In the xy-plane, line l passes through the point \((1, -2)\) and is parallel to the line represented by the...

1. TRANSLATE the problem information

• Given information:
  • Line l passes through point \(\mathrm{(1, -2)}\)
  • Line l is parallel to \(\mathrm{y = -3x + 5}\)
  • Line l also passes through point \(\mathrm{(d, 4)}\)
• What this tells us: Since the lines are parallel, they have the same slope

2. INFER the approach

• First, I need to identify the slope of the given line \(\mathrm{y = -3x + 5}\)
• Since line l is parallel to this line, line l has the same slope
• Then I'll find the equation of line l using the known point \(\mathrm{(1, -2)}\)
• Finally, I'll use the fact that line l passes through \(\mathrm{(d, 4)}\) to solve for d

3. Find the slope of line l

• From \(\mathrm{y = -3x + 5}\), the slope is -3
• Since parallel lines have equal slopes, line l also has slope -3

4. SIMPLIFY to find the equation of line l

• Using point-slope form with point \(\mathrm{(1, -2)}\) and slope -3:
  \(\mathrm{y - (-2) = -3(x - 1)}\)
  \(\mathrm{y + 2 = -3x + 3}\)
  \(\mathrm{y = -3x + 1}\)

5. SIMPLIFY to solve for d

• Since line l passes through \(\mathrm{(d, 4)}\), substitute into the equation:
  \(\mathrm{4 = -3d + 1}\)
  \(\mathrm{3 = -3d}\)
  \(\mathrm{d = -1}\)

Answer: -1

Why Students Usually Falter on This Problem

Most Common Error Path:

Poor TRANSLATE reasoning: Students may not recognize that 'parallel' means the lines have the same slope. They might try to find some other relationship between the lines or get confused about what parallel means algebraically.

This leads to confusion and abandoning a systematic approach, causing them to guess.

Second Most Common Error:

Weak SIMPLIFY execution: Students correctly identify the slope and set up the equation, but make sign errors when manipulating \(\mathrm{y - (-2) = -3(x - 1)}\) or when solving \(\mathrm{4 = -3d + 1}\). Common mistakes include forgetting the negative sign or making arithmetic errors.

This could lead them to get \(\mathrm{d = 1}\) instead of \(\mathrm{d = -1}\).

The Bottom Line:

This problem tests whether students can connect the geometric concept of parallel lines to the algebraic concept of equal slopes, then execute the algebra correctly. The key insight is recognizing that parallel lines give you the slope immediately.