Question:The variables a and b are related by the equations a - b = 5 and ab = 84. If...

1. TRANSLATE the problem information

• Given information:
  • Two equations: \(\mathrm{a - b = 5}\) and \(\mathrm{ab = 84}\)
  • Constraint: \(\mathrm{a}\) is positive
  • Find: the value of \(\mathrm{a}\)

2. INFER the solution approach

• This is a system of equations with one linear and one quadratic relationship
• Substitution works well here since the linear equation easily isolates one variable
• Strategy: Solve for \(\mathrm{b}\) in terms of \(\mathrm{a}\), then substitute into the product equation

3. SIMPLIFY using substitution

• From \(\mathrm{a - b = 5}\), we get: \(\mathrm{b = a - 5}\)
• Substitute into \(\mathrm{ab = 84}\):
  \(\mathrm{a(a - 5) = 84}\)
• Distribute: \(\mathrm{a^2 - 5a = 84}\)
• Rearrange to standard form: \(\mathrm{a^2 - 5a - 84 = 0}\)

4. SIMPLIFY by factoring the quadratic

• Need two numbers that multiply to -84 and add to -5
• Factor pairs of 84: Try 7 and 12 since \(\mathrm{7 × 12 = 84}\)
• For sum of -5: use +7 and -12 (since \(\mathrm{7 + (-12) = -5}\))
• Factored form: \(\mathrm{(a + 7)(a - 12) = 0}\)
• Solutions: \(\mathrm{a = -7}\) or \(\mathrm{a = 12}\)

5. APPLY CONSTRAINTS to select final answer

• Since \(\mathrm{a}\) must be positive (given in problem): \(\mathrm{a = 12}\)
• Verification: If \(\mathrm{a = 12}\), then \(\mathrm{b = 7}\), and \(\mathrm{12 - 7 = 5}\) ✓, \(\mathrm{12 × 7 = 84}\) ✓

Answer: 12

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may not recognize this as a system of equations requiring substitution. Instead, they might try to solve each equation separately or get confused about how to combine the linear and product relationships. This leads to confusion and guessing.

Second Most Common Error:

Poor APPLY CONSTRAINTS reasoning: Students correctly solve the quadratic to get \(\mathrm{a = -7}\) or \(\mathrm{a = 12}\), but forget the constraint that \(\mathrm{a}\) must be positive. They might select the negative solution or become uncertain about which answer to choose, leading to incorrect answer selection of -7.

The Bottom Line:

This problem challenges students to seamlessly blend linear equation manipulation with quadratic solving techniques, while remembering to apply the given constraints. The key insight is recognizing that substitution transforms the system into a manageable quadratic equation.