A landscape designer is creating a garden layout using two types of flower beds: triangular beds and hexagonal beds. The...

1. TRANSLATE the equation components

• Given equation: \(12\mathrm{x} + 18\mathrm{y} = 156\)
• Variables defined:
  • \(\mathrm{x}\) = number of triangular beds
  • \(\mathrm{y}\) = number of hexagonal beds
  • 156 = total area in square feet

2. INFER what coefficients represent in context

• In equations like \(\mathrm{ax} + \mathrm{by} = \mathrm{c}\) where \(\mathrm{c}\) is a total:
  • The coefficient tells us the contribution per unit
  • 12 is multiplied by \(\mathrm{x}\) (triangular beds) → 12 = area per triangular bed
  • 18 is multiplied by \(\mathrm{y}\) (hexagonal beds) → 18 = area per hexagonal bed

3. TRANSLATE back to answer the question

• The question asks about the meaning of 18
• Since 18 is the coefficient of \(\mathrm{y}\) (hexagonal beds), it represents the area each hexagonal bed covers
• Therefore: 18 = area per hexagonal bed = 18 square feet per hexagonal bed

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse coefficients with variables or constants in the equation.

They might think 18 represents the actual number of hexagonal beds (confusing it with the variable \(\mathrm{y}\)) or associate it with triangular beds instead of hexagonal beds. This conceptual mix-up about what different parts of the equation represent leads them to select Choice B (The number of hexagonal beds is 18) or Choice C (Each triangular bed covers 18 square feet).

The Bottom Line:

This problem tests whether students can correctly interpret the structural meaning of coefficients in real-world linear equations - specifically that coefficients represent "per unit" contributions when the equation shows a total.