Which of the following systems of linear equations has no solution?

1. INFER what 'no solution' means

• A system has no solution when the lines are parallel but not identical
• Parallel lines: same slope, different y-intercepts
• This means the lines never intersect

2. INFER the strategy

• Check each system for parallel lines
• If slopes are equal, verify whether the lines are identical or just parallel
• Same slopes + same line = infinite solutions
• Same slopes + different lines = no solution

3. SIMPLIFY each option to find slopes

Option A: \(\mathrm{6x - 3y = 12}\) and \(\mathrm{4x - 2y = 8}\)

• Slopes: \(\mathrm{6/3 = 2}\) and \(\mathrm{4/2 = 2}\) ✓ (same slopes)
• SIMPLIFY to check if same line:
  • First: \(\mathrm{6x - 3y = 12}\) → divide by 3 → \(\mathrm{2x - y = 4}\)
  • Second: \(\mathrm{4x - 2y = 8}\) → divide by 2 → \(\mathrm{2x - y = 4}\)
• Same equation = infinite solutions ❌

Option B: \(\mathrm{6x - 3y = 12}\) and \(\mathrm{4x - 2y = 10}\)

• Slopes: both equal 2 ✓ (same slopes)
• SIMPLIFY to check if same line:
  • First: \(\mathrm{2x - y = 4}\) (divided by 3)
  • Second: \(\mathrm{2x - y = 5}\) (divided by 2)
• Different equations with same slope = no solution ✓

Option C: \(\mathrm{6x - 3y = 12}\) and \(\mathrm{6x + 3y = 18}\)

• Slopes: 2 and -2 (different slopes = intersecting lines) ❌

Option D: \(\mathrm{6x - 3y = 12}\) and \(\mathrm{5x - 2y = 10}\)

• Slopes: 2 and 2.5 (different slopes = intersecting lines) ❌

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students confuse 'no solution' with 'infinite solutions' when they see equations with the same slope.

When they find that both equations in Option A have slope = 2, they might think 'same slope means no solution' and incorrectly select Choice A. They fail to take the crucial next step of checking whether the lines are identical or just parallel.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students make arithmetic errors when converting standard form equations to find slopes or when simplifying to compare equations.

For example, when finding the slope of \(\mathrm{4x - 2y = 10}\), they might incorrectly calculate the slope as \(\mathrm{-4/2 = -2}\) instead of \(\mathrm{4/2 = 2}\), leading them to think the lines intersect and eliminating the correct answer.

The Bottom Line:

This problem tests whether students understand that parallel lines can either be identical (infinite solutions) or distinct (no solution). Success requires both recognizing the concept and executing the algebra accurately to distinguish between these cases.