Line j is defined by the equation 6x + 9y = 18. Line k is perpendicular to line j in...

1. TRANSLATE the equation into usable form

• Given information:
  • Line j: \(\mathrm{6x + 9y = 18}\)
  • Line k is perpendicular to line j
  • Need to find: slope of line k

• To work with slopes, we need the equation in \(\mathrm{y = mx + b}\) form

2. SIMPLIFY to find the slope of line j

• Convert \(\mathrm{6x + 9y = 18}\) to slope-intercept form:
  • \(\mathrm{9y = -6x + 18}\)
  • \(\mathrm{y = \frac{-6}{9}x + \frac{18}{9}}\)
  • \(\mathrm{y = \frac{-2}{3}x + 2}\)

• The slope of line j is \(\mathrm{\frac{-2}{3}}\)

3. INFER the relationship for perpendicular lines

• Perpendicular lines have slopes that are negative reciprocals
• If line j has slope \(\mathrm{\frac{-2}{3}}\), then line k has slope = \(\mathrm{\frac{-1}{\frac{-2}{3}}}\)

4. SIMPLIFY the negative reciprocal calculation

• Slope of line k = \(\mathrm{-1 \div \frac{-2}{3}}\)
• This equals \(\mathrm{-1 \times \frac{-3}{2} = \frac{3}{2}}\)

Answer: D) 3/2

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make sign errors when computing the negative reciprocal. They might calculate \(\mathrm{\frac{-1}{\frac{-2}{3}}}\) incorrectly, getting \(\mathrm{\frac{-3}{2}}\) instead of \(\mathrm{\frac{3}{2}}\), or forget that dividing by a fraction means multiplying by its reciprocal.

This may lead them to select Choice B (\(\mathrm{\frac{-3}{2}}\)).

Second Most Common Error:

Missing conceptual knowledge about perpendicular lines: Students might think that perpendicular lines have slopes that are just reciprocals (not negative reciprocals), or they might confuse parallel lines (same slope) with perpendicular lines.

This could lead them to select Choice C (\(\mathrm{\frac{2}{3}}\)) (just the reciprocal) or Choice A (\(\mathrm{\frac{-2}{3}}\)) (thinking perpendicular means same slope).

The Bottom Line:

This problem tests both algebraic manipulation skills and understanding of geometric relationships. Success requires converting between equation forms AND knowing the specific relationship between perpendicular line slopes.