How Can You Solve the Linear System Ax=b with Given Vectors a1 and a3?

[jtruth914]

Let Ax = b be a linear system whose augmented matrix (A|b) has reduced row echelon form
1 2 0 3 1 ||-2
0 0 1 2 4 || 5
0 0 0 0 0 || 0
0 0 0 0 0 || 0a) Find all solutions to the system

b)If


a1= (1,1,3,4) and a3= (2,-1,1,3)
determine b.
I got part a to be
x1=-2-2r-3t-w
x2=5-2t-4w
x3=r
x4=t
x5=w

I'm having difficulty getting part b. I know a1 and a3 are column vectors and I know Ax=x1a1+x2a2+...+xnan. The textbook solution says its b=(8,-7,-1,7)^T

 

[HallsofIvy]

I have no idea what you mean by "a1" and "a2"! If we call the variables x1, x2, x3, x4, and x5 (I thought at first the you meant "a1" and "a2" to be the first of those but those are, of course, numbers, not vectors) then the two equations become x1+ 2x2+ 3x4+ x5= 2 and x3+ 2x4+ 4x5= 5. From the first, x1= 2- 2x2- 3x4- x5 and from the second, x3= 5- 2x4- 4x5. So you can choose x2, x4, and x5 to be anything you want, then calculate x1 and x3.