\(2\mathrm{x} + 16 = \mathrm{a}(\mathrm{x} + 8)\)In the given equation, a is a constant. If the equation has infinitely many...

1. INFER what "infinitely many solutions" means

• Given: \(\mathrm{2x + 16 = a(x + 8)}\) has infinitely many solutions
• Key insight: For a linear equation to have infinitely many solutions, it must be an identity - meaning both sides are exactly the same for every possible value of x

2. SIMPLIFY the equation to compare both sides

• Expand the right side using distributive property:
  • \(\mathrm{2x + 16 = a(x + 8)}\)
  • \(\mathrm{2x + 16 = ax + 8a}\)

3. INFER the conditions for an identity

• For \(\mathrm{2x + 16 = ax + 8a}\) to be true for ALL values of x, corresponding parts must be equal:
  • Coefficients of x must match: \(\mathrm{2 = a}\)
  • Constant terms must match: \(\mathrm{16 = 8a}\)

4. SIMPLIFY to find the value of a

• From either condition:
  • From coefficients: \(\mathrm{a = 2}\)
  • From constants: \(\mathrm{16 = 8a \rightarrow a = 2}\)
• Both give the same result, confirming our answer

Answer: 2

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't understand what "infinitely many solutions" actually means for a linear equation. They might think it just means "solve for a" or try to find specific x-values instead of recognizing that the equation must be an identity.

This leads to confusion and random guessing rather than systematic analysis.

Second Most Common Error:

Incomplete SIMPLIFY execution: Students expand correctly to get \(\mathrm{2x + 16 = ax + 8a}\), but only check one condition (either coefficients OR constants) instead of verifying both. They might set \(\mathrm{2 = a}\) and stop, or set \(\mathrm{16 = 8a}\) and get \(\mathrm{a = 2}\), without confirming both conditions yield the same result.

This could lead them to doubt their answer or make careless errors in verification.

The Bottom Line:

This problem tests conceptual understanding more than computational skill. The key insight is recognizing that "infinitely many solutions" means the equation is an identity, which requires systematic comparison of all corresponding terms.