Line s is perpendicular to the line defined by y = 3x - 2 and passes through the point \(\mathrm{(9,...

1. TRANSLATE the problem information

• Given information:
  • Line s is perpendicular to \(\mathrm{y = 3x - 2}\)
  • Line s passes through point \(\mathrm{(9, 5)}\)
  • Need to find equation of line s

2. INFER the perpendicular slope relationship

• The given line \(\mathrm{y = 3x - 2}\) has \(\mathrm{slope = 3}\)
• Key insight: Perpendicular lines have slopes that are negative reciprocals
• Therefore, \(\mathrm{slope\ of\ line\ s = -\frac{1}{3}}\)

3. SIMPLIFY using point-slope form

• Use \(\mathrm{y = mx + b}\) with point \(\mathrm{(9, 5)}\) and slope \(\mathrm{-\frac{1}{3}}\):
• Substitute: \(\mathrm{5 = (-\frac{1}{3})(9) + b}\)
• Calculate: \(\mathrm{5 = -3 + b}\)
• Solve: \(\mathrm{b = 8}\)

4. Write the final equation

• Line s: \(\mathrm{y = -\frac{x}{3} + 8}\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing the negative reciprocal relationship for perpendicular slopes

Students might think perpendicular lines have the same slope or opposite slopes (like 3 and -3), rather than negative reciprocals. They might incorrectly use \(\mathrm{slope = -3}\) instead of \(\mathrm{slope = -\frac{1}{3}}\).

This may lead them to select Choice C (\(\mathrm{y = -3x + 32}\)).

Second Most Common Error:

Poor SIMPLIFY execution: Arithmetic errors when solving for the y-intercept

Students correctly identify \(\mathrm{slope = -\frac{1}{3}}\) but make calculation mistakes:

• Wrong: \(\mathrm{5 = -3 + b}\) → \(\mathrm{b = 2}\) (instead of \(\mathrm{b = 8}\))
• Or sign errors in the arithmetic

This may lead them to select Choice D (\(\mathrm{y = -\frac{x}{3} + 6}\)) or cause confusion and guessing.

The Bottom Line:

The key challenge is remembering that perpendicular slopes are negative reciprocals, not just negatives. Students who memorize this relationship and execute careful arithmetic will solve this successfully.