Consider the system of linear equations: ax + 4y = 12 6x + 24y = 35 where a is a...

1. INFER what 'no solution' means geometrically

• A system has no solution when the equations represent parallel but distinct lines
• This happens when:
  • Coefficient ratios are equal (makes lines parallel)
  • Constant ratios are different (makes lines distinct)

2. INFER the strategy for finding \(\mathrm{a}\)

• We need the coefficient ratios to be equal
• From equations \(\mathrm{ax + 4y = 12}\) and \(\mathrm{6x + 24y = 35}\)
• Set up proportion: \(\frac{\mathrm{a}}{6} = \frac{4}{24}\)

3. SIMPLIFY the proportion to find \(\mathrm{a}\)

• Simplify the right side: \(\frac{4}{24} = \frac{1}{6}\)
• So we have: \(\frac{\mathrm{a}}{6} = \frac{1}{6}\)
• Therefore: \(\mathrm{a = 1}\)

4. INFER verification is necessary

• With \(\mathrm{a = 1}\), first equation becomes: \(\mathrm{x + 4y = 12}\)
• Multiply by 6: \(\mathrm{6x + 24y = 72}\)
• Compare with second equation: \(\mathrm{6x + 24y = 35}\)
• Coefficients are identical: \(\mathrm{(6, 24) = (6, 24)}\) ✓
• Constants are different: \(\mathrm{72 ≠ 35}\) ✓

Answer: 1

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't connect 'no solution' to the geometric condition of parallel but distinct lines.

Without this key insight, they might try to solve the system directly or look for when the system becomes inconsistent through substitution/elimination. This leads to confusion about what the question is actually asking, causing them to get stuck and guess randomly.

Second Most Common Error:

Inadequate INFER reasoning: Students correctly identify the need for proportional coefficients but forget to verify that the constant ratios are different.

They might find \(\mathrm{a = 1}\) correctly but fail to check whether this actually produces no solution. Without verification, they may doubt their answer or incorrectly conclude that some other value is needed, leading to confusion and incorrect answer selection.

The Bottom Line:

This problem tests conceptual understanding of when linear systems have no solution, not just algebraic manipulation skills. The key breakthrough is recognizing that 'no solution' has specific geometric meaning that translates to precise algebraic conditions.