Graphical Solution Of Linear Inequalities In Two Variables

Solution:
Let the coefficient matrix be Expanding: Evaluating each minor: So For a unique solution, we need , hence . To determine if the system has infinitely many solutions, we require ( ), and also the system must be consistent in its rank conditions. Substituting into the equations and analyzing the augmented matrix can lead to constraints on . One finds that if and , the system has infinitely many solutions (making (1) true). Checking the other claims shows that (2) and (3) are correct for unique solutions under those specific parameter choices. However, for (4) , it does not provide infinitely many solutions (the system fails to have infinitely many solutions with those values). Hence, statement (4) is not correct.