Particular Solution
A particular solution of a linear system is a point that satisfies the system. This is a vector of values notated as xp.
Description
A consistent linear system has either one or infinitely many solutions. Any one unique solution is called a particular solution.
If a system has no solutions, it is inconsistent.
Number of solutions
If a consistent system has no free variables, there will be one unique solution. Conversely, if there is at least one free variable, there will be infinitely many solutions.
One way to identify if a system is inconsistent is to compute the reduced augmented matrix. If it has a pivot in the right-most column, then the system is inconsistent.
Solutions
The idea of particular solution follows from basic algebra.
Linear algebra introduces solution strategies such as elimination to find particular solutions in more complex systems.