Linear Inequalities

We have the following system of linear equations:

Let's analyze the scenarios provided in List I and match them with List II:

(P) If and , then the system has

When and , the third equation becomes . This implies the system is consistent and has a special form of and , leading to infinitely many solutions as it matches a dependency condition. Thus, it associates with (1) a unique solution, but due to the specific form, it aligns more with (3).

(Q) If and , then the system has

Here, with the condition that , the system becomes inconsistent if derived equations contradict each other, resulting in no solution. Thus, it matches with (2).

(R) If where and , then the system has

With , and , the system is consistent and independent, leading to a unique solution. Thus it matches with (1).

(S) If where and , then the system has

With this condition, we substitute and verify one of the solution propositions whereby, assuming a direct solution trial reveals: , , satisfies all equations. Thus, matches (4).

Hence, the correct matching is: (P) →(3), (Q)→ (2), (R) →(1), (S) →(4)