System solvability check: For 2x + 4y = 6 and 4x + 8y = 6, determine whether the system has a unique solution, no solution, or infinitely many solutions.

Introduction / Context:This problem examines whether two linear equations represent the same line, different parallel lines, or intersecting lines. Comparing proportional coefficients provides a quick test for consistency or inconsistency of the system.

Given Data / Assumptions:

• E1: 2x + 4y = 6
• E2: 4x + 8y = 6

Concept / Approach:If the left-hand sides are proportional but the constants are not in the same ratio, the lines are parallel and distinct, producing no solution. Multiply E1 by 2 to compare fairly with E2.

Step-by-Step Solution:

Verification / Alternative check:Graphically, both lines have slope −(2/4) = −1/2, but different intercepts; hence they never meet.

Why Other Options Are Wrong:Unique or infinite solutions would require equality of constants after scaling; “exactly two solutions” is not a concept in linear pairs in the plane.

Common Pitfalls:Stopping after noticing proportional coefficients and not checking constants, which changes the conclusion from coincident to parallel lines.

Final Answer:no solution