Mathematics
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current location：Equations > Multivariate equations > Answer
Detailed information：
The input equation set is:
-3A + B -1C = 0    (1)
 A + 3B -1C -2D = 0    (2)
-1A -1B + 2C -1E -1F = 0    (3)
-2B + 6D -4F = 15    (4)
-1C + 6E = 20    (5)
-1C -4D + 6F = 0    (6)
Question solving process:

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

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

Multiply both sides of equation (2) by 2
Divide the two sides of equation (2) by 5, the equation can be obtained:
         
4
3
B - 
8
15
C - 
4
5
D = 0    (9)
, then add the two sides of equation (9) to both sides of equation (3), the equations are reduced to:
-3A + B -1C = 0    (1)
 
10
3
B - 
4
3
C -2D = 0    (2)
 
9
5
C - 
4
5
D -1E -1F = 0    (3)
-2B + 6D -4F = 15    (4)
-1C + 6E = 20    (5)
-1C -4D + 6F = 0    (6)

Multiply both sides of equation (2) by 3
Divide the two sides of equation (2) by 5, the equation can be obtained:
         2B - 
4
5
C - 
6
5
D = 0    (10)
, then add the two sides of equation (10) to both sides of equation (4), the equations are reduced to:
-3A + B -1C = 0    (1)
 
10
3
B - 
4
3
C -2D = 0    (2)
 
9
5
C - 
4
5
D -1E -1F = 0    (3)
- 
4
5
C + 
24
5
D -4F = 15    (4)
-1C + 6E = 20    (5)
-1C -4D + 6F = 0    (6)

Multiply both sides of equation (3) by 4
Divide the two sides of equation (3) by 9, the equation can be obtained:
         
4
5
C - 
16
45
D - 
4
9
E - 
4
9
F = 0    (11)
, then add the two sides of equation (11) to both sides of equation (4), the equations are reduced to:
-3A + B -1C = 0    (1)
 
10
3
B - 
4
3
C -2D = 0    (2)
 
9
5
C - 
4
5
D -1E -1F = 0    (3)
 
40
9
D - 
4
9
E - 
40
9
F = 15    (4)
-1C + 6E = 20    (5)
-1C -4D + 6F = 0    (6)

Multiply both sides of equation (3) by 5
Divide the two sides of equation (3) by 9, the equation can be obtained:
         C - 
4
9
D - 
5
9
E - 
5
9
F = 0    (12)
, then add the two sides of equation (12) to both sides of equation (5), the equations are reduced to:
-3A + B -1C = 0    (1)
 
10
3
B - 
4
3
C -2D = 0    (2)
 
9
5
C - 
4
5
D -1E -1F = 0    (3)
 
40
9
D - 
4
9
E - 
40
9
F = 15    (4)
- 
4
9
D + 
49
9
E - 
5
9
F = 20    (5)
-1C -4D + 6F = 0    (6)

Multiply both sides of equation (3) by 5
Divide the two sides of equation (3) by 9, the equation can be obtained:
         C - 
4
9
D - 
5
9
E - 
5
9
F = 0    (13)
, then add the two sides of equation (13) to both sides of equation (6), the equations are reduced to:
-3A + B -1C = 0    (1)
 
10
3
B - 
4
3
C -2D = 0    (2)
 
9
5
C - 
4
5
D -1E -1F = 0    (3)
 
40
9
D - 
4
9
E - 
40
9
F = 15    (4)
- 
4
9
D + 
49
9
E - 
5
9
F = 20    (5)
- 
40
9
D - 
5
9
E + 
49
9
F = 0    (6)

Divide the two sides of equation (4) by 10, the equation can be obtained:
         
4
9
D - 
2
45
E - 
4
9
F = 
3
2
    (14)
, then add the two sides of equation (14) to both sides of equation (5), the equations are reduced to:
-3A + B -1C = 0    (1)
 
10
3
B - 
4
3
C -2D = 0    (2)
 
9
5
C - 
4
5
D -1E -1F = 0    (3)
 
40
9
D - 
4
9
E - 
40
9
F = 15    (4)
 
27
5
E -1F = 
43
2
    (5)
- 
40
9
D - 
5
9
E + 
49
9
F = 0    (6)

Add both sides of equation (4) to both sides of equation (6) ,the equations are reduced to:
-3A + B -1C = 0    (1)
 
10
3
B - 
4
3
C -2D = 0    (2)
 
9
5
C - 
4
5
D -1E -1F = 0    (3)
 
40
9
D - 
4
9
E - 
40
9
F = 15    (4)
 
27
5
E -1F = 
43
2
    (5)
-1E + F = 15    (6)

Multiply both sides of equation (5) by 5
Divide the two sides of equation (5) by 27, the equation can be obtained:
         E - 
5
27
F = 
215
54
    (15)
, then add the two sides of equation (15) to both sides of equation (6), the equations are reduced to:
-3A + B -1C = 0    (1)
 
10
3
B - 
4
3
C -2D = 0    (2)
 
9
5
C - 
4
5
D -1E -1F = 0    (3)
 
40
9
D - 
4
9
E - 
40
9
F = 15    (4)
 
27
5
E -1F = 
43
2
    (5)
 
22
27
F = 
1025
54
    (6)

Multiply both sides of equation (6) by 27
Divide both sides of equation (6) by 22, get the equation：
         F = 
1025
44
    (16)
, then add the two sides of equation (16) to both sides of equation (5), get the equation：
-3A + B -1C = 0    (1)
 
