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

1. TRANSLATE the given line equation to find its slope

• Given information:
  • Line r has equation \(3x + 2y = 12\)
  • Line t is perpendicular to r and passes through \((-4, 1)\)

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

2. INFER the slope of the perpendicular line

• Since perpendicular lines have slopes that are negative reciprocals:
  • Slope of line t = \(-1 \div (-\frac{3}{2}) = \frac{2}{3}\)
• We need to find the equation of a line with slope \(\frac{2}{3}\) passing through \((-4, 1)\)

3. INFER the best approach and apply point-slope form

• Use point-slope form: \(y - y_1 = m(x - x_1)\)
• Substitute: \(y - 1 = \frac{2}{3}(x - (-4))\)
• Simplify: \(y - 1 = \frac{2}{3}(x + 4)\)

4. SIMPLIFY to get slope-intercept form

• Distribute: \(y - 1 = \frac{2}{3}x + \frac{8}{3}\)
• Add 1 to both sides: \(y = \frac{2}{3}x + \frac{8}{3} + 1\)
• Convert 1 to thirds: \(y = \frac{2}{3}x + \frac{8}{3} + \frac{3}{3}\)
• Combine fractions: \(y = \frac{2}{3}x + \frac{11}{3}\)

Answer: C) \(y = \frac{2}{3}x + \frac{11}{3}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students forget that perpendicular lines have slopes that are negative reciprocals, instead thinking they just have opposite signs.

They might think: "If the slope is \(-\frac{3}{2}\), then the perpendicular slope is just \(\frac{3}{2}\) (make it positive)."

Using slope \(\frac{3}{2}\) with point \((-4, 1)\): \(y - 1 = \frac{3}{2}(x + 4)\) leads to \(y = \frac{3}{2}x + 7\).

This doesn't match any answer choice, leading to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly find the perpendicular slope as \(\frac{2}{3}\) but make arithmetic errors when combining fractions to find the y-intercept.

Common mistake: \(y = \frac{2}{3}x + \frac{8}{3} + 1\), then incorrectly calculating \(\frac{8}{3} + 1 = \frac{9}{3} = 3\) instead of \(\frac{11}{3}\).

This leads to \(y = \frac{2}{3}x + 3\), which might cause them to select Choice D (\(y = \frac{2}{3}x - \frac{11}{3}\)) if they also make a sign error.

The Bottom Line:

This problem tests your understanding of the perpendicular slope relationship and requires careful fraction arithmetic. The key insight is remembering that "negative reciprocal" means both flipping the fraction AND changing the sign.