11x + 14y leq 115 Anthony will spend at most $115 to purchase x small cheese pizzas and y large...

1. TRANSLATE the inequality components

• Given information:
  • \(\mathrm{11x + 14y \leq 115}\)
  • \(\mathrm{x}\) = number of small cheese pizzas
  • \(\mathrm{y}\) = number of large cheese pizzas
  • $115 maximum budget
• What this tells us: Each term represents a cost calculation

2. INFER the meaning of each term

• In algebraic expressions like this: coefficient × variable = unit cost × quantity
• So we have:
  • \(\mathrm{11x}\) = $11 per small pizza × \(\mathrm{x}\) small pizzas
  • \(\mathrm{14y}\) = $14 per large pizza × \(\mathrm{y}\) large pizzas

3. TRANSLATE what 14y specifically represents

• \(\mathrm{14y = \$14 \times y}\) large pizzas
• This equals the total dollars spent on large pizzas

4. Eliminate incorrect interpretations

• 14 alone = cost per large pizza (not \(\mathrm{14y}\))
• 11 alone = cost per small pizza (not \(\mathrm{14y}\))
• \(\mathrm{11x}\) = total cost of small pizzas (not \(\mathrm{14y}\))

Answer: C. The total amount, in dollars, Anthony will spend on large cheese pizzas

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students focus only on the coefficient (14) instead of the complete term (\(\mathrm{14y}\)).

They see "14" and think "this is the cost per large pizza" without recognizing that the question asks about "\(\mathrm{14y}\)", not just "14". The coefficient represents unit cost, but the complete term (coefficient × variable) represents total cost.

This may lead them to select Choice A ($14 per large pizza)

Second Most Common Error:

Poor TRANSLATE reasoning: Students mix up which variable represents which type of pizza.

They might associate \(\mathrm{14y}\) with small pizzas instead of large pizzas, or confuse the roles of \(\mathrm{x}\) and \(\mathrm{y}\) in the context. This stems from not carefully tracking which variable represents which quantity.

This may lead them to select Choice D (total spent on small pizzas)

The Bottom Line:

This problem tests whether students can distinguish between unit costs (coefficients) and total costs (coefficient × variable) within the context of a real-world inequality. The key insight is recognizing that \(\mathrm{14y}\) as a complete term represents a total cost calculation, not just the unit price.