Line p is defined by 4y + 8x = 6. Line r is perpendicular to line p in the xy-plane....

1. TRANSLATE the problem information

• Given information:
  • Line p: \(\mathrm{4y + 8x = 6}\)
  • Line r is perpendicular to line p
  • Need to find: slope of line r

• What this tells us: We need the slope of line p first, then use the perpendicular relationship

2. SIMPLIFY to find the slope of line p

• Convert \(\mathrm{4y + 8x = 6}\) to slope-intercept form (\(\mathrm{y = mx + b}\)):
  • Subtract 8x: \(\mathrm{4y = -8x + 6}\)
  • Divide by 4: \(\mathrm{y = -8x/4 + 6/4}\)
  • Simplify: \(\mathrm{y = -2x + 3/2}\)

• From \(\mathrm{y = -2x + 3/2}\), the slope of line p is \(\mathrm{-2}\)

3. INFER the perpendicular relationship

• Perpendicular lines have slopes that are negative reciprocals
• If line p has slope \(\mathrm{-2}\), then line r has slope: \(\mathrm{-1/(-2) = 1/2}\)

Answer: \(\mathrm{1/2}\) (or 0.5 or .5)

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about perpendicular vs parallel lines: Students remember that special line relationships involve slope patterns but mix up which is which. They might think perpendicular lines have the same slope (which is actually true for parallel lines).

This leads them to incorrectly conclude that line r also has slope \(\mathrm{-2}\), causing confusion when this isn't among typical answer choices.

Second Most Common Error:

Weak INFER skill with negative reciprocals: Students remember that perpendicular lines involve reciprocals but forget about the "negative" part. They calculate the reciprocal of \(\mathrm{-2}\) as \(\mathrm{-1/2}\) instead of the negative reciprocal, which is \(\mathrm{1/2}\).

This error stems from not fully understanding that "negative reciprocal" means you flip the fraction AND change the sign.

The Bottom Line:

This problem tests whether students can systematically work through a two-step process: first extract slope information from a non-standard form, then correctly apply the perpendicular line relationship. Success requires both algebraic manipulation skills and solid understanding of geometric relationships.