A company manufactures two types of phone cases: a standard model and a premium model. The equation \(\mathrm{P = 3.5(s...

1. TRANSLATE the problem information

• Given equation: \(\mathrm{P = 3.5(s + p) - 1.2p - 850}\)
• We need to find: profit per standard case (coefficient of s)

2. INFER the approach needed

• The coefficient of variable s will tell us the profit contribution per standard case
• To find this coefficient, we need to simplify the equation into standard form

3. SIMPLIFY by applying the distributive property

• Expand \(\mathrm{3.5(s + p)}\):
  \(\mathrm{3.5(s + p) = 3.5s + 3.5p}\)
• Substitute back: \(\mathrm{P = 3.5s + 3.5p - 1.2p - 850}\)

4. SIMPLIFY by combining like terms

• Combine the p terms: \(\mathrm{3.5p - 1.2p = 2.3p}\)
• Final simplified form: \(\mathrm{P = 3.5s + 2.3p - 850}\)

5. INFER the final answer

• The coefficient of s is \(\mathrm{3.5}\)
• This means each standard case contributes $3.5 to the weekly profit

Answer: C) 3.5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make algebraic errors when distributing or combining like terms. For example, they might incorrectly distribute to get \(\mathrm{3.5s + 3.5p - 1.2s}\) instead of \(\mathrm{3.5s + 3.5p - 1.2p}\), or make sign errors when combining terms. This leads to incorrect coefficients and wrong answer selection.

Second Most Common Error:

Poor INFER reasoning about the question: Students correctly simplify the equation but don't understand that they need the coefficient of s specifically. They might mistakenly think they need to substitute values, or they might confuse which coefficient represents standard case profit versus premium case profit. This may lead them to select Choice B (\(\mathrm{2.3}\)) if they incorrectly identify the premium case coefficient as the answer.

The Bottom Line:

This problem tests whether students can systematically simplify algebraic expressions and interpret coefficients in context. Success requires both solid algebraic manipulation skills and clear understanding of what linear equation coefficients represent in real-world scenarios.