Line p passes through the points \((-3, 1)\) and \((5, -3)\).Line q is parallel to line p in the xy-plane.What...

1. TRANSLATE the problem information

• Given information:
  • Line p passes through points \(\mathrm{(-3, 1)}\) and \(\mathrm{(5, -3)}\)
  • Line q is parallel to line p
  • Need to find: slope of line q

2. INFER the solution strategy

• To find the slope of line q, I need the slope of line p first
• Since parallel lines have identical slopes, once I find line p's slope, that's also line q's slope
• I can find line p's slope using the slope formula with the two given points

3. SIMPLIFY to calculate the slope of line p

• Using the slope formula: \(\mathrm{m = \frac{y_2 - y_1}{x_2 - x_1}}\)
• Substitute the points \(\mathrm{(-3, 1)}\) and \(\mathrm{(5, -3)}\):
  \(\mathrm{m = \frac{-3 - 1}{5 - (-3)}}\)
  \(\mathrm{m = \frac{-4}{8}}\)
  \(\mathrm{m = -\frac{1}{2}}\)

4. INFER the final answer

• Since line q is parallel to line p, they have the same slope
• Therefore, slope of line q = \(\mathrm{-\frac{1}{2}}\)

Answer: \(\mathrm{-\frac{1}{2}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when calculating the slope, particularly with the negative numbers in the coordinate subtraction.

Common mistakes include:

• Computing \(\mathrm{(-3 - 1)}\) as -2 instead of -4
• Computing \(\mathrm{(5 - (-3))}\) as 2 instead of 8
• Sign errors when reducing the fraction

This leads to incorrect slope values and wrong final answers.

Second Most Common Error:

Missing conceptual knowledge about parallel lines: Students find the slope of line p correctly but don't realize that parallel lines have equal slopes.

They may think they need additional information or try to set up equations involving line q, not recognizing that the slope of line p IS the answer.

This causes confusion and may lead to guessing or selecting 'cannot be determined.'

The Bottom Line:

This problem tests both computational accuracy with signed numbers and understanding of the fundamental parallel line property. Success requires careful arithmetic and recognizing that the parallel relationship gives us the answer directly.