Finding Pivot Columns in Matrix A & Solving Ax = b

[physmatics]

Homework Statement

So, I want to find the pivot columns to the matrix A:

A=
-1 1 -5 0 2
3 0 6 1 -1
2 -1 7 2 1
2 -7 25 5 1

and after that I want to solve Ax = b where b is (I kind of want to solve Ax = 0 as well, but let's skip that...):

b=
a+3
2a
-a-1
-9a-5

My thought was to solve everything at the same time, and I started doing that by gausselimination. The problem is that I've been stuck for a while with this:

C=
15 0 -15 0 0
0 3 0 0 0
0 0 -6 -1 0
0 0 0 5 10

and

d=
26a+7
a+2
2a
2a+14

The zero at row 2, column 3 is giving me a headache, and I don't know how to solve it at all. Is it possible I made a mistake while doing these row operations?

I'm very thankful for any kind of advice!

 

[vela]

$$C=\begin{pmatrix} 15 & 0 & -15 & 0 & 0 \\ 0 & 3 & 0 & 0 & 0 \\ 0 & 0 & -6 & -1 & 0 \\ 0 & 0 & 0 & 5 & 10 \end{pmatrix}$$
The zero at row 2, column 3 is giving me a headache, and I don't know how to solve it at all. Is it possible I made a mistake while doing these row operations?

Why is that 0 a problem?

By the way, I think you did make a mistake reducing the matrix somewhere. When I reduced it, there were no fractions in the result, but it looks like you'll end up with some.