Line k is defined by y = 7x + 1/8. Line j is perpendicular to line k in the xy-plane....

1. TRANSLATE the equation to identify the slope

• Given: Line k is defined by \(\mathrm{y = 7x + \frac{1}{8}}\)
• This equation is in slope-intercept form: \(\mathrm{y = mx + b}\)
• The slope of line k is \(\mathrm{m = 7}\)

2. INFER the relationship for perpendicular lines

• When two lines are perpendicular, their slopes are negative reciprocals
• If one line has slope m, the perpendicular line has slope \(\mathrm{-\frac{1}{m}}\)
• Since line k has slope 7, line j has \(\mathrm{slope = -\frac{1}{7}}\)

Answer: B. \(\mathrm{-\frac{1}{7}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students remember that perpendicular lines have "opposite" slopes but apply this incorrectly as just changing the sign.

They think: "Line k has slope \(\mathrm{7}\), so the perpendicular line has slope \(\mathrm{-7}\)."

This may lead them to select Choice A (-8) as the closest negative option, or get confused and guess.

Second Most Common Error:

Incomplete conceptual knowledge about perpendicular slopes: Students remember perpendicular lines involve reciprocals but forget the negative part.

They think: "Line k has slope \(\mathrm{7}\), so the perpendicular line has slope \(\mathrm{\frac{1}{7}}\)."

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

The Bottom Line:

This problem tests whether students understand that "perpendicular slopes" means negative reciprocals, not just negative values or just reciprocals. The key insight is that both operations (negative AND reciprocal) must be applied together.