A line is defined by the parametric equations x = 3t + 5 and y = -2t + 8, where...

1. TRANSLATE the problem information

• Given parametric equations:
  • \(\mathrm{x = 3t + 5}\)
  • \(\mathrm{y = -2t + 8}\)

• Need to find: \(\mathrm{q/p}\) where \(\mathrm{(p,0)}\) is x-intercept and \(\mathrm{(0,q)}\) is y-intercept

• What this tells us: We need \(\mathrm{y = 0}\) to find x-intercept and \(\mathrm{x = 0}\) to find y-intercept

2. INFER the solution strategy

• To find intercepts with parametric equations, we:
  • Set the appropriate coordinate to zero
  • Solve for parameter t
  • Substitute t back to find the other coordinate

• Let's find x-intercept first, then y-intercept

3. SIMPLIFY to find the x-intercept

• Set \(\mathrm{y = 0}\) in the equation \(\mathrm{y = -2t + 8}\):
  \(\mathrm{-2t + 8 = 0}\)
  \(\mathrm{-2t = -8}\)
  \(\mathrm{t = 4}\)

• Substitute \(\mathrm{t = 4}\) into \(\mathrm{x = 3t + 5}\):
  \(\mathrm{x = 3(4) + 5}\)
  \(\mathrm{x = 12 + 5}\)
  \(\mathrm{x = 17}\)

• Therefore \(\mathrm{p = 17}\)

4. SIMPLIFY to find the y-intercept

• Set \(\mathrm{x = 0}\) in the equation \(\mathrm{x = 3t + 5}\):
  \(\mathrm{3t + 5 = 0}\)
  \(\mathrm{3t = -5}\)
  \(\mathrm{t = -5/3}\)

• Substitute \(\mathrm{t = -5/3}\) into \(\mathrm{y = -2t + 8}\):
  \(\mathrm{y = -2(-5/3) + 8}\)
  \(\mathrm{y = 10/3 + 8}\)
  \(\mathrm{y = 10/3 + 24/3}\)
  \(\mathrm{y = 34/3}\)

• Therefore \(\mathrm{q = 34/3}\)

5. SIMPLIFY the final calculation

• \(\mathrm{q/p = (34/3) \div 17}\)
  \(\mathrm{q/p = (34/3) \times (1/17)}\)
  \(\mathrm{q/p = 34/51}\)
  \(\mathrm{q/p = 2/3}\)

Answer: C (2/3)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret what x-intercept and y-intercept mean in the context of parametric equations. They might try to eliminate t algebraically instead of using the intercept conditions.

For example, they might solve \(\mathrm{3t + 5 = x}\) and \(\mathrm{-2t + 8 = y}\) simultaneously, getting lost in unnecessary algebra. This leads to confusion and often causes them to abandon the systematic approach and guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors in the final fraction division \(\mathrm{q/p = (34/3) \div 17}\). They might calculate this as \(\mathrm{(34/3) \times 17}\) instead of \(\mathrm{(34/3) \times (1/17)}\), or mess up the simplification of \(\mathrm{34/51}\).

This computational error often leads them to select Choice A (-3/2) or Choice D (3/2) depending on the nature of their mistake.

The Bottom Line:

This problem tests whether students truly understand that intercepts are found by setting coordinates to zero, not by manipulating the parametric equations algebraically. The key insight is recognizing that parametric equations give you a direct path to intercepts through substitution.