Classifying pairs of lines from ax + by = c and dx + ey = f: I. Parallel if the system has no solution. II. Coincident if the system has a finite number of solutions. III. Intersecting if the system has exactly one solution. Which statements are correct?

Introduction / Context:
This conceptual item links the nature of solutions of a 2×2 linear system to the geometric relationship between its two lines in the plane. Recognizing whether the system is inconsistent, dependent, or independent maps to parallel, coincident, or intersecting lines respectively.

Given Data / Assumptions:

• A pair of linear equations: ax + by = c and dx + ey = f with real coefficients a, b, c, d, e, f.
• Standard solution classifications: no solution, exactly one solution, infinitely many solutions.

Concept / Approach:
Recall: parallel distinct lines ⇒ no solution; coincident lines ⇒ infinitely many solutions; intersecting at a single point ⇒ unique solution. Match each statement to these facts.

Step-by-Step Solution:

Verification / Alternative check:
Determinant method: For ax + by = c and dx + ey = f, a unique solution occurs when ae − bd ≠ 0 (lines intersect). If ae − bd = 0 but ratios of constants disagree, lines are parallel (no solution). If coefficients and constants share the same ratio, lines coincide (infinitely many solutions).

Why Other Options Are Wrong:
Any choice including II is wrong because it misstates the coincident case. “All three” is impossible as I and II contradict each other’s solution counts.

Common Pitfalls:
Confusing “finite” with “infinite” solutions, and forgetting that coincident lines correspond to infinitely many common points (every point on the line).

Final Answer:
Only I and III