Mathematics
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current location：Equations > Multivariate equations > Answer
Detailed information：
The input equation set is:
 3A + B = 1450    (1)
 6B + 8C + D = 2900    (2)
 A + 5B + 2C + 2D = 2650    (3)
 A + 5C + 2D = 800    (4)
Question solving process:

Divide the two sides of equation (1) by 3, the equation can be obtained:
         A + 
1
3
B = 
1450
3
    (5)
, then subtract both sides of equation (5) from both sides of equation (3), the equations are reduced to:
 3A + B = 1450    (1)
 6B + 8C + D = 2900    (2)
 
14
3
B + 2C + 2D = 
6500
3
    (3)
 A + 5C + 2D = 800    (4)

Divide the two sides of equation (1) by 3, the equation can be obtained:
         A + 
1
3
B = 
1450
3
    (6)
, then subtract both sides of equation (6) from both sides of equation (4), the equations are reduced to:
 3A + B = 1450    (1)
 6B + 8C + D = 2900    (2)
 
14
3
B + 2C + 2D = 
6500
3
    (3)
- 
1
3
B + 5C + 2D = 
950
3
    (4)

Multiply both sides of equation (2) by 7
Divide the two sides of equation (2) by 9, the equation can be obtained:
         
14
3
B + 
56
9
C + 
7
9
D = 
20300
9
    (7)
, then subtract both sides of equation (7) from both sides of equation (3), the equations are reduced to:
 3A + B = 1450    (1)
 6B + 8C + D = 2900    (2)
- 
38
9
C + 
11
9
D = - 
800
9
    (3)
- 
1
3
B + 5C + 2D = 
950
3
    (4)

Divide the two sides of equation (2) by 18, the equation can be obtained:
         
1
3
B + 
4
9
C + 
1
18
D = 
1450
9
    (8)
, then add the two sides of equation (8) to both sides of equation (4), the equations are reduced to:
 3A + B = 1450    (1)
 6B + 8C + D = 2900    (2)
- 
38
9
C + 
11
9
D = - 
800
9
    (3)
 
49
9
C + 
37
18
D = 
4300
9
    (4)

Multiply both sides of equation (3) by 49
Divide the two sides of equation (3) by 38, the equation can be obtained:
        - 
49
9
C + 
539
342
D = - 
19600
171
    (9)
, then add the two sides of equation (9) to both sides of equation (4), the equations are reduced to:
 3A + B = 1450    (1)
 6B + 8C + D = 2900    (2)
- 
38
9
C + 
11
9
D = - 
800
9
    (3)
 
69
19
D = 
6900
19
    (4)

Multiply both sides of equation (4) by 209
Divide both sides of equation (4) by 621, get the equation：
         
11
9
D = 
25300
207
    (10)
, then subtract both sides of equation (10) from both sides of equation (3), get the equation：
 3A + B = 1450    (1)
 6B + 8C + D = 2900    (2)
- 
38
9
C = - 
43700
207
    (3)
 
69
19
D = 
6900
19
    (4)

Multiply both sides of equation (4) by 19
Divide both sides of equation (4) by 69, get the equation：
         D = 100    (11)
, then subtract both sides of equation (11) from both sides of equation (2), get the equation：
 3A + B = 1450    (1)
 6B + 8C = 2800    (2)
- 
38
9
C = - 
43700
207
    (3)
 
69
19
D = 
6900
19
    (4)

Multiply both sides of equation (3) by 36
Divide both sides of equation (3) by 19, get the equation：
        -8C = -400    (12)
, then add the two sides of equation (12) to both sides of equation (2), get the equation：
 3A + B = 1450    (1)
 6B = 2400    (2)
- 
38
9
C = - 
43700
207
    (3)
 D = 100    (4)

Divide both sides of equation (2) by 6, get the equation：
         B = 400    (13)
, then subtract both sides of equation (13) from both sides of equation (1), get the equation：
 3A = 1050    (1)
 6B = 2400    (2)
 C = 50    (3)
 D = 100    (4)

The coefficient of the unknown number is reduced to 1, and the equations are reduced to：
 A = 350    (1)
 B = 400    (2)
 C = 50    (3)
 D = 100    (4)


Therefore, the solution of the equation set is:
A = 350
B = 400
C = 50
D = 100

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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