Two lines in the xy-plane are given by the equationsy = (6 + 9x)/py = (9x + 5)/15What is the...

1. TRANSLATE the problem information

• Given information:
  • Line 1: \(\mathrm{y = \frac{6 + 9x}{p}}\)
  • Line 2: \(\mathrm{y = \frac{9x + 5}{15}}\)
  • Need: value of p so lines 'do not intersect'

2. INFER what 'do not intersect' means

• Two lines in a plane either intersect at one point or are parallel
• Since we want them NOT to intersect, they must be parallel
• Parallel lines have the same slope but different y-intercepts

3. TRANSLATE both equations to slope-intercept form

• Rewrite Line 1: \(\mathrm{y = \frac{6 + 9x}{p} = \frac{6}{p} + \frac{9x}{p} = \frac{9}{p}x + \frac{6}{p}}\)
  • Slope: \(\mathrm{\frac{9}{p}}\), y-intercept: \(\mathrm{\frac{6}{p}}\)
• Rewrite Line 2: \(\mathrm{y = \frac{9x + 5}{15} = \frac{9}{15}x + \frac{5}{15} = \frac{3}{5}x + \frac{1}{3}}\)
  • Slope: \(\mathrm{\frac{3}{5}}\), y-intercept: \(\mathrm{\frac{1}{3}}\)

4. INFER the condition for parallel lines

• For parallel lines, slopes must be equal:

\(\mathrm{\frac{9}{p} = \frac{3}{5}}\)

5. SIMPLIFY to solve for p

• Cross multiply: \(\mathrm{9 \times 5 = 3 \times p}\)
• \(\mathrm{45 = 3p}\)
• \(\mathrm{p = 15}\)

6. APPLY CONSTRAINTS to verify the solution

• Check y-intercepts are different when p = 15:
  • Line 1 y-intercept: \(\mathrm{\frac{6}{15} = \frac{2}{5}}\)
  • Line 2 y-intercept: \(\mathrm{\frac{1}{3}}\)
• Since \(\mathrm{\frac{2}{5} \neq \frac{1}{3}}\), lines are indeed parallel

Answer: D) 15

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students misunderstand 'do not intersect' and try to solve the system of equations by setting the expressions equal: \(\mathrm{\frac{6 + 9x}{p} = \frac{9x + 5}{15}}\)

They then solve this equation thinking they need to find when the lines intersect, rather than recognizing they need the lines to be parallel. This algebraic approach leads them to a complex equation that doesn't directly give any of the answer choices, causing confusion and random guessing.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly identify that slopes must be equal but make errors when cross multiplying \(\mathrm{\frac{9}{p} = \frac{3}{5}}\). They might incorrectly multiply to get \(\mathrm{9 \times 3 = 5 \times p}\) (giving p = \(\mathrm{\frac{27}{5}}\), which isn't an answer choice) or make other arithmetic mistakes.

This leads to confusion and guessing among the available choices.

The Bottom Line:

This problem tests whether students understand that 'parallel' is the geometric condition for lines that don't intersect, and whether they can systematically compare slopes rather than trying to solve intersection equations.