced to：
 A = 
45
44
    (1)
 B = 
23
88
    (2)
 C = 
9
22
    (3)
 D = - 
199
88
    (4)
 E = 
24
11
    (5)
 F = 
39
22
    (6)


Therefore, the solution of the equation set is:
A = 
45
44
B = 
23
88
C = 
9
22
D = - 
199
88
E = 
24
11
F = 
39
22


Convert the solution of the equation set to decimals:
A = 1.022727
B = 0.261364
C = 0.409091
D = -2.261364
E = 2.181818
F = 1.772727

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。