Line ell is defined by 3y + 12x = 5. Line n is perpendicular to line ell in the xy-plane....

1. TRANSLATE the given equation to find the slope of line ℓ

• Given: Line ℓ is defined by \(\mathrm{3y + 12x = 5}\)
• To find the slope, I need this in slope-intercept form \(\mathrm{y = mx + b}\)

2. SIMPLIFY the equation algebraically

• Start with: \(\mathrm{3y + 12x = 5}\)
• Subtract 12x from both sides: \(\mathrm{3y = -12x + 5}\)
• Divide everything by 3: \(\mathrm{y = -4x + \frac{5}{3}}\)
• Now I can see that the slope of line ℓ is \(\mathrm{-4}\)

3. INFER the relationship for perpendicular lines

• Since line n is perpendicular to line ℓ, their slopes are negative reciprocals
• If slope of ℓ = \(\mathrm{-4}\), then slope of n = \(\mathrm{\frac{-1}{-4} = \frac{1}{4}}\)

Answer: \(\mathrm{\frac{1}{4}}\) (also acceptable as \(\mathrm{0.25}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skills: Students make algebraic errors when converting to slope-intercept form, such as incorrectly distributing the division by 3 or making sign errors.

For example, they might get \(\mathrm{y = -4x + 5}\) (forgetting to divide the constant term by 3) or \(\mathrm{y = 4x + \frac{5}{3}}\) (sign error). This gives them the wrong slope for line ℓ, leading to an incorrect perpendicular slope. This causes them to select an answer that doesn't match any of the given options and leads to confusion and guessing.

Second Most Common Error:

Conceptual confusion about perpendicular relationships: Students remember that perpendicular lines are related but confuse the relationship, thinking perpendicular slopes are just 'opposites' (changing sign only) rather than negative reciprocals.

With slope of ℓ as \(\mathrm{-4}\), they might think the perpendicular slope is \(\mathrm{+4}\) instead of \(\mathrm{\frac{1}{4}}\). This leads them to an answer not among the choices, causing frustration and random selection.

The Bottom Line:

This problem tests both algebraic manipulation skills and understanding of geometric relationships. Success requires careful algebra followed by correct application of the perpendicular slope rule.