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:
 5A + B + C + D = 1    (1)
 A + 5B + C + D = 5    (2)
 A + B + 5C + D = 25    (3)
 A + B + C + 5D = 125    (4)
Question solving process:

Divide the two sides of equation (1) by 5, the equation can be obtained:
         A + 
1
5
B + 
1
5
C + 
1
5
D = 
1
5
    (5)
, then subtract both sides of equation (5) from both sides of equation (2), the equations are reduced to:
 5A + B + C + D = 1    (1)
 
24
5
B + 
4
5
C + 
4
5
D = 
24
5
    (2)
 A + B + 5C + D = 25    (3)
 A + B + C + 5D = 125    (4)

Divide the two sides of equation (1) by 5, the equation can be obtained:
         A + 
1
5
B + 
1
5
C + 
1
5
D = 
1
5
    (6)
, then subtract both sides of equation (6) from both sides of equation (3), the equations are reduced to:
 5A + B + C + D = 1    (1)
 
24
5
B + 
4
5
C + 
4
5
D = 
24
5
    (2)
 
4
5
B + 
24
5
C + 
4
5
D = 
124
5
    (3)
 A + B + C + 5D = 125    (4)

Divide the two sides of equation (1) by 5, the equation can be obtained:
         A + 
1
5
B + 
1
5
C + 
1
5
D = 
1
5
    (7)
, then subtract both sides of equation (7) from both sides of equation (4), the equations are reduced to:
 5A + B + C + D = 1    (1)
 
24
5
B + 
4
5
C + 
4
5
D = 
24
5
    (2)
 
4
5
B + 
24
5
C + 
4
5
D = 
124
5
    (3)
 
4
5
B + 
4
5
C + 
24
5
D = 
624
5
    (4)

Divide the two sides of equation (2) by 6, the equation can be obtained:
         
4
5
B + 
2
15
C + 
2
15
D = 
4
5
    (8)
, then subtract both sides of equation (8) from both sides of equation (3), the equations are reduced to:
 5A + B + C + D = 1    (1)
 
24
5
B + 
4
5
C + 
4
5
D = 
24
5
    (2)
 
14
3
C + 
2
3
D = 24    (3)
 
4
5
B + 
4
5
C + 
24
5
D = 
624
5
    (4)

Divide the two sides of equation (2) by 6, the equation can be obtained:
         
4
5
B + 
2
15
C + 
2
15
D = 
4
5
    (9)
, then subtract both sides of equation (9) from both sides of equation (4), the equations are reduced to:
 5A + B + C + D = 1    (1)
 
24
5
B + 
4
5
C + 
4
5
D = 
24
5
    (2)
 
14
3
C + 
2
3
D = 24    (3)
 
2
3
C + 
14
3
D = 124    (4)

Divide the two sides of equation (3) by 7, the equation can be obtained:
         
2
3
C + 
2
21
D = 
24
7
    (10)
, then subtract both sides of equation (10) from both sides of equation (4), the equations are reduced to:
 5A + B + C + D = 1    (1)
 
24
5
B + 
4
5
C + 
4
5
D = 
24
5
    (2)
 
14
3
C + 
2
3
D = 24    (3)
 
32
7
D = 
844
7
    (4)

Multiply both sides of equation (4) by 7
Divide both sides of equation (4) by 48, get the equation：
         
2
3
D = 
211
12
    (11)
, then subtract both sides of equation (11) from both sides of equation (3), get the equation：
 5A + B + C + D = 1    (1)
 
24
5
B + 
4
5
C + 
4
5
D = 
24
5
    (2)
 
14
3
C = 
77
12
    (3)
 
32
7
D = 
844
7
    (4)

Multiply both sides of equation (4) by 7
Divide both sides of equation (4) by 40, get the equation：
         
4
5
D = 
211
10
    (12)
, then subtract both sides of equation (12) from both sides of equation (2), get the equation：
 5A + B + C + D = 1    (1)
 
24
5
B + 
4
5
C = - 
163
10
    (2)
 
14
3
C = 
77
12
    (3)
 
32
7
D = 
844
7
    (4)

Multiply both sides of equation (4) by 7
Divide both sides of equation (4) by 32, get the equation：
         D = 
211
8
    (13)
, then subtract both sides of equation (13) from both sides of equation (1), get the equation：
 5A + B + C = - 
203
8
    (1)
 
24
5
B + 
4
5
C = - 
163
10
    (2)
 
14
3
C = 
77
12
    (3)
 
32
7
D = 
844
7
    (4)

Multiply both sides of equation (3) by 6
Divide both sides of equation (3) by 35, get the equation：
         
4
5
C = 
11
10
    (14)
, then subtract both sides of equation (14) from both sides of equation (2), get the equation：
 5A + B + C = - 
203
8
    (1)
 
24
5
B = - 
87
5
    (2)
 
14
3
C = 
77
12
    (3)
 D = 
211
8
    (4)

Multiply both sides of equation (3) by 3
Divide both sides of equation (3) by 14, get the equation：
         C = 
11
8
    (15)
, then subtract both sides of equation (15) from both sides of equation (1), get the equation：
 5A + B = - 
107
4
    (1)
 
24
5
B = - 
87
5
    (2)
 
14
3
C = 
77
12
    (3)
 D = 
211
8
    (4)

Multiply both sides of equation (2) by 5
Divide both sides of equation (2) by 24, get the equation：
         B = - 
29
8
    (16)
, then subtract both sides of equation (16) from both sides of equation (1), get the equation：
 5A = - 
185
8
    (1)
 
24
5
B = - 
87
5
    (2)
 C = 
11
8
    (3)
 D = 
211
8
    (4)

The coefficient of the unknown number is reduced to 1, and the equations are reduced to：
 A = - 
37
8
    (1)
 B = - 
29
8
    (2)
 C = 
11
8
    (3)
 D = 
211
8
    (4)


Therefore, the solution of the equation set is:
A = - 
37
8
B = - 
29
8
C = 
11
8
D = 
211
8


Convert the solution of the equation set to decimals:
A = -4.625000
B = -3.625000
C = 1.375000
D = 26.375000

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。