\(4\mathrm{x} + 12 = \frac{\mathrm{a}(\mathrm{x}+3)}{2}\)In the given equation, a is a constant. If the equation has infinitely many solutions, what...

1. INFER what "infinitely many solutions" means

• Given information:
  • Equation: \(4\mathrm{x} + 12 = \frac{\mathrm{a}(\mathrm{x}+3)}{2}\)
  • The equation has infinitely many solutions
• Key insight: For an equation to have infinitely many solutions, both sides must be equivalent expressions (identical for all values of x)

2. SIMPLIFY the equation to compare both sides

• Multiply both sides by 2 to eliminate the fraction:
  \(2(4\mathrm{x} + 12) = 2 \cdot \frac{\mathrm{a}(\mathrm{x}+3)}{2}\)
  \(8\mathrm{x} + 24 = \mathrm{a}(\mathrm{x} + 3)\)

3. SIMPLIFY further by factoring

• Factor out 8 from the left side:
  \(8\mathrm{x} + 24 = 8(\mathrm{x} + 3)\)
• Now we have: \(8(\mathrm{x} + 3) = \mathrm{a}(\mathrm{x} + 3)\)

4. INFER the value of a

• For these expressions to be equivalent for ALL values of x:
  • The coefficients of \((\mathrm{x} + 3)\) must be equal
  • Therefore: \(\mathrm{a} = 8\)

Answer: C. 8

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not understanding what "infinitely many solutions" means conceptually.

Students may try to solve for x by setting up \(4\mathrm{x} + 12 = \frac{\mathrm{a}(\mathrm{x}+3)}{2}\) as a regular equation, attempting to isolate x. When they get expressions involving both x and a, they become confused about how to proceed. This leads to confusion and guessing rather than recognizing that infinitely many solutions requires equivalent expressions.

Second Most Common Error:

Poor SIMPLIFY execution: Making algebraic errors during manipulation.

Students might correctly understand the concept but make mistakes like:

• Incorrectly distributing when multiplying by 2
• Factoring errors (like getting \(4(\mathrm{x} + 3)\) instead of \(8(\mathrm{x} + 3)\))
• Sign errors during algebraic manipulation

These calculation mistakes lead them to incorrect values of a, potentially selecting Choice B (3) or Choice D (12).

The Bottom Line:

This problem tests both conceptual understanding (what infinitely many solutions means) and algebraic manipulation skills. Students need to recognize that they're not solving for x, but rather finding when two expressions are equivalent.