Line r is defined by the equation 4x - 9y = 3. Line s is parallel to line r in...

1. TRANSLATE the problem information

• Given information:
  • Line r has equation: \(\mathrm{4x - 9y = 3}\)
  • Line s is parallel to line r
  • Need to find: slope of line s

2. INFER the solution strategy

• Key insight: Parallel lines have identical slopes
• Strategy: Find the slope of line r, which equals the slope of line s
• To find slope: Convert the equation to slope-intercept form \(\mathrm{y = mx + b}\)

3. SIMPLIFY the equation to slope-intercept form

• Start with: \(\mathrm{4x - 9y = 3}\)
• Subtract 4x from both sides: \(\mathrm{-9y = -4x + 3}\)
• Divide everything by -9: \(\mathrm{y = \frac{-4x + 3}{-9}}\)
• Simplify: \(\mathrm{y = \frac{4x - 3}{9} = \frac{4}{9}x - \frac{1}{3}}\)

4. INFER the final answer

• From \(\mathrm{y = \frac{4}{9}x - \frac{1}{3}}\), the slope of line r is \(\mathrm{\frac{4}{9}}\)
• Since parallel lines have equal slopes, the slope of line s is also \(\mathrm{\frac{4}{9}}\)

Answer: B. \(\mathrm{\frac{4}{9}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when dividing by -9, especially with the fraction arithmetic. They might incorrectly get \(\mathrm{y = -\frac{4}{9}x + \frac{1}{3}}\) instead of \(\mathrm{y = \frac{4}{9}x - \frac{1}{3}}\), leading them to think the slope is \(\mathrm{-\frac{4}{9}}\). However, since this isn't an answer choice, this leads to confusion and guessing.

Second Most Common Error:

Poor TRANSLATE reasoning: Students might confuse which coefficient represents the slope in standard form \(\mathrm{Ax + By = C}\), thinking they can directly read the slope as either \(\mathrm{\frac{A}{B}}\) or \(\mathrm{\frac{B}{A}}\) without converting to slope-intercept form. This misconception might lead them to select Choice A (\(\mathrm{\frac{9}{4}}\)) by taking the ratio of the y-coefficient to x-coefficient.

The Bottom Line:

This problem tests whether students can systematically convert between forms of linear equations while maintaining conceptual clarity about what parallel lines mean geometrically.