Line p passes through points \((2, 5)\) and \((6, 1)\). Line q has equation kx + 2y = 10, where...

1. TRANSLATE the problem information

• Given information:
  • Line p passes through points \(\mathrm{(2, 5)}\) and \(\mathrm{(6, 1)}\)
  • Line q has equation \(\mathrm{kx + 2y = 10}\)
  • Need to find k value where system has no solution
• What "no solution" tells us: The lines must be parallel but not identical

2. INFER the approach

• To determine when lines are parallel but distinct, we need to compare slopes and y-intercepts
• Strategy: Convert both equations to slope-intercept form \(\mathrm{(y = mx + b)}\), then set slopes equal while ensuring different y-intercepts

3. SIMPLIFY to find line p's equation

• Calculate slope using the two points:
  \(\mathrm{m = \frac{1 - 5}{6 - 2}}\)
  \(\mathrm{m = \frac{-4}{4}}\)
  \(\mathrm{m = -1}\)
• Use point-slope form with \(\mathrm{(2, 5)}\):
  \(\mathrm{y - 5 = -1(x - 2)}\)
  \(\mathrm{y - 5 = -x + 2}\)
  \(\mathrm{y = -x + 7}\)

4. SIMPLIFY to convert line q to slope-intercept form

• Starting with \(\mathrm{kx + 2y = 10}\):
  \(\mathrm{2y = -kx + 10}\)
  \(\mathrm{y = \frac{-k}{2}x + 5}\)

5. INFER the condition for parallel lines

• Line p has slope \(\mathrm{-1}\)
• Line q has slope \(\mathrm{\frac{-k}{2}}\)
• For parallel lines: \(\mathrm{\frac{-k}{2} = -1}\)
• Solving: \(\mathrm{k = 2}\)

6. APPLY CONSTRAINTS to verify no solution

• When \(\mathrm{k = 2}\):
  • Line p: \(\mathrm{y = -x + 7}\) (y-intercept = 7)
  • Line q: \(\mathrm{y = -x + 5}\) (y-intercept = 5)
• Same slope \(\mathrm{(-1)}\) but different y-intercepts \(\mathrm{(7 \neq 5)}\) confirms parallel but distinct lines

Answer: D) 2

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Confusing "no solution" with "infinitely many solutions"

Students may think "no solution" means the lines are identical, leading them to set both slopes AND y-intercepts equal. This would require \(\mathrm{\frac{-k}{2} = -1}\) AND \(\mathrm{5 = 7}\), which is impossible. Realizing this contradiction, they might conclude no value of k works and randomly guess among the choices.

Second Most Common Error:

Inadequate SIMPLIFY execution: Making algebraic errors when converting equations to slope-intercept form

Students might incorrectly manipulate \(\mathrm{kx + 2y = 10}\), perhaps getting \(\mathrm{y = -kx + 5}\) instead of \(\mathrm{y = \frac{-k}{2}x + 5}\). This leads to setting \(\mathrm{-k = -1}\), giving \(\mathrm{k = 1}\). Since 1 isn't among the choices, they're forced to guess or might select the closest value.

The Bottom Line:

This problem tests whether students truly understand what different types of solutions mean geometrically. The key insight is that "no solution" specifically means parallel but distinct lines - a concept that requires both procedural skill in manipulating equations and conceptual understanding of how algebraic conditions translate to geometric relationships.