In the xy-plane, line r has equation 3x + 2y = 12. Line t is perpendicular to line r and...

1. TRANSLATE the given information

• Given information:
  • Line r: \(\mathrm{3x + 2y = 12}\)
  • Line t is perpendicular to line r
  • Line t passes through point (-4, 1)
  • Need to find equation of line t

2. INFER the solution approach

• To find a perpendicular line, I need:
  • The slope of the original line r
  • Apply the perpendicular slope relationship
  • Use the given point to complete the equation

3. TRANSLATE line r's equation to find its slope

• Convert \(\mathrm{3x + 2y = 12}\) to slope-intercept form:
  • \(\mathrm{2y = -3x + 12}\)
  • \(\mathrm{y = \frac{-3}{2}x + 6}\)
• Slope of line r = \(\mathrm{\frac{-3}{2}}\)

4. INFER the slope of perpendicular line t

• Perpendicular lines have negative reciprocal slopes
• If line r has slope \(\mathrm{\frac{-3}{2}}\), then line t has slope: \(\mathrm{\frac{-1}{\frac{-3}{2}} = \frac{2}{3}}\)

5. SIMPLIFY to find the complete equation

• Use \(\mathrm{y = mx + b}\) with \(\mathrm{m = \frac{2}{3}}\) and point (-4, 1):

\(\mathrm{1 = \frac{2}{3}(-4) + b}\)

\(\mathrm{1 = \frac{-8}{3} + b}\)

\(\mathrm{b = 1 + \frac{8}{3} = \frac{3}{3} + \frac{8}{3} = \frac{11}{3}}\)

• Therefore: \(\mathrm{y = \frac{2}{3}x + \frac{11}{3}}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students confuse perpendicular vs parallel slope relationships and use the same slope (\(\mathrm{\frac{-3}{2}}\)) instead of the negative reciprocal.

They might think "perpendicular means same slope" or get confused about the negative reciprocal concept. Using slope \(\mathrm{\frac{-3}{2}}\), they would get \(\mathrm{y = \frac{-3}{2}x + b}\), and after substituting (-4, 1): \(\mathrm{1 = \frac{-3}{2}(-4) + b = 6 + b}\), so \(\mathrm{b = -5}\). This gives them \(\mathrm{y = \frac{-3}{2}x - 5}\), which isn't among the choices, leading to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the perpendicular slope as \(\mathrm{\frac{2}{3}}\) but make arithmetic errors when finding the y-intercept.

A common mistake is: \(\mathrm{1 = \frac{2}{3}(-4) + b}\) → \(\mathrm{1 = \frac{-8}{3} + b}\) → \(\mathrm{b = 1 - \frac{8}{3} = \frac{-5}{3}}\) (instead of adding \(\mathrm{\frac{8}{3}}\)). This leads them to \(\mathrm{y = \frac{2}{3}x - \frac{5}{3}}\), and they might select the closest-looking answer or get confused since this exact form isn't available.

The Bottom Line:

This problem tests whether students truly understand the perpendicular slope relationship (negative reciprocals) and can execute multi-step algebraic procedures accurately. The key insight is recognizing that "perpendicular" creates a specific mathematical relationship between slopes, not just "different."