In the xy-plane, line L has equation y = -3x + 7. Line k is the vertical line with equation...

1. TRANSLATE the problem information

• Given information:
  • Line L has equation \(\mathrm{y = -3x + 7}\)
  • Line k is vertical with equation \(\mathrm{x = -2}\)
• What we need: The intersection point of these two lines

2. INFER the solution approach

• At an intersection point, both equations must be satisfied simultaneously
• Since line k tells us \(\mathrm{x = -2}\), we can substitute this x-value directly into line L's equation
• This will give us the y-coordinate of the intersection point

3. SIMPLIFY by substituting and calculating

• Substitute \(\mathrm{x = -2}\) into \(\mathrm{y = -3x + 7}\):
  • \(\mathrm{y = -3(-2) + 7}\)
  • \(\mathrm{y = 6 + 7}\)
  • \(\mathrm{y = 13}\)
• The intersection point is \(\mathrm{(-2, 13)}\)

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Sign error when computing \(\mathrm{-3(-2)}\)

Students often incorrectly calculate \(\mathrm{-3(-2) = -6}\) instead of +6. This leads to:
\(\mathrm{y = -6 + 7 = 1}\)

This may lead them to select Choice B (\(\mathrm{(-2, 1)}\))

Second Most Common Error:

Incomplete SIMPLIFY execution: Stopping the calculation too early

Some students correctly find \(\mathrm{-3(-2) = 6}\) but then forget to add the constant term, thinking the answer is just 6.

This may lead them to select Choice C (\(\mathrm{(-2, 6)}\))

The Bottom Line:

This problem tests careful arithmetic execution more than conceptual understanding. The setup is straightforward, but sign errors and incomplete calculations are the main pitfalls.