General equations

STEP 1: Construct the augmented matrix  for the matrix equation  , then produce a reduced echelon form (REF). For example,

STEP 2: Consider free variables ( and in the above example) as general parameters (for example, and ). Then basic variables (, , , and in the above example) can be solved with these parameters.

STEP 3: Together with free variables, express general solutions to  in terms of parametric vector form. In the above example, we obtain

EXAMPLE 2. Describe all solutions of with

and

Department of Mathematics