What is the equation of the line that passes through the point \((1, 4)\) and is perpendicular to the graph...

1. TRANSLATE the problem information

• Given information:
  • Point the line passes through: \(\mathrm{(1, 4)}\)
  • Line it's perpendicular to: \(\mathrm{y = -\frac{1}{2}x + 3}\)
• What we need to find: equation of the perpendicular line

2. INFER the key relationship

• The given line \(\mathrm{y = -\frac{1}{2}x + 3}\) has slope \(\mathrm{-\frac{1}{2}}\)
• Perpendicular lines have slopes that are negative reciprocals
• The negative reciprocal of \(\mathrm{-\frac{1}{2}}\) is 2
• So our perpendicular line has slope \(\mathrm{m = 2}\)

3. INFER the best approach

• We have a point \(\mathrm{(1, 4)}\) and a slope \(\mathrm{m = 2}\)
• Point-slope form is perfect for this situation: \(\mathrm{y - y_1 = m(x - x_1)}\)

4. SIMPLIFY using point-slope form

• Substitute our values: \(\mathrm{y - 4 = 2(x - 1)}\)
• Distribute: \(\mathrm{y - 4 = 2x - 2}\)
• Add 4 to both sides: \(\mathrm{y = 2x - 2 + 4}\)
• Combine like terms: \(\mathrm{y = 2x + 2}\)

Answer: A (\(\mathrm{y = 2x + 2}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about perpendicular slopes: Many students remember that perpendicular lines have "opposite" slopes but forget about the reciprocal part. They might think if the given slope is \(\mathrm{-\frac{1}{2}}\), then the perpendicular slope is just \(\mathrm{\frac{1}{2}}\) (changing only the sign).

Following this incorrect reasoning with slope \(\mathrm{\frac{1}{2}}\):

• \(\mathrm{y - 4 = \frac{1}{2}(x - 1)}\)
• \(\mathrm{y - 4 = \frac{1}{2}x - \frac{1}{2}}\)
• \(\mathrm{y = \frac{1}{2}x - \frac{1}{2} + 4}\)
• \(\mathrm{y = \frac{1}{2}x + \frac{7}{2}}\)

This may lead them to select Choice C (\(\mathrm{y = -\frac{1}{2}x + \frac{7}{2}}\)) after additional sign errors, or get confused and guess.

Second Most Common Error:

Weak SIMPLIFY execution: Students correctly identify the perpendicular slope as 2 and set up the point-slope form correctly, but make arithmetic errors during the simplification process. Common mistakes include sign errors when distributing or combining like terms.

This leads to getting stuck partway through the algebra and potentially selecting the wrong answer choice or abandoning the systematic solution.

The Bottom Line:

Success on this problem hinges on knowing the negative reciprocal relationship for perpendicular slopes AND executing clean algebraic manipulation. Missing either piece typically leads to an incorrect answer.