In the xy-plane, line l passes through the point \((0, 0)\) and is perpendicular to the line represented by the...

1. TRANSLATE the problem information

• Given information:
  • Line l passes through origin (0, 0)
  • Line l is perpendicular to \(\mathrm{y = -2x + 7}\)
  • Line l passes through point (4, k)
• What we need: Find the value of k

2. INFER the approach

• To find k, we need line l's equation
• To get line l's equation, we need its slope
• Since l is perpendicular to \(\mathrm{y = -2x + 7}\), we can find l's slope using the perpendicular slope relationship

3. TRANSLATE the slope from the given line

• The line \(\mathrm{y = -2x + 7}\) has slope \(\mathrm{-2}\) (coefficient of x)

4. INFER the slope of line l

• Perpendicular lines have slopes that are negative reciprocals
• Slope of line l = negative reciprocal of \(\mathrm{-2}\) = \(\mathrm{-1/(-2) = 1/2}\)

5. INFER the equation of line l

• Line l passes through origin (0, 0) with slope \(\mathrm{1/2}\)
• Equation: \(\mathrm{y = (1/2)x}\)

6. SIMPLIFY to find k

• Line l passes through (4, k), so substitute \(\mathrm{x = 4}\):
• \(\mathrm{k = (1/2)(4) = 2}\)

Answer: 2

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students confuse the perpendicular slope relationship and think perpendicular lines have slopes that are just negatives of each other (instead of negative reciprocals).

They calculate: slope of line l = \(\mathrm{-(-2) = 2}\), leading to equation \(\mathrm{y = 2x}\), then \(\mathrm{k = 2(4) = 8}\).

This may lead them to select an incorrect answer or get confused when 8 isn't among typical answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify that the slope should be \(\mathrm{1/2}\), but make arithmetic errors either in finding the negative reciprocal or in the final multiplication.

Common mistakes: calculating \(\mathrm{-1/(-2)}\) incorrectly, or errors in \(\mathrm{(1/2)(4)}\). This leads to various incorrect values for k and potential guessing.

The Bottom Line:

Success requires both understanding the geometric relationship between perpendicular lines AND executing the algebraic steps correctly. The perpendicular slope relationship is the key conceptual hurdle most students miss.