7/5y - 3/5x = 4/5 - 7/5y5/4x + 7/4 = py + 15/4In the given system of equations, p is...

1. TRANSLATE the problem information

• Given system:
  • \(\frac{7}{8}\mathrm{y} - \frac{5}{8}\mathrm{x} = \frac{4}{7} - \frac{7}{8}\mathrm{y}\)
  • \(\frac{5}{4}\mathrm{x} + \frac{7}{4} = \mathrm{py} + \frac{15}{4}\)
• Need to find: value of p that makes the system have no solution

2. INFER what "no solution" means

• No solution occurs when the system represents two parallel but distinct lines
• Parallel lines have the same slope but different y-intercepts
• Mathematically: coefficients must be proportional but constants different

3. SIMPLIFY both equations to standard form

First equation: \(\frac{7}{8}\mathrm{y} - \frac{5}{8}\mathrm{x} = \frac{4}{7} - \frac{7}{8}\mathrm{y}\)

• Add \(\frac{7}{8}\mathrm{y}\) to both sides: \(\frac{14}{8}\mathrm{y} - \frac{5}{8}\mathrm{x} = \frac{4}{7}\)
• Simplify: \(\frac{7}{4}\mathrm{y} - \frac{5}{8}\mathrm{x} = \frac{4}{7}\)
• Multiply by 8: \(-5\mathrm{x} + 14\mathrm{y} = \frac{32}{7}\)

Second equation: \(\frac{5}{4}\mathrm{x} + \frac{7}{4} = \mathrm{py} + \frac{15}{4}\)

• Subtract \(\frac{7}{4}\) from both sides: \(\frac{5}{4}\mathrm{x} = \mathrm{py} + \frac{8}{4}\)
• Rearrange: \(\frac{5}{4}\mathrm{x} - \mathrm{py} = 2\)
• Multiply by 4: \(5\mathrm{x} - 4\mathrm{py} = 8\)

4. INFER the parallel line condition

• Standard forms: \(-5\mathrm{x} + 14\mathrm{y} = \frac{32}{7}\) and \(5\mathrm{x} - 4\mathrm{py} = 8\)
• For parallel lines: coefficient ratios must be equal
• Set up proportion: \(\frac{-5}{5} = \frac{14}{-4\mathrm{p}}\)

5. SIMPLIFY to solve for p

• \(\frac{-5}{5} = \frac{14}{-4\mathrm{p}}\)
• \(-1 = \frac{-14}{4\mathrm{p}}\)
• Cross multiply: \(-4\mathrm{p} = -14\)
• Divide by -4: \(\mathrm{p} = \frac{14}{4} = \frac{7}{2}\)

Answer: \(\frac{7}{2}\) (or 3.5)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning about no solution condition: Many students know that "no solution" is bad but don't connect it to the geometric concept of parallel lines. They might try to solve the system directly or set it equal to zero, missing that they need to analyze the relationship between coefficients.

This leads to confusion and random guessing among the answer choices.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students often make algebraic errors when converting to standard form, especially with the first equation that has y terms on both sides. A common mistake is forgetting to combine like terms or making sign errors during the rearrangement.

This causes them to set up incorrect proportions and get wrong values for p.

The Bottom Line:

This problem requires students to shift from "solving" mode to "analyzing" mode - understanding that sometimes we don't solve for x and y, but instead analyze the structure of the system itself.