Row reduction with variables in augmented column

Question

I came across a problem where the linear system of equations were equal to different known quantities (represented as variables). After using Gauss-Jordan elimination, I came across two different scenarios...

1) In the last row, I had something like 0 = a + b + c (some linear combination of known variables)

2) In the last row, I had something like 2 = a + b + c (some linear combination of known variables)

I can interpret #1 as the system being inconsistent if a + b + c is not equal to 0, but I'm not sure how to interpret #2 in terms of free variables, pivot variables, etc.

Answer 1

For the first case you are right. In the second case I presume that the value 2 is associated to an unknown, in this case a+b+c is just twice the value of that unknown.

Answer 2

Interpret the results in terms of ranks of the matrices: the minear system is consistent if and only if the matrix of the augmented matrix has the same rank as the matrix of the left-hand side .