For a non-homogeneous linear system Ax=b whose right-hand side contains free parameters, both the existence and the form of the solution depend on those parameters: Gaussian elimination on the augmented matrix produces a row whose left side vanishes, yielding a solvability condition (a linear relationship among the parameters) that must hold for any solution to exist. When that condition is satisfied, the complete solution set is expressed as the sum of one particular solution (obtained by setting free variables to zero) and an arbitrary linear combination of special solutions (obtained by setting one free variable to one and the rest to zero), one special solution per free variable. This is a topic within linear algebra concerning the structure of solution sets to linear systems and their dependence on row-reduced echelon form, pivot versus free variables, and consistency conditions.
For a non-homogeneous linear system Ax=b whose right-hand side contains free parameters, both the existence and the form of the solution depend on those parameters: Gaussian elimination on the augmen…