Line ℓ in the xy-plane is perpendicular to the line with equation x = 2. What is the slope of...

1. TRANSLATE the problem information

• Given information:
  • Line \(\mathrm{ℓ}\) is perpendicular to the line \(\mathrm{x = 2}\)
  • Need to find the slope of line \(\mathrm{ℓ}\)

2. INFER what type of line x = 2 represents

• The equation \(\mathrm{x = 2}\) means x is always 2, regardless of what y equals
• This describes a vertical line passing through points like \(\mathrm{(2, 0)}\), \(\mathrm{(2, 1)}\), \(\mathrm{(2, -3)}\), etc.
• Vertical lines have undefined slope

3. INFER the relationship for perpendicular lines

• When one line is vertical, its perpendicular line must be horizontal
• This is a special case - we can't use the typical "slopes multiply to -1" rule because vertical lines have undefined slope
• Horizontal lines have the form \(\mathrm{y = k}\) (constant)

4. INFER the slope of a horizontal line

• Horizontal lines have slope = 0
• This is because slope = \(\mathrm{\frac{\Delta y}{\Delta x}}\), and horizontal lines have no change in y

Answer: A. 0

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students try to apply the standard perpendicular slopes formula (\(\mathrm{m_1 \times m_2 = -1}\)) without recognizing that \(\mathrm{x = 2}\) is a vertical line with undefined slope.

They might think: "If the slope is 2, then the perpendicular slope is -1/2" or "If the slope is -2, then the perpendicular slope is 1/2." This leads them to incorrectly associate the number 2 in the equation \(\mathrm{x = 2}\) with the slope value.

This may lead them to select Choice B (-1/2) or get confused about which perpendicular relationship to use.

Second Most Common Error:

Conceptual confusion about line equations: Students don't recognize that \(\mathrm{x = 2}\) represents a vertical line and instead think it's somehow related to a slope of 2.

This confusion about the fundamental difference between \(\mathrm{x = k}\) (vertical) and \(\mathrm{y = mx + b}\) (non-vertical) equations causes them to misapply slope concepts entirely.

This may lead them to select Choice C (-2) or Choice D by incorrectly thinking line \(\mathrm{ℓ}\) itself has undefined slope.

The Bottom Line:

This problem tests whether students can distinguish between the special case of vertical/horizontal perpendicularity versus the general perpendicular slopes relationship. The key insight is recognizing that \(\mathrm{x = 2}\) describes a vertical line, making this a geometry problem rather than an algebraic slope calculation.