Line k is defined by y = -17x/3 + 5. Line j is perpendicular to line k in the xy-plane....

1. TRANSLATE the given information

• Given information:
  • Line k: \(\mathrm{y = -\frac{17x}{3} + 5}\)
  • Line j is perpendicular to line k
  • Need to find: slope of line j

2. TRANSLATE the slope from line k's equation

• The equation \(\mathrm{y = -\frac{17x}{3} + 5}\) is in slope-intercept form \(\mathrm{y = mx + b}\)
• This means the slope of line k is \(\mathrm{-\frac{17}{3}}\)

3. INFER the relationship for perpendicular lines

• Perpendicular lines have slopes that are negative reciprocals
• If line k has slope \(\mathrm{-\frac{17}{3}}\), then line j has slope = \(\mathrm{-\frac{1}{(-\frac{17}{3})}}\)

4. SIMPLIFY to find the negative reciprocal

• \(\mathrm{-\frac{1}{(-\frac{17}{3})} = -1 \times (-\frac{3}{17}) = \frac{3}{17}}\)

Answer: \(\mathrm{\frac{3}{17}}\) (also acceptable as \(\mathrm{0.1764, 0.1765, \text{ or } 0.176}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about perpendicular vs parallel lines: Students remember that parallel lines have the same slope, but confuse this with perpendicular lines. They incorrectly think perpendicular lines also have the same slope.

This leads them to answer \(\mathrm{-\frac{17}{3}}\) instead of \(\mathrm{\frac{3}{17}}\), missing the negative reciprocal relationship entirely.

Second Most Common Error:

Weak SIMPLIFY execution: Students correctly identify that they need the negative reciprocal of \(\mathrm{-\frac{17}{3}}\), but make arithmetic errors in the calculation. They might calculate \(\mathrm{-\frac{1}{(-\frac{17}{3})}}\) incorrectly, perhaps getting \(\mathrm{-\frac{3}{17}}\) (forgetting the negative of the negative) or \(\mathrm{\frac{17}{3}}\) (inverting without the negative).

This leads to selecting incorrect numerical values that don't match the true answer of \(\mathrm{\frac{3}{17}}\).

The Bottom Line:

This problem tests whether students can distinguish between parallel and perpendicular line relationships, and whether they can execute the negative reciprocal calculation accurately. The key insight is recognizing that "perpendicular" specifically means negative reciprocal slopes, not just any related slope.