In the xy-plane, line m passes through the point \((0, -1)\) and is parallel to the line that passes through...

1. TRANSLATE the problem information

• Given information:
  • Line m passes through point (0, -1)
  • Line m is parallel to a line through points (1, 3) and (3, 17)
  • Need to find the equation of line m

2. INFER the solution strategy

• To find line m's equation, I need its slope and y-intercept
• Since m is parallel to the given line, they have the same slope
• Strategy: Find the slope of the line through (1, 3) and (3, 17) first

3. SIMPLIFY to find the reference line's slope

• Using slope formula: \(\mathrm{m = (y_2 - y_1)/(x_2 - x_1)}\)
• \(\mathrm{m = (17 - 3)/(3 - 1)}\)
• \(\mathrm{= 14/2}\)
• \(\mathrm{= 7}\)

4. INFER line m's slope and y-intercept

• Since line m is parallel: slope of m = 7
• Line m passes through (0, -1), so y-intercept = -1

5. TRANSLATE into slope-intercept form

• Using \(\mathrm{y = mx + b}\) with \(\mathrm{m = 7}\) and \(\mathrm{b = -1}\)
• \(\mathrm{y = 7x - 1}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Making arithmetic errors when calculating the slope

Students might compute (17 - 3)/(3 - 1) incorrectly, getting slopes like 14, 1/7, or -7. For example, if they mistakenly calculate 14/2 as 1/7 or forget the division entirely, this leads them to select Choice A (\(\mathrm{y = (1/7)x - 1}\)).

Second Most Common Error:

Missing conceptual knowledge: Not understanding that parallel lines have equal slopes

Some students might find the slope of the reference line correctly (slope = 7) but then think parallel lines have opposite slopes or reciprocal slopes. This confusion about the parallel line relationship could lead them to select Choice D (\(\mathrm{y = -7x - 1}\)) thinking they need the negative slope.

The Bottom Line:

This problem tests both computational accuracy and understanding of parallel line properties. Success requires careful arithmetic with the slope formula and clear conceptual knowledge about what "parallel" means for line equations.