Question

解非齐次线性方程组： 2X1-4X2+5X3+3X4=7 3X1-6X2+4X3+2X4=7 4X1-8X2+17X3+11X4=21 X1-2X2+13X3+9X4=14

Answer 1

解：
2X1-4X2+5X3+3X4=7
3X1-6X2+4X3+2X4=7
4X1-8X2+17X3+11X4=21
X1-2X2+13X3+9X4=14

增广矩阵B=（A，b）=
2 -4 5 3 7
3 -6 4 2 7
4 -8 17 11 21
1 -2 13 9 14

r1<-->r4
1 -2 13 9 14

2 -4 5 3 7
3 -6 4 2 7
4 -8 17 11 21

r2-2r1，r3-3r1，r4-4r1
1 -2 13 9 14
0 0 -21 -15 -21
0 0 -35 -25 -35
0 0 -35 -25 -35

r4-r3
1 -2 13 9 14
0 0 -21 -15 -21
0 0 -35 -25 -35
0 0 0 0 0

r3-5/3 r2
1 -2 13 9 14
0 0 -21 -15 -21
0 0 0 0 0
0 0 0 0 0

-1/21 r2
1 -2 13 9 14
0 0 1 5/7 1
0 0 0 0 0
0 0 0 0 0

r1-13r2
1 -2 0 -2/7 1
0 0 1 5/7 1
0 0 0 0 0
0 0 0 0 0

所以方程为：x1-2x2-2/7 x4=1
x3+5/7x4=1
故方程组的解为x1=1+2x2+2/7 x4
x3=1-5/7 x4
【其中x2和x4为任意变量】

Answer 2

2X1-4X2+5X3+3X4=7 3X1-6X2+4X3+2X4=7 4X1-8X2+17X3+11X4=21 x1-2x2+13x3+9x4=14