Question:\(\frac{2}{3}(\mathrm{ax} - 12) = 4\mathrm{x} + \mathrm{b}\)In the given equation, a and b are constants. The equation has no solution,...

1. TRANSLATE the problem information

• Given equation: \(\frac{2}{3}(\mathrm{ax} - 12) = 4\mathrm{x} + \mathrm{b}\)
• Key conditions: The equation has no solution, and \(\mathrm{b} \lt -10\)
• Find: The value of a

2. INFER the mathematical condition for no solution

• For a linear equation to have no solution, two conditions must be met:
  • The coefficients of the variable (x) on both sides must be equal
  • The constant terms on both sides must be unequal
• This creates a contradiction: same slope but different y-intercepts

3. SIMPLIFY the left side by distributing

• Apply distributive property: \(\frac{2}{3}(\mathrm{ax} - 12) = \frac{2\mathrm{a}}{3}\mathrm{x} - 8\)
• Rewritten equation: \(\frac{2\mathrm{a}}{3}\mathrm{x} - 8 = 4\mathrm{x} + \mathrm{b}\)

4. INFER and apply the coefficient condition

• Left side coefficient of x: \(\frac{2\mathrm{a}}{3}\)
• Right side coefficient of x: 4
• Set equal: \(\frac{2\mathrm{a}}{3} = 4\)

5. SIMPLIFY to solve for a

• Multiply both sides by 3: \(2\mathrm{a} = 12\)
• Divide by 2: \(\mathrm{a} = 6\)

6. APPLY CONSTRAINTS to verify the constant terms condition

• Left side constant: \(-8\)
• Right side constant: b
• For no solution: \(-8 \neq \mathrm{b}\)
• Since \(\mathrm{b} \lt -10\), this inequality is automatically satisfied

Answer: C (6)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not understanding what "no solution" means mathematically

Students might think "no solution" simply means the equation can't be solved, leading them to set up incorrect conditions or try to make the entire equation impossible rather than focusing on the specific coefficient/constant relationship. This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Conceptual confusion about solution types: Mixing up "no solution" with "infinite solutions"

Students might incorrectly set up the condition where both coefficients AND constants are equal (which creates infinite solutions, not no solution). This would lead them to solve \(\frac{2\mathrm{a}}{3} = 4\) and \(-8 = \mathrm{b}\) simultaneously, creating contradictory conditions with the given \(\mathrm{b} \lt -10\) constraint. This causes them to get stuck and randomly select an answer.

The Bottom Line:

This problem requires clear understanding of what different solution types mean algebraically. The key insight is that "no solution" creates a specific mathematical contradiction that students must recognize and set up correctly.