(1) Augmented Matrix:
For System A:
| 1 2 3 |
| 1 2 4 |
For System B:
| 1 1 3 |
| 2 2 6 |
(2) Echelon Form:
For System A:
| 1 2 3 |
| 0 0 2 |
For System B:
| 1 1 3 |
| 0 0 0 |
(3) Reduced Echelon Form:
For System A:
| 1 2 3 |
| 0 0 2 |
For System B:
| 1 1 3 |
| 0 0 0 |
(4) Consistency:
A system of linear equations is consistent if it has at least one solution. If a system of linear equations is in reduced echelon form and has a row of zeros on the right-hand side, then it is inconsistent (no solutions). If the right-hand side is not all zeros, then it is consistent (has at least one solution).
For System A: consistent (has at least one solution)
For System B: inconsistent (no solutions)
(5) Set of Solutions (in parametric form):
For System A:
x1 = t
x2 = 3
t is a free variable and can take any value, so the set of solutions is:
x1 = t, x2 = 3
For System B:
The system is inconsistent, so there is no set of solutions.
(6) Number of Solutions:
For System A: infinitely many solutions
For System B: no solutions
(7) Geometric Interpretation:
The geometric interpretation of the set of solutions is a line (for System A) or the empty set (for System B) in two-dimensional space. The variables x1 and x2 can be thought of as coordinates in a two-dimensional plane, and each equation in the system represents a constraint on the values that x1 and x2 can take. In System A, the set of solutions represents all the points in the two-dimensional plane that satisfy both constraints, which is a line. In System B, the constraints are incompatible, so there is no point in the two-dimensional plane that satisfies both, which is represented by the empty set.