已知线性方程组:x1+x2+x3+x4+x5=a,3x1+2x2+x3+x4-3x5=0,x2+2x3+2x4+6x5=b,5x1+4x2+3x3+3x4-x5=2.求当a,b

Question

线性代数:已知线性方程组:x1+x2+x3+x4+x5=a,3x1+2x2+x3+x4-3x5=0,x2+2x3+2x4+6x5=b,5x1+4x2+3x3+3x4-x5=2.求1)当a,b为何值时，方程组有解？
2)方程有解时，写出它的通解结构。

Answer 1

:x1+x2+x3+x4+x5=a,①
3x1+2x2+x3+x4-3x5=0,②
x2+2x3+2x4+6x5=b,③
5x1+4x2+3x3+3x4-x5=2④
先消去x1,①*3-②，x2+2x3+2x4+6x5=3a,
①*5-④，x2+2x3+2x4+6x5=5a-2,
∴方程组有解时5a-2=3a,a=1.由③，b=3.
(2)x3=m,x4=n,x5=p,其中m,n,p是任意数，
由③，x2=3-2m-2n-6p,
代入①，x1=m+n+5p-2.