Question:Line m passes through points \((2, 5)\) and \((6, 7)\). Line n is parallel to line m in the xy-plane....

1. TRANSLATE the problem information

• Given information:
  • Line m passes through points \(\mathrm{(2, 5)}\) and \(\mathrm{(6, 7)}\)
  • Line n is parallel to line m
  • Need to find the slope of line n

• What this tells us: We need to find a slope, and we have two points on the related line m.

2. INFER the approach

• Key insight: Since line n is parallel to line m, we need to find the slope of line m first
• Strategy: Use the slope formula on the given points, then apply the parallel lines property
• What to do first: Calculate slope of line m using the two given points

3. SIMPLIFY to find the slope of line m

• Apply slope formula: slope = \(\mathrm{\frac{y_2 - y_1}{x_2 - x_1}}\)
• Using points \(\mathrm{(2, 5)}\) and \(\mathrm{(6, 7)}\):
  • slope = \(\mathrm{\frac{7 - 5}{6 - 2}}\) = \(\mathrm{\frac{2}{4}}\) = \(\mathrm{\frac{1}{2}}\)
• Be careful with the arithmetic - reduce the fraction completely

4. INFER the final answer

• Since parallel lines have equal slopes, line n has the same slope as line m
• Therefore: slope of line n = \(\mathrm{\frac{1}{2}}\)

Answer: \(\mathrm{\frac{1}{2}}\) (or 0.5)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students calculate \(\mathrm{\frac{7-5}{6-2}}\) = \(\mathrm{\frac{2}{4}}\) but forget to reduce the fraction to lowest terms, leaving their answer as \(\mathrm{\frac{2}{4}}\) instead of \(\mathrm{\frac{1}{2}}\). Some students might also make basic subtraction errors like \(\mathrm{(7-5) = 3}\) or \(\mathrm{(6-2) = 3}\).

This leads to confusion when trying to match their answer to the expected form, causing them to second-guess their approach or guess randomly.

Second Most Common Error:

Missing conceptual knowledge: Students don't remember or understand that parallel lines have equal slopes. They might try to use both sets of coordinates somehow or think they need additional information about line n's position.

This causes them to get stuck after finding the slope of line m, leading to abandoning systematic solution and guessing.

The Bottom Line:

This problem tests whether students can connect the geometric concept of parallel lines to the algebraic concept of equal slopes. The calculation itself is straightforward, but students often falter on either the arithmetic simplification or the conceptual connection.