\(2(\mathrm{p} + 1) + 8(\mathrm{p} - 1) = 5\mathrm{p}\) What value of p is the solution of the equation above?...

1. TRANSLATE the problem information

• Given equation: \(2(\mathrm{p} + 1) + 8(\mathrm{p} - 1) = 5\mathrm{p}\)
• Find: The value of p that makes this equation true

2. INFER the solution strategy

• This is a linear equation with parentheses, so we need to:
  • First distribute to eliminate parentheses
  • Then combine like terms
  • Finally isolate the variable

3. SIMPLIFY by distributing

• Apply distributive property to both terms:
  • \(2(\mathrm{p} + 1) = 2\mathrm{p} + 2\)
  • \(8(\mathrm{p} - 1) = 8\mathrm{p} - 8\)
• Equation becomes: \(2\mathrm{p} + 2 + 8\mathrm{p} - 8 = 5\mathrm{p}\)

4. SIMPLIFY by combining like terms

• On the left side, combine:
  • Variable terms: \(2\mathrm{p} + 8\mathrm{p} = 10\mathrm{p}\)
  • Constant terms: \(2 + (-8) = -6\)
• Simplified equation: \(10\mathrm{p} - 6 = 5\mathrm{p}\)

5. SIMPLIFY by solving for p

• Move all p terms to one side: \(10\mathrm{p} - 5\mathrm{p} = 6\)
• Combine: \(5\mathrm{p} = 6\)
• Divide both sides by 5: \(\mathrm{p} = \frac{6}{5} = 1.2\)

Answer: 1.2 (or \(\frac{6}{5}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Making sign errors when distributing or combining terms

Students often struggle with \(8(\mathrm{p} - 1)\), writing \(8\mathrm{p} - 1\) instead of \(8\mathrm{p} - 8\), or make errors when combining \(2 + (-8) = -6\). These algebraic mistakes cascade through the remaining steps, leading to incorrect final answers.

This leads to confusion and incorrect answer selection.

Second Most Common Error:

Incomplete SIMPLIFY process: Stopping before fully isolating the variable

Some students correctly get to \(5\mathrm{p} = 6\) but then leave their answer as this equation rather than completing the division to find \(\mathrm{p} = 1.2\). Others might write \(\mathrm{p} = \frac{5}{6}\) by inverting the fraction incorrectly.

This causes them to provide an incomplete or incorrect final answer.

The Bottom Line:

This problem tests sustained algebraic accuracy across multiple steps. Success requires careful attention to sign changes and systematic progression through the solution process.