If 2 + x = 60, what is the value of 16 + 8x?

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{2 + x = 60}\)
• Need to find: \(\mathrm{16 + 8x}\)

2. INFER the most efficient approach

• Instead of solving for x first, look for a pattern
• Notice that \(\mathrm{16 + 8x}\) can be factored: \(\mathrm{16 + 8x = 8(2 + x)}\)
• Since we know \(\mathrm{2 + x = 60}\), we can use this directly!

3. SIMPLIFY by applying the relationship

• Multiply both sides of the given equation by 8:
  \(\mathrm{8(2 + x) = 8(60)}\)
• The left side becomes:
  \(\mathrm{8(2) + 8(x) = 16 + 8x}\)
• The right side becomes:
  \(\mathrm{8(60) = 480}\)
• Therefore:
  \(\mathrm{16 + 8x = 480}\)

Answer: 480

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the direct relationship between the given equation and the target expression

Many students will solve for x first: \(\mathrm{2 + x = 60}\), so \(\mathrm{x = 58}\). Then substitute: \(\mathrm{16 + 8(58) = 16 + 464 = 480}\). While this gives the correct answer, it's a longer approach and more prone to arithmetic errors.

Second Most Common Error:

Poor SIMPLIFY execution: Students recognize the pattern but make arithmetic mistakes

Students might correctly set up \(\mathrm{8(2 + x) = 8(60)}\) but then calculate \(\mathrm{8 \times 60}\) incorrectly, or make errors when distributing the 8.

The Bottom Line:

The key insight is recognizing that algebraic expressions can often be manipulated directly without solving for individual variables first. This 'work with what you have' approach is both more elegant and less error-prone.