System consistency check: For the equations 3x + y − 4 = 0 and 6x + 2y − 8 = 0, determine the nature of solutions (unique, none, or infinitely many).

Introduction / Context:
This question tests your ability to recognize dependent (coincident) linear equations. If one equation is a constant multiple of the other, the two lines coincide and the system has infinitely many solutions (every point on the line satisfies both).

Given Data / Assumptions:

• E1: 3x + y − 4 = 0
• E2: 6x + 2y − 8 = 0

Concept / Approach:
Compare coefficients: if E2 equals k * E1 for some constant k (same k on x, y, and constant term), then lines are coincident. Here, check k = 2 carefully across all terms including the constant term.

Step-by-Step Solution:

Verification / Alternative check:
Pick a point satisfying E1. For instance, x = 1, y = 1 gives 3*1 + 1 − 4 = 0; substitute into E2: 6*1 + 2*1 − 8 = 0. Works for infinitely many such points along the line.

Why Other Options Are Wrong:
“No solution” would require parallel but distinct lines (not multiples). Unique solutions (specific ordered pairs) are impossible when equations are identical.

Common Pitfalls:
Forgetting to check the constant term when deciding if equations are multiples; mismatched constants would indicate parallel distinct lines (no solution), not coincident lines.

Final Answer:
infinite solutions