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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:
 A + 4B + 16C + D + 4E + F = 40    (1)
 9A + 12B + 16C + 3D + 4E + F = 24    (2)
 16A + 12B + 9C + 4D + 3E + F = 30    (3)
 A + 2B + 4C + D + 2E + F = 24    (4)
 4A + 2B + C + 2D + E + F = 32    (5)
 16A + 4B + C + 4D + E + F = 25    (6)
Question solving process:

Multiply both sides of equation (1) by 9, the equation can be obtained:
         9A + 36B + 144C + 9D + 36E + 9F = 360    (7)
, then subtract both sides of equation (7) from both sides of equation (2), the equations are reduced to:
 A + 4B + 16C + D + 4E + F = 40    (1)
-24B -128C -6D -32E -8F = -336    (2)
 16A + 12B + 9C + 4D + 3E + F = 30    (3)
 A + 2B + 4C + D + 2E + F = 24    (4)
 4A + 2B + C + 2D + E + F = 32    (5)
 16A + 4B + C + 4D + E + F = 25    (6)

Multiply both sides of equation (1) by 16, the equation can be obtained:
         16A + 64B + 256C + 16D + 64E + 16F = 640    (8)
, then subtract both sides of equation (8) from both sides of equation (3), the equations are reduced to:
 A + 4B + 16C + D + 4E + F = 40    (1)
-24B -128C -6D -32E -8F = -336    (2)
-52B -247C -12D -61E -15F = -610    (3)
 A + 2B + 4C + D + 2E + F = 24    (4)
 4A + 2B + C + 2D + E + F = 32    (5)
 16A + 4B + C + 4D + E + F = 25    (6)

Subtract both sides of equation (1) from both sides of equation (4) ,the equations are reduced to:
 A + 4B + 16C + D + 4E + F = 40    (1)
-24B -128C -6D -32E -8F = -336    (2)
-52B -247C -12D -61E -15F = -610    (3)
-2B -12C -2E = -16    (4)
 4A + 2B + C + 2D + E + F = 32    (5)
 16A + 4B + C + 4D + E + F = 25    (6)

Multiply both sides of equation (1) by 4, the equation can be obtained:
         4A + 16B + 64C + 4D + 16E + 4F = 160    (9)
, then subtract both sides of equation (9) from both sides of equation (5), the equations are reduced to:
 A + 4B + 16C + D + 4E + F = 40    (1)
-24B -128C -6D -32E -8F = -336    (2)
-52B -247C -12D -61E -15F = -610    (3)
-2B -12C -2E = -16    (4)
-14B -63C -2D -15E -3F = -128    (5)
 16A + 4B + C + 4D + E + F = 25    (6)

Multiply both sides of equation (1) by 16, the equation can be obtained:
         16A + 64B + 256C + 16D + 64E + 16F = 640    (10)
, then subtract both sides of equation (10) from both sides of equation (6), the equations are reduced to:
 A + 4B + 16C + D + 4E + F = 40    (1)
-24B -128C -6D -32E -8F = -336    (2)
-52B -247C -12D -61E -15F = -610    (3)
-2B -12C -2E = -16    (4)
-14B -63C -2D -15E -3F = -128    (5)
-60B -255C -12D -63E -15F = -615    (6)

Multiply both sides of equation (2) by 13
Divide the two sides of equation (2) by 6, the equation can be obtained:
        -52B - 
832
3
C -13D - 
208
3
E - 
52
3
F = -728    (11)
, then subtract both sides of equation (11) from both sides of equation (3), the equations are reduced to:
 A + 4B + 16C + D + 4E + F = 40    (1)
-24B -128C -6D -32E -8F = -336    (2)
 
91
3
C + D + 
25
3
E + 
7
3
F = 118    (3)
-2B -12C -2E = -16    (4)
-14B -63C -2D -15E -3F = -128    (5)
-60B -255C -12D -63E -15F = -615    (6)

Divide the two sides of equation (2) by 12, the equation can be obtained:
        -2B - 
32
3
C - 
1
2
D - 
8
3
E - 
2
3
F = -28    (12)
, then subtract both sides of equation (12) from both sides of equation (4), the equations are reduced to:
 A + 4B + 16C + D + 4E + F = 40    (1)
-24B -128C -6D -32E -8F = -336    (2)
 
91
3
C + D + 
25
3
E + 
7
3
F = 118    (3)
- 
4
3
C + 
1
2
D + 
2
3
E + 
2
3
F = 12    (4)
-14B -63C -2D -15E -3F = -128    (5)
-60B -255C -12D -63E -15F = -615    (6)

Multiply both sides of equation (2) by 7
Divide the two sides of equation (2) by 12, the equation can be obtained:
        -14B - 
224
3
C - 
7
2
D - 
56
3
E - 
14
3
F = -196    (13)
, then subtract both sides of equation (13) from both sides of equation (5), the equations are reduced to:
 A + 4B + 16C + D + 4E + F = 40    (1)
-24B -128C -6D -32E -8F = -336    (2)
 
91
3
C + D + 
25
3
E + 
7
3
F = 118    (3)
- 
4
3
C + 
1
2
D + 
2
3
E + 
2
3
F = 12    (4)
 
