Question:2x - 5y = 7y + 3hy = 4 + 2xIn the given system of equations, h is a constant....

1. SIMPLIFY both equations into standard form

• First equation: \(\mathrm{2x - 5y = 7y + 3}\)
  • Move 7y to the left side: \(\mathrm{2x - 5y - 7y = 3}\)
  • Combine like terms: \(\mathrm{2x - 12y = 3}\)
• Second equation: \(\mathrm{hy = 4 + 2x}\)
  • Rearrange to standard form: \(\mathrm{-2x + hy = 4}\)

2. INFER what "no solution" means geometrically

• No solution occurs when lines are parallel (same slope, different y-intercepts)
• For equations \(\mathrm{Ax + By = C}\) and \(\mathrm{Dx + Ey = F}\), this happens when:
  • Coefficient ratios are equal: \(\mathrm{A/D = B/E}\)
  • BUT constant ratios are different: \(\mathrm{A/D \neq C/F}\)

3. SIMPLIFY by setting up the proportion

• From our system:
  • \(\mathrm{2x - 12y = 3}\)
  • \(\mathrm{-2x + hy = 4}\)
• Set up coefficient ratio equation: \(\mathrm{2/(-2) = (-12)/h}\)
• This gives us: \(\mathrm{-1 = -12/h}\)

4. SIMPLIFY to solve for h

• From \(\mathrm{-1 = -12/h}\):
• Cross multiply: \(\mathrm{-h = -12}\)
• Therefore: \(\mathrm{h = 12}\)

5. INFER by verifying our answer

• With \(\mathrm{h = 12}\), coefficient ratios: \(\mathrm{2/(-2) = -12/12 = -1}\) ✓
• Constant ratios: \(\mathrm{3/4 = 0.75}\)
• Since \(\mathrm{-1 \neq 0.75}\), we have no solution ✓

Answer: D (12)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't connect "no solution" to the parallel lines condition. They may try to solve the system directly by substitution or elimination, getting confused when they reach a contradiction like \(\mathrm{0 = 7}\). Without understanding what causes no solution, they can't work backwards to find h.

This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students make sign errors when rearranging equations, particularly with the second equation. Writing \(\mathrm{hy = 4 + 2x}\) as \(\mathrm{2x + hy = 4}\) instead of \(\mathrm{-2x + hy = 4}\) leads to incorrect coefficient ratios and the wrong value of h.

This may lead them to select Choice C (6) or other incorrect answers.

The Bottom Line:

This problem requires recognizing that "no solution" is a geometric concept (parallel lines) that translates to specific algebraic conditions about coefficient ratios. Students who only know mechanical solving techniques without understanding the deeper meaning of different solution types will struggle.