10
3
B - 
4
3
C -2D = 0    (2)
 
9
5
C - 
4
5
D -1E -1F = 0    (3)
 
40
9
D - 
4
9
E - 
40
9
F = 15    (4)
 
27
5
E = 
1971
44
    (5)
 
22
27
F = 
1025
54
    (6)

Multiply both sides of equation (6) by 60
Divide both sides of equation (6) by 11, get the equation：
         
40
9
F = 
10250
99
    (17)
, then add the two sides of equation (17) to both sides of equation (4), get the equation：
-3A + B -1C = 0    (1)
 
10
3
B - 
4
3
C -2D = 0    (2)
 
9
5
C - 
4
5
D -1E -1F = 0    (3)
 
40
9
D - 
4
9
E = 
11735
99
    (4)
 
27
5
E = 
1971
44
    (5)
 
22
27
F = 
1025
54
    (6)

Multiply both sides of equation (6) by 27
Divide both sides of equation (6) by 22, get the equation：
         F = 
1025
44
    (18)
, then add the two sides of equation (18) to both sides of equation (3), get the equation：
-3A + B -1C = 0    (1)
 
10
3
B - 
4
3
C -2D = 0    (2)
 
9
5
C - 
4
5
D -1E = 
1025
44
    (3)
 
40
9
D - 
4
9
E = 
11735
99
    (4)
 
27
5
E = 
1971
44
    (5)
 
22
27
F = 
1025
54
    (6)

Multiply both sides of equation (5) by 20
Divide both sides of equation (5) by 243, get the equation：
         
4
9
E = 
365
99
    (19)
, then add the two sides of equation (19) to both sides of equation (4), get the equation：
-3A + B -1C = 0    (1)
 
10
3
B - 
4
3
C -2D = 0    (2)
 
9
5
C - 
4
5
D -1E = 
1025
44
    (3)
 
40
9
D = 
1100
9
    (4)
 
27
5
E = 
1971
44
    (5)
 F = 
1025
44
    (6)

Multiply both sides of equation (5) by 5
Divide both sides of equation (5) by 27, get the equation：
         E = 
365
44
    (20)
, then add the two sides of equation (20) to both sides of equation (3), get the equation：
-3A + B -1C = 0    (1)
 
10
3
B - 
4
3
C -2D = 0    (2)
 
9
5
C - 
4
5
D = 
695
22
    (3)
 
40
9
D = 
1100
9
    (4)
 
27
5
E = 
1971
44
    (5)
 F = 
1025
44
    (6)

Multiply both sides of equation (4) by 9
Divide both sides of equation (4) by 50, get the equation：
         
4
5
D = 22    (21)
, then add the two sides of equation (21) to both sides of equation (3), get the equation：
-3A + B -1C = 0    (1)
 
10
3
B - 
4
3
C -2D = 0    (2)
 
9
5
C = 
1179
22
    (3)
 
40
9
D = 
1100
9
    (4)
 E = 
365
44
    (5)
 F = 
1025
44
    (6)

Multiply both sides of equation (4) by 9
Divide both sides of equation (4) by 20, get the equation：
         2D = 55    (22)
, then add the two sides of equation (22) to both sides of equation (2), get the equation：
-3A + B -1C = 0    (1)
 
10
3
B - 
4
3
C = 55    (2)
 
9
5
C = 
1179
22
    (3)
 
40
9
D = 
1100
9
    (4)
 E = 
365
44
    (5)
 F = 
1025
44
    (6)

Multiply both sides of equation (3) by 20
Divide both sides of equation (3) by 27, get the equation：
         
4
3
C = 
1310
33
    (23)
, then add the two sides of equation (23) to both sides of equation (2), get the equation：
-3A + B -1C = 0    (1)
 
10
3
B = 
3125
33
    (2)
 
9
5
C = 
1179
22
    (3)
 D = 
55
2
    (4)
 E = 
365
44
    (5)
 F = 
1025
44
    (6)

Multiply both sides of equation (3) by 5
Divide both sides of equation (3) by 9, get the equation：
         C = 
655
22
    (24)
, then add the two sides of equation (24) to both sides of equation (1), get the equation：
-3A + B = 
655
22
    (1)
 
10
3
B = 
3125
33
    (2)
 
9
5
C = 
1179
22
    (3)
 D = 
55
2
    (4)
 E = 
365
44
    (5)
 F = 
1025
44
    (6)

Multiply both sides of equation (2) by 3
Divide both sides of equation (2) by 10, get the equation：
         B = 
625
22
    (25)
, then subtract both sides of equation (25) from both sides of equation (1), get the equation：
-3A = 
15
11
    (1)
 
10
3
B = 
3125
33
    (2)
 C = 
655
22
    (3)
 D = 
55
2
    (4)
 E = 
365
44
    (5)
 F = 
1025
44
    (6)

The coefficient of the unknown number is reduced to 1, and the equations are reduced to：
 A = - 
5
11
    (1)
 B = 
625
22
    (2)
 C = 
655
22
    (3)
 D = 
55
2
    (4)
 E = 
365
44
    (5)
 F = 
1025
44
    (6)


Therefore, the solution of the equation set is:
A = - 
5
11
B = 
625
22
C = 
655
22
D = 
55
2
E = 
365
44
F = 
1025
44


Convert the solution of the equation set to decimals:
A = -0.454545
B = 28.409091
C = 29.772727
D = 27.500000
E = 8.295455
F = 23.295455

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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