Two lines intersect in the coordinate plane: L_1 has equation x + y = 6 and L_2 has equation 2x...

1. TRANSLATE the problem information

• Given information:
  • Line \(\mathrm{L_1}\): \(\mathrm{x + y = 6}\)
  • Line \(\mathrm{L_2}\): \(\mathrm{2x - y = 3}\)
  • Lines intersect at point \(\mathrm{(p, q)}\)
  • Need to find: \(\mathrm{q/p}\)

2. INFER the solution approach

• To find intersection point \(\mathrm{(p, q)}\), I need to solve the system of two equations
• I can use either elimination or substitution method
• Elimination looks efficient here since the y-terms have opposite signs

3. SIMPLIFY using elimination method

• Add the two equations to eliminate y:

\(\mathrm{(x + y) + (2x - y) = 6 + 3}\)
\(\mathrm{3x = 9}\)
\(\mathrm{x = 3}\)

• So \(\mathrm{p = 3}\)

4. SIMPLIFY to find the second coordinate

• Substitute \(\mathrm{x = 3}\) into the first equation:

\(\mathrm{3 + y = 6}\)
\(\mathrm{y = 3}\)

• So \(\mathrm{q = 3}\)

5. SIMPLIFY to calculate the final ratio

• \(\mathrm{q/p = 3/3 = 1}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Making arithmetic errors during the elimination or substitution process

Students might make sign errors when adding equations, or calculation mistakes when substituting back. For instance, they might incorrectly calculate \(\mathrm{3x = 9}\) as \(\mathrm{x = 2}\), or make errors when substituting to find y. These arithmetic slips can lead to incorrect values for p and q, resulting in wrong ratios like \(\mathrm{q/p = 3/2}\) or \(\mathrm{q/p = 2/3}\).

This may lead them to select Choice D (3/2) or Choice B (2/3)

Second Most Common Error:

Poor TRANSLATE reasoning: Misunderstanding what the problem is asking for at the end

Even after correctly finding the intersection point \(\mathrm{(3, 3)}\), some students calculate \(\mathrm{p/q}\) instead of \(\mathrm{q/p}\), or get confused about which coordinate represents p versus q. While this particular error wouldn't affect the final answer in this case since \(\mathrm{p = q}\), it represents a fundamental misreading that could be costly in other problems.

The Bottom Line:

This problem tests whether students can systematically solve a linear system without making computational errors. The key insight is that intersection problems always require solving systems of equations, and careful arithmetic execution is essential for success.