35
3
C + 
3
2
D + 
11
3
E + 
5
3
F = 68    (5)
-60B -255C -12D -63E -15F = -615    (6)

Multiply both sides of equation (2) by 5
Divide the two sides of equation (2) by 2, the equation can be obtained:
        -60B -320C -15D -80E -20F = -840    (14)
, then subtract both sides of equation (14) from both sides of equation (6), the equations are reduced to:
 A + 4B + 16C + D + 4E + F = 40    (1)
-24B -128C -6D -32E -8F = -336    (2)
 
91
3
C + D + 
25
3
E + 
7
3
F = 118    (3)
- 
4
3
C + 
1
2
D + 
2
3
E + 
2
3
F = 12    (4)
 
35
3
C + 
3
2
D + 
11
3
E + 
5
3
F = 68    (5)
 65C + 3D + 17E + 5F = 225    (6)

Multiply both sides of equation (3) by 4
Divide the two sides of equation (3) by 91, the equation can be obtained:
         
4
3
C + 
4
91
D + 
100
273
E + 
4
39
F = 
472
91
    (15)
, then add the two sides of equation (15) to both sides of equation (4), the equations are reduced to:
 A + 4B + 16C + D + 4E + F = 40    (1)
-24B -128C -6D -32E -8F = -336    (2)
 
91
3
C + D + 
25
3
E + 
7
3
F = 118    (3)
 
99
182
D + 
94
91
E + 
10
13
F = 
1564
91
    (4)
 
35
3
C + 
3
2
D + 
11
3
E + 
5
3
F = 68    (5)
 65C + 3D + 17E + 5F = 225    (6)

Multiply both sides of equation (3) by 5
Divide the two sides of equation (3) by 13, the equation can be obtained:
         
35
3
C + 
5
13
D + 
125
39
E + 
35
39
F = 
590
13
    (16)
, then subtract both sides of equation (16) from both sides of equation (5), the equations are reduced to:
 A + 4B + 16C + D + 4E + F = 40    (1)
-24B -128C -6D -32E -8F = -336    (2)
 
91
3
C + D + 
25
3
E + 
7
3
F = 118    (3)
 
99
182
D + 
94
91
E + 
10
13
F = 
1564
91
    (4)
 
29
26
D + 
6
13
E + 
10
13
F = 
294
13
    (5)
 65C + 3D + 17E + 5F = 225    (6)

Multiply both sides of equation (3) by 15
Divide the two sides of equation (3) by 7, the equation can be obtained:
         65C + 
15
7
D + 
125
7
E + 5F = 
1770
7
    (17)
, then subtract both sides of equation (17) from both sides of equation (6), the equations are reduced to:
 A + 4B + 16C + D + 4E + F = 40    (1)
-24B -128C -6D -32E -8F = -336    (2)
 
91
3
C + D + 
25
3
E + 
7
3
F = 118    (3)
 
99
182
D + 
94
91
E + 
10
13
F = 
1564
91
    (4)
 
29
26
D + 
6
13
E + 
10
13
F = 
294
13
    (5)
 
6
7
D - 
6
7
E = - 
195
7
    (6)

Multiply both sides of equation (4) by 203
Divide the two sides of equation (4) by 99, the equation can be obtained:
         
29
26
D + 
2726
1287
E + 
2030
1287
F = 
45356
1287
    (18)
, then subtract both sides of equation (18) from both sides of equation (5), the equations are reduced to:
 A + 4B + 16C + D + 4E + F = 40    (1)
-24B -128C -6D -32E -8F = -336    (2)
 
91
3
C + D + 
25
3
E + 
7
3
F = 118    (3)
 
99
182
D + 
94
91
E + 
10
13
F = 
1564
91
    (4)
- 
164
99
E - 
80
99
F = - 
1250
99
    (5)
 
6
7
D - 
6
7
E = - 
195
7
    (6)

Multiply both sides of equation (4) by 52
Divide the two sides of equation (4) by 33, the equation can be obtained:
         
6
7
D + 
376
231
E + 
40
33
F = 
6256
231
    (19)
, then subtract both sides of equation (19) from both sides of equation (6), the equations are reduced to:
 A + 4B + 16C + D + 4E + F = 40    (1)
-24B -128C -6D -32E -8F = -336    (2)
 
91
3
C + D + 
25
3
E + 
7
3
F = 118    (3)
 
99
182
D + 
94
91
E + 
10
13
F = 
1564
91
    (4)
- 
164
99
E - 
80
99
F = - 
1250
99
    (5)
- 
82
33
E - 
40
33
F = - 
1813
33
    (6)

Multiply both sides of equation (5) by 3
Divide the two sides of equation (5) by 2, the equation can be obtained:
        - 
82
33
E - 
40
33
F = - 
625
33
    (20)
, then subtract both sides of equation (20) from both sides of equation (6), the equations are reduced to:
 A + 4B + 16C + D + 4E + F = 40    (1)
-24B -128C -6D -32E -8F = -336    (2)
 
91
3
C + D + 
25
3
E + 
7
3
F = 118    (3)
 
99
182
D + 
94
91
E + 
10
13
F = 
1564
91
    (4)
- 
164
99
E - 
80
99
F = - 
1250
99
    (5)
0 = -36    (6)

由第(6)方程式可以看出，该方程组无解。

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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