3/2y - 1/4x = 2/3 - 3/2y 1/2x + 3/2 = py + 9/2 In the given system of equations,...

1. TRANSLATE the problem requirements

• Given information:
  • System of two linear equations with parameter p
  • System has no solution
• What this means: The lines are parallel but distinct (different)

2. INFER the mathematical condition for no solution

• For a system to have no solution, the lines must be parallel and distinct
• This happens when coefficient ratios are equal but constant ratios are different
• We need both equations in standard form \(\mathrm{Ax} + \mathrm{By} = \mathrm{C}\) to compare ratios

3. SIMPLIFY the first equation to standard form

Starting with: \(\frac{3}{2}\mathrm{y} - \frac{1}{4}\mathrm{x} = \frac{2}{3} - \frac{3}{2}\mathrm{y}\)

• Add \(\frac{3}{2}\mathrm{y}\) to both sides: \(3\mathrm{y} - \frac{1}{4}\mathrm{x} = \frac{2}{3}\)
• Multiply by 4: \(12\mathrm{y} - \mathrm{x} = \frac{8}{3}\)
• Multiply by 3: \(36\mathrm{y} - 3\mathrm{x} = 8\)
• Rearrange to standard form: \(-3\mathrm{x} + 36\mathrm{y} = 8\)

4. SIMPLIFY the second equation to standard form

Starting with: \(\frac{1}{2}\mathrm{x} + \frac{3}{2} = \mathrm{py} + \frac{9}{2}\)

• Subtract py from both sides: \(\frac{1}{2}\mathrm{x} - \mathrm{py} = \frac{9}{2} - \frac{3}{2}\)
• Simplify right side: \(\frac{1}{2}\mathrm{x} - \mathrm{py} = 3\)
• Multiply by 2: \(\mathrm{x} - 2\mathrm{py} = 6\)

5. APPLY the no-solution condition

• For parallel lines: coefficient ratios must be equal
• Set up proportion: \(\frac{-3}{1} = \frac{36}{-2\mathrm{p}}\)
• Solve: \(-3 = \frac{-18}{\mathrm{p}}\)
• Therefore: \(3\mathrm{p} = 18\), so \(\mathrm{p} = 6\)

6. VERIFY the solution

When \(\mathrm{p} = 6\), the equations become:
\(-3\mathrm{x} + 36\mathrm{y} = 8\) and \(\mathrm{x} - 12\mathrm{y} = 6\)

• Coefficient ratios: \(\frac{-3}{1} = \frac{36}{-12} = -3\) ✓ (equal - parallel lines)
• Constant ratios: \(\frac{8}{6} \neq -3\) ✓ (unequal - distinct lines)

Answer: 6

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't connect "no solution" to the specific mathematical condition about parallel and distinct lines. They may try to solve the system directly or set it equal to zero, not realizing they need to analyze coefficient relationships.

This leads to confusion and guessing rather than systematic analysis.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when converting to standard form, especially when dealing with multiple fractions. Common mistakes include sign errors when moving terms or incorrect fraction operations.

This may lead them to get incorrect coefficient ratios and calculate the wrong value of p.

The Bottom Line:

This problem requires understanding the deeper concept that "no solution" has a precise mathematical meaning in terms of line relationships, not just that equations can't be solved. Students must connect this conceptual understanding to the mechanical process of comparing coefficient ratios.