In the xy-plane, line k is defined by x + y = 0. Line j is perpendicular to line k,...

1. TRANSLATE the problem information

• Given information:
  • Line k: \(\mathrm{x + y = 0}\)
  • Line j is perpendicular to line k
  • Line j has y-intercept (0, 3)
  • Need equation of line j

2. INFER the approach

• To find line j's equation, I need its slope and y-intercept
• I already have the y-intercept (0, 3)
• To find the slope, I need to use the perpendicular relationship with line k
• This means I first need to find line k's slope

3. SIMPLIFY line k to find its slope

• Rewrite \(\mathrm{x + y = 0}\) in slope-intercept form:
  \(\mathrm{x + y = 0}\)
  \(\mathrm{y = -x}\)

• The slope of line k is \(\mathrm{-1}\)

4. INFER the slope of line j

• Since line j is perpendicular to line k, their slopes are negative reciprocals
• If line k has slope \(\mathrm{-1}\), then line j has slope: \(\mathrm{-1/(-1) = 1}\)

5. INFER the equation of line j

• Line j has slope 1 and y-intercept (0, 3)
• Using slope-intercept form: \(\mathrm{y = 1x + 3}\), or \(\mathrm{y = x + 3}\)

6. SIMPLIFY to match answer format

• Convert \(\mathrm{y = x + 3}\) to standard form:
  \(\mathrm{y = x + 3}\)
  \(\mathrm{x - y = -3}\)

Answer: D. \(\mathrm{x - y = -3}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that perpendicular lines have slopes that are negative reciprocals—they think perpendicular lines have the same slope (like parallel lines do).

With slope \(\mathrm{-1}\) for both lines and y-intercept (0, 3), they get \(\mathrm{y = -x + 3}\), which converts to \(\mathrm{x + y = -3}\).

This may lead them to select Choice B (\(\mathrm{x + y = -3}\))

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly find that line j should have slope 1 and get \(\mathrm{y = x + 3}\), but make sign errors when converting to standard form.

They might rearrange as \(\mathrm{x - y = 3}\) instead of \(\mathrm{x - y = -3}\).

This may lead them to select Choice C (\(\mathrm{x - y = 3}\))

The Bottom Line:

This problem tests whether students truly understand the perpendicular line relationship (negative reciprocals, not same slopes) and can accurately convert between equation forms without sign errors.